ANALYSIS OF BUCKLING AND POST BUCKLING OF ......2.1. Buckling of piles Buckling is understood as a loss of stability of a flexible structure that may result in an abrupt and catastrophic

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International Journal of Civil Engineering and Technology (IJCIET)

Volume 10, Issue 05, May 2019, pp. 559-570, Article ID: IJCIET_10_05_060

Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJCIET&VType=10&IType=5

ISSN Print: 0976-6308 and ISSN Online: 0976-6316

© IAEME Publication

ANALYSIS OF BUCKLING AND POST

BUCKLING OF PILES FULLY EMBEDDED IN

GROUND

Vlora Shatri

Faculty of Civil Engineering & Architecture, University of Hasan Prishtina, 10000 Kosovo

Laur Haxhiu

Faculty of Civil Engineering & Architecture, University of Hasan Prishtina, 10000 Kosovo

ABSTRACT

This paper work aims to present analytical and numerical analysis, linear and

nonlinear buckling and post buckling behaviour of piles fully embedded in ground. In

the first step, we have used the Galerkin’s Method for linear buckling and nonlinear

post buckling analysis of piles with linear-elastic material behaviour. Then is

continued with numerical buckling and post buckling analysis of the piles with

nonlinear elasto-plastic material behaviour. For this, the Riks Method is used.

Numerical analysis was performed in software Abaqus 6.13. This paper finally shows

that the post buckling behaviour of the pile with ideal load stiffness increase after

bifurcation point.

Key words: Piles, Buckling, Post buckling, Linear analysis, Nonlinear analysis.

Cite this Article: Vlora Shatri and Laur Haxhiu, Analysis of Buckling and Post

Buckling of Piles Fully Embedded in Ground, International Journal of Civil

Engineering and Technology 10(5), 2019, pp. 559-570.

http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=5

1. INTRODUCTION

Stability of the structures falls within the field of mechanics dealing with the behavior of

structures subject to axial compression forces and is analyzed through calculation of critical

buckling force. The critical buckling force is the force that corresponds to the situation in

which a perturbation of the deformation state does not disturb the equilibrium between the

external and internal forces (Silva, 2006) [1] and is always calculated based on the eigen

values of the linear analysis of buckling. At the problem of eigen values of buckling we

determine the force for which the model stiffness matrix is singular. Their respond usually

includes small deformations before buckling. However, the problem of the stability of the

structures is nonlinear by nature hence it falls within the problems of nonlinear analysis. If we

want to make a detail analysis of a structure then it is a requirement to use a nonlinear

analysis that includes a post buckling behavior with elasto-plastic behavior of materials. A

post buckling analysis is always required when a concern of the material nonlinearity,

geometric nonlinearity, or of the unstable post buckling behavior exists [2].

Vlora Shatri and Laur Haxhiu


With an aim of buckling and post buckling analysis of piles, in this paper we deal with

eigenvalue linear buckling analysis and nonlinear post buckling analysis using the Galerkin’s

Method with linear-elastic behaviour of material and nonlinear buckling analysis using

modified Rik’s Method with elasto-plastic behaviour of material.

2. BUCKLING AND POST BUCKLING OF PILES WITH LINEAR-

ELASTIC BEHAVIOUR OF MATERIAL

2.1. Buckling of piles

Buckling is understood as a loss of stability of a flexible structure that may result in an abrupt

and catastrophic collapse, as is the complete falling or breaking of a structure [3]. Stability

shall be analysed for two types of elements:

Ideal elements, with no imperfection, that we do not meet in reality but only

theoretically and their theoretical solution leads us to the problem of stability (Ugural

and Fenster, 1987) [3]. The phenomena of stability of ideal elements is analyzed and

resolved by Euler (1774).

Real elements, with various types of imperfection that are met in everyday life, lead to

buckling resistance.

The problem of pile buckling is closely connected with the problem of a beam in an

elastic base (Hetenyi, 1960) [4]. Beam deformations are influenced by an axial force the beam

may be subjected, hence with an aim of calculating the response of a vertical pile fully

embedded in ground and externally loaded, the pile will be considered as a beam of an elastic

foundation (Figure 1), (Reese & Matlock, 1960) [5].

Figure 1 Vertical pile fully embedded in ground and externally loaded

Differential equation of beam buckling:

According to statics, the flexural moment at section "x" of the pile (Figure 1) is:

Analysis of Buckling and Post Buckling of Piles Fully Embedded in Ground


∫ ∫

where: - p(x)=ky(x) is soil pressure. Whereas the equation of flexure of a pile subject to axial

compression force only may be written in the following shape:

(

)

where: -EI is pile stiffness, P is axial force on a pile, and K is modulus of horizontal soil

reaction.

Analytical calculation of critical buckling force of a pile based on the theory of elasticity

may be found through the solution of a differential equation or through Energy Method,

Galerkin’s Method, Finite Element Method, Finite Difference Method, and the Rayleigh-Ritz

Method.

2.1.1. Linear analysis of pile buckling- Galerkin’s Method

The Galerkin’s Method belongs to the group of analytical approximate methods. According to

this method, one choice is to approximate the virtual displacement functions with those of the

same basis as the real displacement function.

For a pile with linear-elastic behaviour of material, with the end conditions, pinned at the

head and pinned at the tip, (p-p), fully embedded in ground of constant stiffness, k, assume a

mode that satisfies the boundary conditions:

where

q1-unknown coefficient

sin(πx/L) -assumed mode shape satisfies all boundary conditions

Using the principle of virtual work according to Galerkin’s Method, we write:

∫

*

(

)

+

After the required mathematical operations are performed in Eq. (5), the critical bucking

force of a (p-p) pile, fully embedded in grounds of the horizontal modulus of soil reaction, k,

constant along the pile length, is obtained:

From equation (6) we notice that the critical buckling force of the pile, Pcrit(p-p) is a

function of the pile characteristics and of the soil. First part of the Eq. (6) corresponds to the

Euler equation for buckling of the column (p-p) of elastic material while the second part

reflects the contribution of lateral restraint caused by the ground.



Figure 2 Dependence of buckling force of a pile with end support conditions, pinned at the head –

pinned at the tip, Pcrit (p-p) and the pile length, L, for a pile of C25/30 concrete class, diameter, D=30cm,

fully embedded in grounds of the horizontal modulus of soil reaction, k0=1000kN/m2

The pile buckling force calculated based on linear analysis, in general is greater than the

real buckling force of the pile. This because at linear analysis of pile buckling the

imperfection is not included but in reality, is actually present. In Figure 2, the dependence

between the critical buckling force of the pile, Pcrit and the pile length, L, is given (as per Eq.

(6)).

2.2. Post buckling behaviour of the piles

With the phenomena of post buckling we understand the continuation of the buckling

phenomena. After the load on the structure has reached the value of buckling force, then the

deflections will continue to raise while the load may remain unchanged or may even start to

decline. After a certain deformation the structure continues to receive greater loading, as a

result of which we have the deflection increase, and eventually entering into a second

buckling cycle as a result.

Out of the post buckling nonlinear analysis, we obtain much more information compared

with the linear analysis of Eigen-values. The nonlinear buckling analysis of a structure is the

simulating procedure that allows greater deformations and geometric or material nonlinearity

where at material nonlinearity-the material properties are a function of stresses of

deformations while at the geometric nonlinearity-the deformations are greater enough so as

the deflections may not be assumed to be small with and intention of neglecting the same.

2.2.1. Nonlinear analysis of post buckling behaviour of a pile with linear-elastic behaviour

of material

In this paperwork, the Galerkin’s Method is used for analytic analysis of post buckling

behaviour of a pile with the linear-elastic material. Here, the pile is adopted as fully embedded

in ground of the constant modulus of soil reaction along the pile length, k, and of the end

conditions of a pile, pinned at the head and pinned at the tip, p-p. Out of the axial

compression force loading, deformations are caused in a pile that are not small compared with

the characteristic dimensions of the pile, hence as such may not be neglected, and due to this,

the geometric nonlinearity is involved in calculation. The moment–curvature equation for

large deflections, in a section "s" of the pile is:

Pcrit [kN]

L [m]



where: -1/ρ=dφ/ds is curvature. Out of the pile section, we write:

After derivation of Eq. (8) we will have obtained:

In addition, based on the Figure 1, we may write:

√ √ (

)

Out of the first substantial trigonometric identity we write:

√

So, by substituting the Eq. (10) in Eq. (9) we write:

√ (

)

Using the Mackler series we resolve:

√ (

)

(

)

(

)

(

)

We substitute the Eq. (13) in the Eq. (12) and we have:

*

(

)

(

)

(

)

+

The flexural moment in a section "s" of the pile (Figure 1) may be written in the following

form:

∫ ∫

Out of Eq. (7), Eq. (14) and Eq. (15) we may write:

*

(

)

(

)

(

)

+

∫ ∫

where the nonlinear members are:

(

)

(

)

(

)

Eq. (16) can also be written in the shape as:

(*

(

)

(

)

(

)

+

)

We use the Galerkin’s Method for solution of the Eq. (18), and we get:



∫

*

((

(

)

(

)

(

)

)

)

+

This equation prevails worth for the case when wc/L>0.3, in contrary, for wc/L<0.3, we

have right to neglect the nonlinear members of higher ranges, so the Eq. (19) will take this

form:

∫

*

((

(

)

)

)

+

Substituting the Eq. (4) in the Eq. (20) as well as after integration we obtain:

(

)

From the Eq. (21) we observe two cases:

I. First case

q1=0 (trivial solution, y=0) out of where the critical buckling force of the pile with the support

conditions, pinned at both the head and the tip, (p-p), fully embedded in grounds of the

constant horizontal soil reaction along the pile length, (k) is:

II. Second case

√

Figure 3 Representation of load versus center deflection for post-buckled pile based on Galerkin

Method

Eq. (24) is only valid for P/Pcrit>1. In Figure 3, is given the dependence between the pile

deflection yc(q1) and the P/Pcrit ratio for post buckled (p-p) pile, of a length L=25m, C25/30

concrete class, D=30cm diameter, fully embedded in grounds of the modulus of horizontal

soil reaction, k0=1000kN/m2, based on the Galerkin’s Method.

0

0.5

1

1.5

2

2.5

0.0 L 0.5 L 1.0 L 1.5 L 2.0 L 2.5 L 3.0 L 3.5 L

P/P

crit

Wc



3. NONLINEAR ANALYSIS OF POST BUCKLING BEHAVIOUR OF A

PILE WITH NONLINEAR ELASTO-PLASTIC BEHAVIOUR OF

MATERIAL

For nonlinear analysis of a post buckling behaviour of a pile with nonlinear elasto-plastic

behaviour of material the Arc-Length Method is used. Also called the Modified Riks Method

initially developed by Riks [6], and Wempner [7]. More useful Arc-length method to record

the thorough stable and unstable buckling equilibrium route is introduced by Riks. [6]. As a

method it is efficient in predicting of the instable behaviour of the structures with large

deflections. While iterating in order to reach the solution this method puts an additional

restraint that allows us to reduce the applied load when calculating and so find the

equilibrium. Such a feature of this method makes possible to research the state of the

behaviour after the limit point is reached, although the predetermined stiffness matrix being

negative.

Since the subsequent step in Riks analysis is a continuation of a previous one, the applied

load in a structure in the previous step is considered as a dead load, P0 out of the Riks

analysis. The load specified in the Riks analysis step is considered as the reference load, Pref.

The force proportionally increases during the analysis from the initial dead load up to the

reference force, where during this the ratio of proportionality of the force is calculated for

each step, ζ (LPF).

According to the manual of Abaqus [2], the load in each step of the Rik’s solution scheme

is proportional with the amount of the actual load:

( )

The Riks method can also be used to solve post buckling problems, both with stable and

unstable post buckling behavior.

4. FE MODELLING

The FE Model of a pile was done in software package Abaqus 6.13 [8]. The 2D-FE Model of

pile buckling is built for a RC pile of diameter, D=0.3m, pile length, L=25m, C25/30 concrete

class, of end conditions pinned at the head and pinned at the tip, (p-p) and fully embedded in

very soft grounds with the elastic spring stiffness as given in Table 1.

Table 1 Stiffness’s of elastic springs

For analysis, the pile is divided in beam elements of linear hexahedral type C3D8R by

division in a total of 3000 members (Figure 4.a). Each beam element (linear hexahedral

C3D8R) has a length of 2.5m and all are connected with the elastic spring elements (Figure

Height

[m]

Spring K,

[N/m]

Individual Spring

K

0 0 0 K1

2.5 158.8545084 39.71363 K2

5 677.1152319 169.2788 K3

7.5 2529.855115 632.4638 K4

10 6024.738707 1506.185 K5

12.5 7364.197896 1841.049 K6

15 8652.084235 2163.021 K7

17.5 10556.228115 2639.057 K8

20 13127.35843 3281.84 K9

22.5 16109.13014 4027.283 K10

25 9674.409814 2418.602 K11



4.b). With an aim of discretization of the interaction pile-soil, elastic springs are assumed in

all four sides of the pile (ten lines in total, Figure 5.a). The members of the spring are four in a

plane and ten in the pile length (Figure 5.b).

.

a) b)

Figure 4 Model 2D-FE of a pile buckling: a) division of the pile in beam elements of linear hexaedral

C3D8R type; b) discretization of pile-soil interaction through elastic springs

a) b)

Figure 5 Model 2D-FE of a pile buckling: a) the top view; b) the members of the spring in a plane



4.1. Results of linear eigenvalue analysis

a) b) c)

Figure 6 Three initial modal shapes and the pile buckling forces under the end conditions, pinned at

the head and pined at the tip, (p-p), fully embedded in ground: a) 1st Mode; b) 2

nd Mode; c) 3

rd Mode

In Figure 6 are given the three essential modal shapes and buckling forces of pile buckling

with the end conditions pinned at the head and pinned at the tip, (p-p), fully embedded in

ground with the given elastic spring stiffness’s, Table 1. The critical buckling force resulted

out of Abaqus eigenvalue linear buckling analysis is Pcrit(p-p)=342,804kN (Figure 6.a). Out of

the linear-elastic FEM model the same are obtained as of the eigenvalue linear analysis in

terms of the critical buckling load.

4.2. Results of linear-elastic and nonlinear elasto-plastic FEM model with the

Riks method

Analyses with linear-elastic and nonlinear elasto-plastic material were done with the Arc-

length method and through the Abaqus command *STATIC, RIKS*. At the Arc-length

method the force LPF ratio is modified for each iteration in order the solution to follow up a

specific path until the convergence is achieved. Since in this method the force ratio (LPF) is

treated as a variable then another unknown is added to the equilibrium equations that result

out of procedure of finite elements reaching the actual number of unknowns, n+1, where n is

the number of elements in the deflection vector. Additional equations for constraint are

required to define the n +1 unknowns written as a function of actual displacement, load ratio,

and arc-length. Two approaches are used in general, one when the arc length is kept fixed and

the second one when the arch length changes. In these analyses the second approach is used

while the load acting in an ideal state (i. e., in the centre of the bar). The way of completion

shall be specified to finish the Rik’s analysis since the magnitude of the load is part of the

solution. A maximum value of the LPF may be specified or a maximum displacement value

for a specific degree of freedom. In this case the LPF of 25 is specified.

The dependence curve of the force ratio, LPF as well as the arc-length of the linear-elastic

and nonlinear elasto-plastic FEM model are shown and as per the history plot of reference

nodes (e.g. the load applied nodes) as shown in Figure 8, whereas the buckling modes of a

pile (p-p) are given in Figure 7, for the case when the LPF having different values.

1st Mode (342.804kN)

2nd Mode (882kN)

3rd Mode (1711kN)



With linear-elastic FEM model results are the same as for eigenvalue linear analysis in

way of critical buckling load. Post buckling behaviour for linear model shows that the pile

cannot increase its strength beyond bifurcation point because it has zero stiffness after

buckling has occurred.

When the problem of buckling of a (p-p) pile, fully embedded in very soft grounds with

the elastic spring stiffness given in Table1, the results obtained out of nonlinear elasto-plastic

FEM model analysis give the much smaller critical buckling (Figure 8), (i.e., Pcrit(p-p)≈290kN).

For a pile with ideal load the post buckling behaviour shows an increase of stiffness beyond

the bifurcation point. It is observed that when the pile approached the elasto-plastic area, an

expressive reduction of critical buckling load is caused by the respective stiffness loss

comparing with linear model (i.e. a decreasing from Pcrit(p-p)=342,804kN to Pcrit(p-p)≈290kN).

So, according to nonlinear elasto-plastic FEM model the (p-p) pile critical buckling force is

approximately 15,4% smaller than the buckling critical buckling force of a (p-p) pile

according to linear-elastic FEM model.

Figure 7 Pile buckling modes for various values of the force factor (LPF): a) LPF=0.75; b) LPF=1.25;

c) LPF=1.75; d) LPF=2.0

a) b)

c) d)

A B

C D



Figure 8 Load proportional factor (LPF)-Arc length curve of linear and nonlinear FEM model

5. CONCLUSION

Linear eigenvalue buckling analysis is used to determine the critical buckling force of a pile,

where, through this analysis the critical buckling force of a pile is quickly determined. The

problem of stability of the structure is a nonlinear phenomena by its nature hence it belongs to

nonlinear analysis of problems. If we want to do a detail analysis of a structure then using the

nonlinear analysis that includes the post buckling behaviour with elasto-plastic behaviour of

material is a requirement.

For buckling analysis and post buckling behaviour of a pile with linear-elastic material, in

this work the Galerkin’s Method is used, whereas the buckling analysis and post buckling

behaviour with nonlinear elasto-plastic material were done with the Arc-length method and

through the Abaqus command *STATIC, RIKS*. Out of the numerical analysis, we may

conclude:

Linear eigenvalue analysis may only be used to determine the critical buckling force of

the pile.

For the evaluation of the post buckling response the Rik’s method may be used.

Rather than physically, buckling is easier explained mathematically with failure modes of

mathematically expressed failing shapes. It appears that deflection increases rapidly with

load.

Stresses and deflections are induced by imperfections. A failure before buckling is a

consequence of this.

The pile ability to sustain stresses and deflections may be improved by additional

stiffening for extensive to this purpose.

Given the ideal load is applied only, the post buckling response among the elasto-plastic

material turn to be stable.

At post buckling response of elasto-plastic material behaviour we note that the critical

bucking force of the pile is directly influenced by the imperfection, meaning, smaller

critical force for larger deflection.

A

B

C

D

LPF=1.25

LPF=1.75

LPF=2.0

LPF=1.0 →342804N (1st critical force)

← LPF=0.75 approximately 75% critical buckle force pile is getting loose bearing capability



When entering elasto-plastic zone the corresponding stiffness loss causses considerable reduction of critical buckling force at deformable structures subject to compression axial load.

REFERENCES

[1] Silva, V. D. Mechanics and Strength of Materials. Springer, Netherland, 2006, pp. 391.

[2] Abaqus. Analysis User's Manual, Volume II, Version 6.13, Dassault Systéms Simulia

Corp., USA, 2013.

[3] Ugural, A. C., and Fenster, S. K. Advanced Strength and Applied Elasticity. Elsevier

Science Publishing Co., Inc., New York, 1987.

[4] https://doi.org/10.1002/qre.4680040324

[5] Hetenyi, M. I. Beam on Elastic Foundation, University of Michigan Press., Ann Arbor, 1960.

[6] Reese, L. C. & Matlock H., Generalized Solutions for Laterally Loaded Piles. Journal of

the Soil Mechanics and Foundations Division, 86 (5), 1960, pp.63-94.

[7] Riks, E. An incremental approach to the solution of snapping and buckling problems.

International Journal of Solids and Structures, 15, 1979, pp. 529-551.

[8] https://doi.org/10.1016/0020-7683(79)90081-7

[9] Wempner, G.A. Discrete Approximation Related to Nonlinear Theories of Solids.

International Journal of Solids and Structures, 17, 1971, pp. 1581-1599.

[10] https://doi.org/10.1016/0020-7683(71)90038-2

[11] ABAQUS 6.13 [Computer software]. Providence, RI, Dassault Systèmes Simulia.

Documents

ANALYSIS OF BUCKLING AND POST BUCKLING OF ......2.1. Buckling of piles Buckling is understood as a loss of stability of a flexible structure that may result in an abrupt and catastrophic