J. Cent. South Univ. Technol. (2007)06-0853-05 

DOI: 10.1007/s11771-007-0162-5 

Perturbation analysis on post-buckling behavior of pile

Zhao Ming-hua(赵明华)¹, He Wei(贺  炜)², WANG Hong-hua(王泓华)²

(1. Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China;

2. School of Bridge and Structure Engineering, Changsha University of Science and Technology, Changsha 410000, China)

Abstract: The nonlinear large deflection differential equation, based on the assumption that the subsoil coefficient is the 2nd root of the depth, was established by energy method. The perturbation parameter was introduced to transform the equation to a series of linear differential equations to be solved, and the deflection function according with the boundary condition was considered. Then, the nonlinear higher-order asymptotic solution of post-buckling behavior of a pile was obtained by parameter-substituting. The influencing factors such as bury-depth ratio and stiffness ratio of soil to pile, slenderness ratio on the post-buckling behavior of a pile were analyzed. The results show that the pile is more unstable when the bury-depth ratio and stiffness ratio of soil to pile increase, and although the buckling load increases with the stiffness of soil, the pile may ruin for its brittleness. Thus, in the region where buckling behavior of pile must be taken into account, the high grade concrete is supposed to be applied, and the dynamic buckling behavior of pile needs to be further studied.

Key words: pile foundation; stable loading path; perturbation approach; bury-depth ratio; stiffness ratio of pile to soil

1 Introduction

A slightly lateral deflection frequently causes a vertical loaded pile buckling, when the pile is of  large slenderness ratio^[1], little bury-depth ratio or in extra-weak soil, and increasing length of pile cannot enhance bearing capacity of the pile at this situation, because it is limited by buckling load. Many scholars did researches on the calculating method of buckling load of pile, such as the TIMOSHENKO’s solution in Ref.[2], in which elastic module of soil was considered as a constant, the simulation computer solution by DAVISSION and ROBINSON^[3], in which the layer stratum was taken into account, the power series solution by HEELIS et al^[4], the experience method and Galerkin method by PENG^[5] in China, the energy method by ZHAO et al^[6-9]. These solutions assumed that the strain was so minor that it could be neglected, therefore, the deflection will remain increasing after the loading on pile top reaches buckling, so they cannot be used to analyze the post-buckling behavior of a pile.

BUDKOWSKA and SZYMCZAK^[10] studied the stable loading path of partially embedded pile. However, they simply used two items from the power series of the deflection function, and presumed that there was only one semi-wave in the deflection function of pile, which was great different from that in practice. HE et al^[11] studied the stable loading path of entirely embedded pile by dual-perturbation approach. The result shows that the stable loading path is closely relevant to slenderness ratio of pile and stiffness ratio of pile to soil. Nevertheless, the entirely embedded pile has too large buckling load to be unstable. Hence the nonlinear large deflection differential equation for partially embedded pile was established by energy method and C method herein, based on which the stable loading path was deduced by perturbation method.

2 Mechanical models of soil around pile

The calculating models for soil around pile can be classified into three kinds: 1) the method supposing that maximum resistance of subsoil exerts; 2) the method supposing that the foundation is completely elastic; 3) the method with advantages of 1) and 2). And the second method is widely applied in engineering practice because it is simple to deal with in mathematics, which is based on Winkler-model.

2.1 Method ZHANG

This method was put forward by ZHANG^[1], which assumed that the coefficient of subsoil kept unchangeable along pile shaft (Fig.1(a)). And it is used very popularly in Japan.

It has been proved by experiments that the resistance of cohesionless soil or normally consolidated cohesion soil is almost equal to zero on the ground, butresult calculated by method ZHANG is on the opposition. The subsoil coefficient does not vary with the depth only in the hard rock. This method has not been used for quite a long time.

  Fig.1 Sketch map of subsoil coefficient

(a) Method ZHANG; (b) Method M; (c) Method C

2.2 Method M

Method M was introduced into China by SILIN^[1], which supposed that the subsoil coefficient increased linearly with the depth along pile shaft (Fig.1(b)).

Many scholars agree that the subsoil coefficient of cohesionless soil or normally consolidated cohesion soil can be analyzed with method M. Presently, the pile foundation specifications for railway, highway and bridge and the current national standards all recommend this method.

2.3 Method C

Method C was put forward by KOUICHI^[1], which was proposed that the subsoil coefficient increased, conforming to parabolic curve with the depth along pile shaft (Fig.1(c)).

The researchers in Shanxi Highway and Transport Institute in China found that the subsoil coefficient increased in 0.1-0.5 power with the depth through analyzing the tests results of many piles^[1]. The standards recommend method C as well as method M.

3 Solution of perturbation method for pile

In recent years, long concrete piles are applied to engineering practice in a large amount. In method M, it is assumed that the subsoil coefficient keeps increasing with the depth, so the method turns to method C. In order to simplify the problem, taking the two-pinned elastic pile into account, and the mechanic model is shown in Fig.2.

Fig.2 Mechanical model of pile

A great number of calculating results show that, self-weight and side resistance have little effect on buckling load^[6-7], so the non-load-transfer is assumed. The total potential energy is obtained by bending potential energy of pile, elastic potential energy of soil and potential energy of exterior load:

 (1)

where  Π is the total potential energy, N·m; E is the elastic modulus of the pile, Pa; I is the inertial moment of section of the pile, m⁴; k is the stiffness of subsoil, N/m⁴; b is the calculating width of the pile, m; y  is the deflection function of the pile; l  is the length of the pile, m.

When the potential energy reaches the maximum or minimum, that is, the situationget the non- dimensional indexes as follows:

，，

By using the symbols unchanging and through Eula-formula, it leads to:

(2)

where

  and .

Boundary condition can be described as

        (3)

Eqn.(2) can be written by power series^[12-14]:

        (4)

where  ε is a perturbation parameter that is not given meaning for the moment.

Substituting Eqn.(4) into Eqn.(2), perturbation formula in each order can be obtained as

O(ε):

Considering the boundary condition, the solution can be written as

           (5)

Substituting Eqn.(5) into the first order perturbation formula, it leads to:

              (6)

O(ε²):                   (7)

Assuming that and substituting it into Eqn.(7), then P₁=0 and y₂=0 can be obtained

O(ε³):    (8)

Assuming that

       (9)

Substituting Eqn.(9) into Eqn.(8) yields:

        (10)

O(ε⁴):  (11)

Just like the second order, there exist P₃=0 and y₄=0

O(ε⁵):

  (12)

Assuming that

  (13)

Substituting Eqn.(13) into Eqn.(12) leads to:

        (14)

O(ε⁶)…

So the asymptotic solution of Eqn.(2) can be obtained:

  (15)

         (16)

where all coefficients can be described as a function of :

               (17)

       (18)

    (19)

    (20)

             (21)

             (22)

           (23)

           (24)

              (25)

    (26)

    (27)

     (28)

        (29)

Substituting  into Eqn.(15), the maximum deflection of the pile can be got:

  (30)

where

 ; .

Otherwise

 (31)

Eqn.(16) represents the behavior of load distribution along the pile shaft when deflection fits sine curve. Assuming that the load at the topes pile is the average of load distribution along pile shaft, it leads to:

            (32)

Substituting Eqns.(16) and (31) into Eqn.(32) yields

      (33)

where

 ,

,

.

Let y_max＝0 in Eqn.(33), then

     (34)

Eqn.(34) is the buckling load of vertical loaded pile.

Analyzing unconfined compressed pole, the results are shown in Fig.3. When there is no deflection, P equals 1, which means that the buckling load is equal to the classic Eula-solution. When y_max equals 0.05, P turns to 1.003, the result of the analytical solution is P=1.003, there is no error^[14]; when y_max equals 0.1, P turns to 1.013, and the result of the analytical solution is P=1.012, and the error is 0.098%; when y_max equals 0.2, the result turns to P=1.055, the result of the analytical solution is P=1.057, and the error is 0.189%. It can be seen that in the situation of minor deflection, the perturbation solution can give an accurate result.

Fig.3 Post-buckling equilibrium path of pinned pole

4 Influencing factors

According to Ref.[14]，the subsoil coefficient (c) can be obtained and shown in Table 1, and elastic modulus of pile can be got and shown in Table 2. Let the pile be 30 m in length, and 1 m in diameter, assembling the data in Tables 1 and 2, the stiffness ratios of pile to soil can be seen in the range of 0.307-0.664.

Under conditions of the stiffness ratios of 0.307 and 0.664 the post-buckling behavior of pile was analyzed, and compared with the results when stiffness ratio equals 0, 0.1 and 100. Fig.4 shows the situation when the bury-  depth ratio(h/l) is 0.4. As increasing the stiffness ratio, the ascending curve descends gradually. When analyzing the pile with stiffness ratio recommended in Ref.[15], The results are almost the same as those when the stiffness ratio reaches the infinity. Figs.5 and 6 respectively show the post-buckling behavior of pile when bury-depth ratio is 0.65 and 0.90, which are similar to Fig.4. The difference is that as the bury-depth ratio increases, the ascending curve turns to descend.

Figs.7 and 8 show the post-buckling behavior of pile with stiffness ratio of 0.100 and 0.307, respectively. When α=0.100, as increasing bury-depth ratio, the ascending post-buckling behavior curve turns to descend. When α=0.307, as increasing the bury-depth ratio, the post-buckling behavior curves all descend, and curve with larger bury-depth ratio is more obvious in descending.

Table 1 Subsoil coefficients (c) soil

Table 2 Elastic module of concrete

Fig.4 Effect of stiffness ratio of pile shaft to soil on post-buckling behavior at h/l=0.4(δ is the deflection at point of the pile)

Fig.5 Effect of stiffness ratio of pile shaft to soil on post-buckling behavior at h/l=0.65

Fig.6 Effect of stiffness ratio of pile shaft to soil on post-buckling behavior at h/l=0.9

Fig.7 Effect of bury-depth ratio on post-buckling behavior at ?α=0.100

Fig.8 Effect of bury-depth ratio on post-buckling behavior at ?α=0.307

Ascending curve indicates that the structure is ductile. When the structure reaches the limited state, it will keep on absorbing energy, instead of ruining immediately. But the descending one is on the contrast. For the pile with large bury-depth ratio, although the post-buckling behavior is unstable, the buckling load is very large, so its behavior is good. The pile will fail in the form of brittleness when it is acted by huge instantaneous load and absorbs huge energy, so the problem is meaningful in dynamic buckling. Besides, in the region of weak subsoil, higher grade concrete is supposed to be applied. In this way, the buckling load can be enhanced, as well as ductility in post-buckling.

5 Conclusions

1) Based on energy method and method C, the nonlinear differential equation for large deflection vertical loaded pile is established. Then, the equation is transformed to a series of linear differential equations and solved by perturbation parameter. By substituting the perturbation parameters, and taking the deflection function according with the boundary condition into account, the nonlinear higher-order asymptotic solution of post-buckling behavior of a pile is obtained.

2) When the bury-depth ratio increases, the buckling load of pile becomes larger, and the post-buckling behavior of pile turns to be more unstable.

3) When the stiffness ratio of pile to soil increases, the buckling load of pile becomes larger, and the post-buckling behavior turns to be more unstable.

4) The buckling load increases with the increase of stiffness of soil, but the pile may ruin for its brittleness. Thus, in the region where buckling behavior of pile must be considered, the high grade concrete is supposed to be applied, and it is significant to study the dynamic buckling behavior of pile.

References

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(Edited by CHEN Wei-ping)

Foundation item: Project (50378036) supported by the National Natural Science Foundation of China

Received date: 2007-04-22; Accepted date: 2007-05-30

Corresponding author: ZHAO Ming-hua, Professor, PhD; Tel: +86-731-8821590; E-mail: mhzhaohd@21cn.com