Research Article | Open Access

Volume 2021 |Article ID 5560790 | https://doi.org/10.1155/2021/5560790

Rui Zhang, Chuanxun Li, Dandan Jin, "Buckling Stability Analysis for Piles in the Slope Foundation Based on Cusp Catastrophe Theory", Mathematical Problems in Engineering, vol. 2021, Article ID 5560790, 11 pages, 2021. https://doi.org/10.1155/2021/5560790

# Buckling Stability Analysis for Piles in the Slope Foundation Based on Cusp Catastrophe Theory

Academic Editor: Jose Renato de Sousa
Revised05 May 2021
Accepted12 Jun 2021
Published21 Jun 2021

#### Abstract

The aim of this paper is to analyze the buckling stability problem for piles in the slope foundation based on cusp catastrophe theory. Formulation of critical buckling load of piles in the slope foundation is obtained. The influential factors of slope angle, distribution of landslide thrust behind the pile, pile-embedded ratio, pile constraints, pile-side friction, pile-side soil resistance, and pile socketed ratio upon buckling stability characteristic for piles in the slope foundation are examined. The results reveal that when pile diameter remains unchanged, critical buckling load increases with the increase of pile length when pile-embedded ratio reaches 60%. When pile length remains unchanged, critical buckling load increases with the increase of pile diameter. Critical buckling load with the assumption of nonlinear horizontal elastic resistance of pile-side soil in the paper is more close to the value based on horizontal elastic resistance of pile-side soil suggested in the code. When slope angle increases, decreased extent of buckling critical load for piles 30–60 m in length is more obvious than the piles which are 10–30 m in length. Strengthening of pile constraints and increase of pile-embedded ratio and socketed ratio are helpful to pile critical buckling load increase. The influential factors of pile-side friction and landslide thrust behind the pile upon pile critical buckling load are tiny and can be neglected.

#### 1. Introduction

During the 14th Five-Year Plan Period, the pace of construction and upgrading will be promoted continuously in China. Meanwhile, as a mountainous country, mountainous and hilly areas cover two-thirds of the total land area. Therefore, with the continuous impetus of national infrastructure, as well as increasing piles of civil buildings, roads and bridges will be located in slope foundation. Piles in slope foundation are influenced by asymmetric pile-soil interaction, slope sliding force, and nonlinear pile-side soil resistance, whose buckling stability research is very vital. It is of important theoretical and practical significance to make deep research on buckling stability for piles in slope foundation .

Cusp catastrophe theory  is a branch on discontinuity, which has developed greatly during the last 30 years and had a wide range of applications in pile ultimate bearing capacity prediction . Some scholars have studied the buckling stability of piles based on cusp catastrophe theory. With static load test, Zhang and Cui  derived computing formula on ultimate vertical bearing capacity for a single end bearing pile and found that pile displacement value of instability point could be calculated out with parabolic load-displacement curve. Chen et al.  discussed buckling stability for Y-shape piles and established cusp catastrophe model for piles whose tips were socketed and tops were free; critical buckling loads of Y-shape piles were higher than those of uniform piles by 30%. Li et al.  performed buckling stability analysis for tapered piles considering self-weight influence and concluded that self-weight of piles had a great influence on critical buckling load for tapered piles. When embedded ratio was less than 40% of pile length, larger critical buckling load would be obtained, which would cause unsafe bearing capacity designed according to the strength of pile body.

Based on the analysis of the existing literature, there are still some challenges on buckling stability analysis. (1) Cusp catastrophe theory is seldom adopted in the existing articles for in-slope pile buckling stability analysis. (2) Potential energy function is not complete, which may lead to incomprehensive influential factors discussion on pile buckling stability analysis.

The objective of this study is to analyze buckling stability for piles in slope foundation based on cusp catastrophe theory. The paper establishes total potential energy equation with assumptions of nonlinear distributed horizontal elastic resistance of pile-side soil and landslide thrust behind pile. Furthermore, the paper discusses influences of slope angle, landslide thrust distribution behind pile, pile-embedded ratio, pile constraints, pile-side soil friction, pile-side soil resistance, and pile socketed ration upon buckling stability for piles in slope foundation.

#### 2. Cusp Catastrophe Theory

Potential function of cusp catastrophe model is defined as where x is the system state variable and u and are system control variables.

Making the derivation of potential function V (x) zero, we can get equilibrium surface of cusp catastrophe model:

This surface is called catastrophe manifold, which indicates condition of system equilibrium state. The corresponding point is called critical or extreme point in mathematics.

System singular point sets satisfy equation (2) and the second derivative of potential function ; that is,

Eliminate x through the combination of equations (2) and (3), and parametric plane equation is obtained as

Equation (4) is called system bifurcation sets, which denotes projection on the plane determined by control variables u and of singular point sets. When system control variables satisfy equation (4), the system is in critical equilibrium state and will jump to a stable equilibrium state finally to achieve system catastrophe.

Curve D is called bifurcation sets, whose discriminant is written as .

When D > 0, the system is in a stable state. When D < 0, the system is in an unstable state. Catastrophe occurs when u < 0 and D = 0. In this state, there are three real roots for equation (4).

#### 3. Computation Model and Basic Assumptions

Assuming that L is the total pile length, it can be divided into three sections: socketed section l1, slope influenced section l2, and free section l; that is, L = l1 + l2 + l. Vertical load on pile top is P. Pile embedded depth is h: h = l1 + l2. The coordinate origin o is at the center of pile bottom. Simplified computation model for buckling stability analysis of an in-slope pile is shown in Figure 1.

Pile bottom is a fixed constraint, pile top is free, α is the slope angle, d is the pile diameter, Q(x) is the landslide thrust behind the pile, l2 = 4dtanα , and τ is the ultimate friction strength between pile and soil.

Basic assumptions in the computation are listed as follows:(1)Ultimate friction strength between pile and soil τ is a constant.(2)Horizontal elastic resistance of pile-side soil q (x, y) is nonlinearly distributed within the range of embedded depth of pile, whose expression is regarded as follows :where m is proportional coefficient of horizontal elastic resistance of pile-side soil. b0 is calculated width of piles. For circular piles, b0 = 0.9 (1.5 d + 0.5), when d < 1.0 m. b0 = 0.9 (d + 1), when d ≥ 1.0 m . Generalized expression of landslide thrust behind pile Q (x) is regarded as where a1, b1, and c1 are undetermined parameters related to the resultant force of landslide thrust F.

#### 4. Solution of Critical Buckling Load

##### 4.1. Establishment of Deflection Functions

According to the boundary conditions shown in Figure 1, the deflection function for a pile whose tip is socketed and top is free is established as [12, 20]where n is half-wave number of the deflection function. Cn is undetermined constant term. Under the small deformation assumption of pile-soil system, take n = 1. Therefore, pile deflection function is expressed as

From equation (8), the first and second derivatives of pile deflection function are obtained as

##### 4.2. Solution Based on Cusp Catastrophe Theory

The total potential energy equation of pile-soil system is established as follows [12, 20]:where UP is the bending strain energy of the pile, US is the elastic deformation energy of soil beside pile, VQ is the potential energy of landslide thrust behind the pile, VP is the potential energy of pile top load, VG is the potential energy of pile self-weight, and Wf is the potential energy of pile-side friction embedded in soil. Substituting equations (8)∼(10), deduction of each potential energy item is listed as follows.

Bending strain energy of pile UP is as follows:where E is the elastic modulus of the pile and I is the moment of inertia of the pile section.

Deformation energy of soil beside pile US is as follows:where

Potential energy of landslide thrust behind pile VQ is as follows:where

Potential energy of pile top load VP is as follows:

Potential energy of pile self-weight VG is as follows:where γ is the pile unit weight and A is the pile section area.

Potential energy of pile-side friction Wf is as follows:where U is the pile section circumference. The expressions of Cf and are written as

Substituting equations (12)–(19) into equation (11), the total potential energy equation of pile-soil system is expressed as

From equations (12)–(19), we can see that each expression of partial potential energy is the function of C2 and C4, so . The expressions of can be written as

Substituting variables of C, let , , and the total potential energy equation (21) is expressed aswhere and .

According to cusp catastrophe theory, the discriminant is written as

When D = 0, system catastrophe occurs. Therefore, it is assumed that

From equation (25), the expression of pile critical buckling load Pcr is obtained as

#### 5. Correctness Verification of Critical Buckling Load Expression

According to critical buckling load expression of equal section pile in level ground from , suppose that slope angle α = 0, and horizontal elastic resistance of pile-side soil q (x, y) is adopted as q (x, y) = mb0 (h − x) y. In the meantime, the influences of ultimate friction strength between pile and soil τ and landslide thrust behind pile are neglected, and formula (26) can be written as follows:

The derived critical buckling load expression of equal section pile in level ground from formula (26) is the same as that in .

Furthermore, considering pile self-weight zero and pile-embedded depth h = 0, expression (27) can be simplified as

According to expression (28), we obtain critical buckling load of ordinary compressed bar whose tip is socketed and top is free. Therefore, critical buckling load expression (26) for piles in slope foundation derived in the paper is reliable.

#### 6. Case Study

The pile is in slope foundation, pile concrete strength grade is C25, slope soil is soft plastic clay, and slope angle α is 30°. Pile-embedded ratio h/L is 0.6. Landslide thrust behind pile Q (x) is assumed as triangular distribution. Composition of landslide thrust behind pile is 80 kN/m. Other pile and soil parameters are obtained from the work of Zhao et al.  and listed in Table 1. Physical meanings of the input parameters are explained in the previous section.

 E (MPa) γ (kN∙m−3) m (MN∙m−4) τ (kPa) 2.38 × 104 25 5.0 20
##### 6.1. Variation Analysis of Pile Buckling Load with Different Pile Lengths

Critical buckling load changes for piles in slope foundation with varying pile diameter and length are shown in Figure 2. Pile length ranges from 10 m to 60 m. Pile diameter d is 0.5 m, 0.75 m, 1.0 m, 1.5 m, and 2.0 m. Pile-embedded ratio h/L is 0.6.

It can be seen from Figure 2 that when pile diameter d remains unchanged, with the increase of pile length L, pile socketed length increases, and critical buckling load of piles in slope foundation increases accordingly. When pile length L remains unchanged, with the increase of pile diameter d, the pile slenderness ratio L/d decreases, and critical buckling load of piles in slope foundation increases accordingly.

##### 6.2. Influence of Slope Angle

With slope angle α being 0°, 10°, 20°, 30°, 40°, 50°, and 60°, the influence of slope angle on critical buckling load of piles in slope foundation is discussed. Critical buckling load change curves of piles in slope foundation with varying slope angle are shown in Figure 3.

It can be seen from Figure 3 that, compared with pile in level ground (α = 0°), in-slope pile critical buckling load decreases. For medium-long piles in the range of 10–20 m, critical buckling load reduction is small when slope angle increases. For long piles in the range of 30–40 m and superlong piles over 50 m, critical buckling load reduction is obvious when slope angle increases. Regardless of the pile length, critical buckling load decreases proportionally with the angle approximately. That is because when pile-embedded ratio is constant, socketed length l1 decreases proportionally with slope angle approximately.

##### 6.3. Influence of Landslide Thrust behind the Pile

According to landslide thrust distribution function table in , parabola distribution, linear distribution, trapezoidal distribution, triangular distribution, and no thrust distribution of landslide thrust are employed, and the influence of landslide thrust behind pile upon critical buckling load is discussed. Composition of landslide thrust behind pile is 80 kN/m. Critical buckling load change curves of piles in the slope foundation with varying landslide thrust are listed in Table 2.

 L (m) Parabola distribution Linear distribution Trapezoidal distribution Triangular distribution No thrust distribution 10 36.108 36.099 36.112 36.114 36.153 20 68.221 68.208 68.227 68.229 68.363 30 240.959 240.946 240.965 240.968 241.206 40 632.638 632.627 632.646 632.648 632.996 50 1336.587 1336.578 1336.600 1336.602 1336.991 60 2447.879 2447.877 2447.899 2447.901 2448.464

It can be seen from Table 2 that when landslide thrust behind pile is considered, critical buckling load of piles in slope foundation decreases with tiny reduction. Compared with critical buckling load without thrust distribution, the reduction is less than 0.5%. Critical buckling load varies with tiny reduction with parabola, linear, trapezoidal, and triangular distribution. Therefore, for in-slope pile buckling stability analysis, influence of landslide thrust behind pile upon critical buckling load is tiny and can be neglected.

##### 6.4. Influence of Pile-Embedded Ratio

Pile-embedded ratio h/L is 0.15, 0.20, 0.30, 0.40, 0.50, 0.60, 0.70, and 0.75, and the influence of pile-embedded ratio upon critical buckling load is discussed. Critical buckling load change curves of piles in slope foundation with varying pile-embedded ratio are shown in Figure 4.

It can be seen from Figure 4 that when pile-embedded ratio h/L ≤ 0.20, critical buckling load of piles in slope foundation decreases gradually with the increase of pile length. At this moment, critical buckling point is in the free section of the pile. When pile-embedded ratio h/L is in the range of 0.30 to 0.50, critical buckling load of piles in slope foundation decreases first and increases afterwards with the increase of pile length. Critical buckling point transits from free section to influenced section of piles. When pile-embedded ratio h/L ≥ 0.60, critical buckling load of piles in slope foundation increases with the increase of pile length. Therefore, pile-embedded ratio h/L should reach 60% for piles in slope foundation.

##### 6.5. Influence of Pile Constraints

Constraint types of pile tip and top include free, elastic embedded, articulated, and fixed. Suppose that pile tip constraint is fixed and pile top constraint is one of the above four constraints. Combine the constraint conditions of pile tip and pile top afterwards. With the assumption of small deformation of pile-soil system, influence of pile constraints upon critical buckling load of piles in slope foundation is discussed. The corresponding pile deflection functions are listed in Table 3 .

 Pile top constraint Deflection function Free Elastic embedded Articulated Fixed

Critical buckling load change curves of piles in slope foundation with varying pile top constraints are shown in Figure 5.

It can be seen from Figure 5 that critical buckling load is of the minimum value when pile top is free. Piles are most prone to buckling instability. Critical buckling load is of the maximum value when pile top is fixed. Piles are the least prone to buckling instability. Critical buckling load is between the previous values when pile top is elastic embedded or articulated. Critical buckling load with articulated pile top is a little more than the value with elastic embedded pile top. Therefore, when pile tip is fixed, critical buckling load of piles in slope foundation can be improved effectively with strengthening of pile top constraints.

##### 6.6. Influence of Pile-Side Friction

Let ultimate friction strength between pile and soil τ be 0, 30, and 60 kPa. Pile-embedded ratio h/L is 0.6. The influence of pile-side friction upon critical buckling load is discussed. Critical buckling loads with different ultimate friction strength between pile and soil and pile length are shown in Table 4.

 L (m) τ = 0 kPa τ = 30 kPa τ = 60 kPa 10 36.084 36.129 36.174 20 68.169 68.259 68.348 30 240.879 241.013 241.147 40 632.531 632.710 632.888 50 1336.45 1336.67 1336.90 60 2447.71 2447.98 2448.25

It can be seen from Table 4 that when pile-side friction is considered, critical buckling load of piles in slope foundation increases. But the extent of increase is less than 0.5%. Therefore, the influence of pile-side friction upon critical buckling load can be neglected.

##### 6.7. Influence of Horizontal Elastic Resistance of Pile-Side Soil

The typical methods to calculate horizontal elastic resistance of pile-side soil include Constant method, m method, and c method. Among them, m method is the method adopted in the current national code  to calculate foundation counterforce. At present, some scholars combine m method and c method with Constant method to calculate pile-side soil resistance. In this section, critical buckling load of piles in slope foundation is obtained with different pile-side soil resistance computation methods. The expressions assumed in the above methods are listed in Table 5 (the expressions assumed in the paper are given in equation (5)).

 Method name q(x, y) 0 ≤ x < l1 l1≤x ≤ h m method c method m-constant method c-constant method

q(x, y) can be expressed aswhere K (x) is pile-side soil resistance coefficient. Therefore, pile-side soil resistance distribution along pile length is different for the above methods. Pile-side soil resistance is linear distribution along pile length for m method. Pile-side soil resistance is 0.5 power exponential distribution for c method. For m-Constant method and c-Constant method, pile-side soil resistance is constant when 0 ≤ x < l1 and linear or 0.5 power exponential distribution when l1 ≤ x ≤ h. As shown in expression (5), for pile-side soil resistance suggested in the paper, it is linear distribution when 0 ≤ x < l1 and square exponential distribution when l1 ≤ x ≤ h.

Critical buckling load change curves of piles with varying pile-side soil resistance computation methods are shown in Figure 6.

It can be seen from Figure 6 that, for medium-long piles in the range of 10 m–20 m, difference of critical buckling load with varying pile-side soil resistance computation methods is not wide. For long piles in the range of 30–40 m and superlong piles over 50 m, difference of critical buckling load with varying pile-side soil resistance computation methods appears gradually. Critical buckling load obtained with pile-side soil resistance computation method suggested in the paper is close to the value with m method adopted in the current national code. Therefore, pile-side soil resistance computation method suggested in the paper is of certain usability.

##### 6.8. Influence of Pile Socketed Ratio

For piles in slope foundation with different lengths, the influence of pile socketed ratio l1/L upon critical buckling load is discussed through increase of pile socketed ratio. Critical buckling load change curves of piles in slope foundation with different lengths and pile socketed ratio are shown in Figure 7.

It can be seen from Figure 7 that critical buckling load of piles in slope foundation increases with the increase of pile socketed ratio. That is to say, appropriate increase of pile socketed length is helpful to increase pile critical buckling load. Furthermore, the more of pile length, the more of increase latitude of pile critical buckling load. There is especially obvious increase latitude for superlong piles over 50 m. Critical buckling load change curves will become flat when pile socketed ratio reaches a certain value. That is to say, there is a critical fixed depth . Critical buckling load change curves of medium-long piles in the range of 10 m–20 m tend to be flat earlier.

#### Data Availability

The data that support the findings of this study are available from the corresponding author upon reasonable request.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

This research was funded by the National Natural Science Foundation of China (Grant no. 51708256), Jiangsu Province Postdoctoral Foundation (no. 1301067B), Open Innovation Foundation of Changjiang Institute of Survey, Planning, Design and Research (Grant no. CX2020K08), and Advanced Expert Foundation of Jiangsu University (Grant no. 09JDG058).

1. H. Zhao, P.-B. Yin, and X.-B. Li, “Mechanical response of bridge piles in high-steep slopes and sensitivity study,” Journal of Central South University of Central, vol. 22, no. 10, pp. 4043–4048, 2015. View at: Publisher Site | Google Scholar
2. N. P. Kurian and M. S. Srinivas, “Studies on the behaviour of axially loaded tapered piles by the finite element method,” International Journal for Numerical and Analytical Methods in Geomechanics, vol. 19, no. 12, pp. 869–888, 1995. View at: Publisher Site | Google Scholar
3. B. B. Budkowska and C. Szymczak, “Partially embedded piles subjected to critical buckling load-sensitivity analysis,” Computers & Structures, vol. 61, no. 1, pp. 193–196, 1996. View at: Publisher Site | Google Scholar
4. M. E. Heelis, M. N. Pavlović, and R. P. West, “The analytical prediction of the buckling loads of fully and partially embedded piles,” Géotechnique, vol. 54, no. 6, pp. 363–373, 2004. View at: Publisher Site | Google Scholar
5. J. Lee, K. Paik, D. Kim, and S. Hwang, “Estimation of axial load capacity for bored tapered piles using CPT results in sand,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 135, no. 9, pp. 1285–1294, 2009. View at: Publisher Site | Google Scholar
6. L. W. Ren and G. Y. Wang, “Practical variational analysis on vertical behavior of jet grouting soil-cement-pile strengthened pile,” ASCE GSP, vol. 220, pp. 126–134, 2011. View at: Google Scholar
7. X.-J. Zou and M.-H. Zhao, “Axial bearing behavior of super-long piles in deep soft clay over stiff layers,” Journal of Central South University, vol. 20, no. 7, pp. 2008–2016, 2013. View at: Publisher Site | Google Scholar
8. S. Manandhar and N. Yasufuku, “Vertical bearing capacity of tapered piles in sands using cavity expansion theory,” Soils and Foundations, vol. 53, no. 6, pp. 853–867, 2013. View at: Publisher Site | Google Scholar
9. G. Q. Kong, Q. Yang, H. L. Liu, and R. Y. Liang, “Numerical study of a new belled wedge pile type under different loading modes,” European Journal of Environmental and Civil Engineering, vol. 17, pp. 65–82, 2013. View at: Publisher Site | Google Scholar
10. S. Bhattacharya and K. Goda, “Probabilistic buckling analysis of axially loaded piles in liquefiable soils,” Soil Dynamics and Earthquake Engineering, vol. 45, pp. 13–24, 2013. View at: Publisher Site | Google Scholar
11. M. Nadeem, T. Chakraborty, and V. Matsagar, “Nonlinear buckling analysis of slender piles with geometric imperfections,” Journal of Geotechnical and Geoenvironmental Engineering, vol. 141, Article ID 6014014, 2015. View at: Google Scholar
12. M. H. Zhao, W. H. Liu, P. B. Yin, C. W. Yang, and H. Zhao, “Buckling analysis of bridge piles in steep slopes based on energy method,” Journal of Central South University (Science and Technology), vol. 47, no. 2, pp. 586–592, 2016. View at: Google Scholar
13. P. B. Yin, Y. Yang, W. He, X. X. Liu, and H. Zhao, “Analysis of critical buckling loads of piles considering slope effect,” Rock and Soil Mechanics, vol. 38, no. 9, pp. 2662–2669, 2017. View at: Google Scholar
14. W. Lu and D. Zhao, “Analysis on calculated length for buckling stability of steel pipe pile based on energy method,” The Open Civil Engineering Journal, vol. 11, no. 1, pp. 167–175, 2017. View at: Publisher Site | Google Scholar
15. J. K. Lee, S. Jeong, and Y. Kim, “Buckling of tapered friction piles in inhomogeneous soil,” Computers and Geotechnics, vol. 97, pp. 1–6, 2018. View at: Publisher Site | Google Scholar
16. R. Thom, Structural Stability and Morphogenesis, W A Benjamin, Washington, NY, USA, 1978.
17. K. Yang, T. Wang, and Z. Ma, “Application of cusp catastrophe theory to reliability analysis of slopes in open-pit mines,” Mining Science and Technology (China), vol. 20, no. 1, pp. 71–75, 2010. View at: Publisher Site | Google Scholar
18. Y. F. Zhang and S. Q. Cui, “Analysis on vertical bearing capacity of end-bearing pile foundation based on the catastrophe theory,” Rock and Soil Mechanics, vol. 28, pp. 901–904, 2007. View at: Google Scholar
19. Y. H. Chen, X. Q. Wang, and H. L. Liu, “Buckling critical load analysis of Y style vibro-pile based on cusp catastrophe theory,” Engineering Mechanics, vol. 26, no. 4, pp. 119–127, 2009. View at: Google Scholar
20. C. X. Li, X. Z. Liu, and W. B. Wu, “Buckling analysis for tapered pile considering self-weight of pile,” Journal of Southwest Jiaotong University, vol. 52, no. 6, pp. 1130–1138, 2016. View at: Google Scholar
21. Z. H. Dai, “Study on distribution laws of landslide-thrust and resistance of sliding mass acting on antislide piles,” Chinese Journal of Rock Mechanics and Engineering, vol. 21, no. 4, pp. 517–521, 2002. View at: Google Scholar
22. H. F. Shan, T. D. Xia, F. Yu, and J. H. Hu, “Critical buckling capacity of piles with different pile-head constraints for excavation beneath existing foundation,” Chinese Journal of Geotechnical Engineering, vol. 39, pp. 49–52, 2017. View at: Google Scholar
23. The Professional Standards Compilation Group of People’s Republic of China, JGJ 94-2008. Technical Code for Building Pile Foundations, China Architecture and Building Press, Beijing, China, 2008.