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://iaeme.com/Home/issue/IJCIET?Volume=10&Issue=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].
With an aim of buckling and post buckling analysis of piles, in this paper we deal with
Method with linear-elastic behaviour of material and nonlinear buckling analysis using -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 of the pile (Figure 1) is: "x"
Analysis of Buckling and Post Buckling of Piles Fully Embedded in Ground
where: - p(x)=ky(x) soil pressure. Whereas the equation of flexure of a pile subject to axial iscompression force only may be written in the following shape:
where: - is pile stiffness, P axial force on a pile, and K is modulus of horizontal soil EI is 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,
-Ritz Method.
2.1.1. Linear analysis of pile buckling- Galerkin’s Method
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 -assumed mode shape satisfies all boundary conditions
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, k 0=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. buckling behaviour of the piles Post 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
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 " of the pile is: s"
P [kN] crit
L [m]
Analysis of Buckling and Post Buckling of Piles Fully Embedded in Ground
This equation prevails worth for the case when wc/L>0.3 in contrary, for w, c/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,
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 c
rit
Wc
Analysis of Buckling and Post Buckling of Piles Fully Embedded in Ground
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
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
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
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
Analysis of Buckling and Post Buckling of Piles Fully Embedded in Ground
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) 2nd Mode; c) 3rd 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 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- -plastic FEM model with the elastic and nonlinear elastoRiks 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 ratio is modified for each iteration in order the solution to follow up a LPFspecific 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
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.
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., P 290kN). crit(p-p)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 P =342,804kN to P 290kN). crit(p-p) crit(p-p)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
Analysis of Buckling and Post Buckling of Piles Fully Embedded in Ground
Figure 8 Load proportional factor ( )-Arc length curve of linear and nonlinear FEM model LPF
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
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. 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
st critical force)
LPF=0.75 approximately 75% critical buckle force pile is getting loose bearing capability
When entering elasto-plas c zone the corresponding s ness loss causses considerable reduc on of cri cal 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.
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[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, (5), 1960, pp.63-94. 86
[7] Riks, An incremental approach to the solution of snapping and buckling problems. E.International Journal of Solids and Structures, , 1979, pp. 529-551. 15
[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.
