J. Cent. South Univ. Technol. (2007)05-0725-05
DOI: 10.1007/s11771-007-0138-5 
Catastrophic model for stability analysis of high pile-column bridge pier
ZHAO Ming-hua(赵明华), JIANG Chong(蒋 冲), CAO Wen-gui(曹文贵)，LIU Jian-hua(刘建华)
(Institute of Geotechnical Engineering, Hunan University, Changsha 410082, China)
Abstract：According to the engineering features of higher pile-column bridge pier in mountainous area, a clamped beam mechanical model was set up by synthetically analyzing the higher pile-column bridge pier buckling mechanism. Based on the catastrophe theory, the cusp catastrophe model of higher pile-column bridge pier was established by the determination of its potential function and bifurcation set equation, the necessary instability conditions of high pile-column bridge pier were deduced, and the determination method for column-buckling and lateral displacement of high pile-column bridge pier was derived. The comparison between the experimental and calculated results show that the calculated curves agree with testing curves and the method is reasonable and effective.
Key words：stability; pier; pile foundation; catastrophe theory; buckling; critical load
1 Introduction
There are lots of solutions about bulking analysis of pile foundation, such as solution considering foundation coefficient as constant[1], simulating computing solution considering layer characteristics of foundation coefficient[2], power series solution[3], and research on partially embedding pile foundation with initial post buckling condition[4]. After 1870’s, with the rapid development of pile foundation in hydraulic engineering and bridge engineering in China, and based on assimilating experience of theory and experiment analysis, the calculation methods were presented. Fox example, energy method was presented by ZHAO et al[5-9], critical load formula through Galerkin method was presented by PENG10], and few references reported on buckling analysis of high pier.
The calculation methods based on elasticity theory are generally used to analyze stability about high pile-column bridge pier[11]. For the low pier with small slenderness ratio, small deviation comes into being. But calculation result will bring great deviation with horizontal force P and vertical force F interaction for larger slenderness ratio, because the conventional method has not consider lateral displacement and influence of vertical force[12]. The nonlinear characteristics in bulking destruction was considered in Refs.[13] and [14], but stress conditions and buckling mechanisms were not fully realized on high pile-column bridge pier in a mountainous area and pile foundations were comparatively considered and the difference between calculation and test result exists.
In this paper, based on analyzing buckling destruction mechanisms and according to catastrophe characteristics with discontinuousness and asymmetry of high pile-column bridge pier in a mountainous area, the nonlinear science—catastrophe theory was introduced[15-17], and the cusp catastrophic model of instability of high pile-column bridge pier was built. Then, the destruction condition was solved. In the end, the new method for determining elastic destruction load and lateral displacement was established according to the condition.
2 Instability mechanism analysis of higher pile-column bridge pier
According to the variable characteristics of engineering structure in destabilization critical state, stability problems occur in the higher pile-column bridge pier and can be divided into two groups as follows.
2.1 Destabilization of equilibrium bifurcation—the first class of stability problem
When a load acting on the top of a pier is less than the critical value, the erect stability equilibrium state of pile foundation and pier is always kept. When the load value comes to the critical value, the pier shaft will be bent, which is named buckling or instability. And now, the equilibrium state of high pier comes from erecting a micro bends, just comes from simple compressive equilibrium state changing to bending-compressive equilibrium. When the load excesses the critical value, just as bifurcation, the curves of load-deflection show two possible equilibrium approaches and that load can be referred to as bulking critical load or critical load. Because the equilibrium bifurcation phenomenon appears at the same load value, the instability state can be referred to as equilibrium bifurcation instability, i.e. the first class instability.
2. Destabilization of extremum point—the second
class of stability problem
The pier with ideal condition actually does not exist and eccentric compression state with initial fault and residual remainder stress always occurs in it. With the horizontal loads including longitudinal braking force of automobile and temperature etc acting on rubber bearings at the top of pier, and considering the elastic resistance of soil in the lateral pile foundation under the pier, the work performance of high pier and pile foundation between the axial compression and eccentric compression has not fundamental difference except the eccentric degree of load. At the first time, the pier locates the bending-compressive equilibrium and the horizontal displacement increases with accretion of loads. When the load acting at the top of pier reaches a certain critical value, the pier is immediately destroyed because of instability of equilibrium with disturbing of pier shaft. Thereby, the descending segment curve is difficultly drawn, the point is named as extremum point, and the corresponding load is named as stable limit load of pile foundation or crushing load. This can be seen that the eccentric compression pier with extremum point instability only has extremum point, and the pier shaft is always on the condition of bending-compressive equilibrium without bifurcation point including two different deformation states at the same load point. Therefore, the load of instability is named as extremum point of the high pile-column bridge pier or the second class of stability problem.
3 Cusp catastrophe model for stability
analysis of higher pile-column bridge pier
3.1 Mechanical model
In order to simplify calculation, the hypothesis is drawn: do not consider the rotation of pier bottom and regard boundary condition of the bottom as fixing; presume member moving in one plane, just consider the deformation and displacement of high bridge pier on the plane of longitudinal; presume the cross section is always perpendicular at axial deformation. Therefore, the high pier and pile foundation can be simplified as mechanical model as shown in Fig.1, with the axial compressive member model of fixing at the bottom and free at the top under the concentrative load PF acting on the top (including the load from superstructure and self- weight of pier), horizontal concentrative load PP (automobile braking force) and horizontal distributing load Pq.
Fig.1 Mechanic model of high pier and pile foundation
3.2 Potential function
After the establishment of mechanical model above, the next key task is to deduce its potential energy and establish the total potential function of the mechanical model, and then translate the potential function into normal form of cusp catastrophe model by mathematical method.
The total potential function of higher pile-column bridge pier is made up of its bend strain energy and the corresponding displacement power.
(1)
where U is the bend strain energy, V is the loading potential energy and 
is the external force work.
(2)
(3)
(4)
If command x/l=z; ω=y/l, integrate the integrand into Taylor series and neglect the item above fourth power, integrating can obtain:
(5)
3.3 Bifurcation point set equation
Based on the geometric boundary condition of higher pile-column bridge pier, the hypothetic deflection function expression is
(6)
where δ is the deflection amplitude in the empirical formulation.
Substituting Eqn.(6) into Eqn.(5) and integrating can obtain:
(7)
where 
, if command
(8)
(9)
(10)
(11)
then the potential energy of higher pile-column bridge pier is
(12)
And the equation of equilibrium surface is
(13)
(14)
And the equation of bifurcation point set is
(15)
Substituting Eqns.(10) and (11) into Eqn.(15) and the equation of bifurcation point set is
(16)
3.4 Elasticity failure load of higher pile-column bridge pier
When Pq=0 and PP=0，it can obtain α=1 based on Eqn.(16), so
(17)
This is the exact value of critical load for elastic columns, of which one end is fixed and the other is free, because the hypothetic curve is the actual deflection curve of elastic columns instability under axial compression, whose one end is fixed and the other is free.
Based on the basic principle of catastrophe theory and before the stability of higher pile-column bridge pier, u≤0, so
≤1 (18)
Therefore, the elastic failure load of higher pile-column bridge pier with one end fixed and the other end free and subjecting lateral uniformly distributed loading is
PF≤ (19)
3.5 Lateral displacement of higher pile-column bridge pier
1) When 
＞0, the real root for Eqn.(13) is
(20)
It is the dimensionless variety between maximum of lateral displacement and load of higher pile-column bridge pier after instability, then
＜0 (21)
So it is proved that instability of higher pile-column bridge pier is at unequilibrium state. But the equilibrium positions is exclusive.
2) When ＜0，the maxima of lateral displacements at three different unequilibrium states of higher pile-column bridge pier are as follows:
＞0 (22)
＞0 (23)
＜0 (24)
＜y3＜0 (25)
＜0 (26)
＜0 (27)
where 
0＜β＜π. And substituting δ1, δ2 and δ3 into Eqn.(7) respectively, then we can obtain
＜0 (28)
＜0 (29)
＞0 (30)
This means that the higher pile-column bridge pier under lateral uniformly distributed loading can be at three different equilibrium states before it is unstable. When the control point is in the middle of equilibrium surface, the higher pile-column bridge pier is at stability equilibrium state and its lateral maximum displacement is y3. When the control point is up of equilibrium surface, the higher pile-column bridge pie is at unstable equilibrium state and its lateral maximum displacement is y1. When the control point is under the equilibrium surface, the higher pile-column bridge pier is at unstable equilibrium state and its lateral maximum displacement is y2.
3) When =0, the maximum of lateral displacement at two different equilibrium states of higher pile-column bridge pier is
, Pq＞0 (31)
＞0 (32)
＜0 (33)
＞0 (34)
Substituting Eqns.(31) and (32) into Eqn.(7) respectively, and then we can also easily obtain
＜0 (35)
(36)
So when control point is at the juncture of above and middle of equilibrium surface, the higher pile-column bridge pier is at critical state.
4 Comparative analysis of theoretical calculation and laboratory experiment
In order to draw all kinds of balanced variety between lateral displacement and load of higher pile-column bridge pier, Figs.2-6 show the relation curves for kinds of balanced states of lateral displacement(y1- y5)(absolute value) and Pq of lateral load when F is constant and α is 0.8.
Fig.2 Relation curve between lateral displacement (y1) and lateral load (Pq)
Fig.3 Relation curve between lateral displacement (y2) and lateral load (Pq)
Fig.4 Relation curve between lateral displacement (y3) and lateral load (Pq)
Fig.5 Relation curve between lateral displacement (y4) and lateral load (Pq)
Fig.6 Relation curve between lateral displacement (y5 or y6 ) and lateral load (Pq)
By analyzing Figs.2-6, the following conclusions can be drawn. The theoretical calculated curves of higher pile-column bridge pier between lateral displacement and lateral load are in good agreement with the experimental curves, the lateral displacement is not positively correlated with lateral load when axial load is constant, whether control point is above, middle or under equilibrium surface .
5 Conclusions
1) The stability mechanism of higher pile-column bridge pier in a mountainous area based on catastrophe theory was deeply analyzed and the cusp catastrophic model of instability of higher pile-column bridge pier was built.
2) Based on the catastrophe theory, all kinds of unstable states of higher pile-column bridge pier were analyzed, the method for determining elastic destruction load and lateral displacement was proposed for all kinds of balanced states.
3) The comparison between the experimental and calculated results shows that the calculated curves are in good accordance with testing curves, indicating that the method is reasonable and effective.
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Foundation item: Project(50578060) supported by the National Natural Science Foundation of China
Received date: 2007-03-20; Accepted date: 2007-05-18
Corresponding author: ZHAO Ming-hua, Professor; Tel: +86-731-8821590; E-mail: mhzhaohd@21cn.com
(Edited by YANG Hua)
