J. Cent. South Univ. Technol. (2008) 15(s1): 411-414

DOI: 10.1007/s11771-008-390-3

Viscoelastic behavior of concrete pile

DING Ke(丁 科)^{1, 2}, TANG Xiao-di(唐小弟)^{1, 2}

(1. College of Civil Engineering and Mechanics, Central South University of Forestry and Technology,Changsha 410004, China;

2. Institute of Rheological Mechanics and Material Engineering,Central South University of Forestry and Technology, Changsha 410004, China)

Abstract: Based on constitutive theory of viscoelasticity, the viscoelastic behaviour of concrete pile was investigated. The influence of viscosity coefficient on the stress, displacement and velocity response was discussed. With the increase of viscosity coefficient, the amplitude of stress wave decreases, and the maximum value of the stress wave shifts to deeper position of the pile. In other words, the viscosity coefficient behaves as lag effect to stress wave.

Key words: concrete pile; viscoelasticity; stress; viscosity coefficient

1 Introduction

For worldwide researchers, the vibration theory of pile is always an important research topic. It provides not only the basic academic basis for earthquake resistance of pile foundation and anti-seismic design but also the theoretical basis of pile dynamic testing. At the present, the popular point of view is to regard pile as 1D elastic pole and then study the propagation law of stress wave. According to the viewpoint of material mechanics, solid materials are highly close to viscoelastic bodies, which have elasticity and viscosity at the same time. The viscosity of materials can be assumed to be the internal friction in solids. When external force deforms the solids, it must make works to overcome the internal friction, which is consistent with the fact that the energy of wave will be attenuated during its propagation in the viscosity solids.

In recent years, some researchers start to investigate pile as the viscoelastic pole. CEDERBAUM and MOND^[1] studied the stability of viscoelastic pole under periodic forces. SUN^[2] discussed the influence of external damp on kinetic stability of elastic pile. LIU et al^[3] analyzed the free vibration of elastic beam in viscoelastic foundation. LI^[4] studied the stress wave in viscoelastic pile, but he did not considered the  influence of soil around pile on wave. FENG et al^[5] analyzed the kinetic stability of viscoelastic pile. QUE and WANG^[6] studied the longitudinal vibration with 3D

effect of soil. In this work, the viscoelastic behaviours of concrete pile with the consideration of the friction resistance of soil around pile were investigated and discussed.

2 Mathematical model of dynamic response about viscoelastic pile

If a transient force F(t) is applied to the top of pile, the stress wave is induced in the pile and then propagates along the pile. Its propagating law obeys 1D wave equation. The wave’s actions can be observed from two different points of view. One is the axial movement for each section of pile, which will lead to corresponding displacement u(x, t), velocity v(x, t) and acceleration a(x, t). The other is the axial force applied to each section, resulting in stress σ(x, t) and strain ε(x, t). By setting the top of pile to be the coordinate’s origin and the downward direction of the pile to be axis x, the vibration equation of pile can be expressed as

             (1)

where  is the velocity of longitudinal wave in the pile, ξ=η/E is the relaxation time of the viscoelastic body, τ=ξC²/2 is equivalent relaxation time of the viscoelastic body, β=hC²/(2EA) is the equivalent viscous damping coefficient of soil around the pile. In the above equations, E is the elasticity modulus of pile, ρ is the density of pile, A is the lateral section area of pile, η is the viscous coefficient of concrete pile, and h is the damping coefficient of around the pile. Without considering the viscous behavior of the pile, the second item in Eqn.(1) can be deleted.

If a transient exciting force with the impulse I is applied to the pile with the mass of m at time t=0, the initial conditions can be expressed as

  (0≤x≤L)                  (2)

where  δ(x) is the Dirac function. Then the boundary conditions can be expressed as

                    (3)

where  K₀ is the unit rigidity of the soil below the pile.

According to the initial conditions and the boundary conditions, the solution of Eqn.(1) can be determined to be^[7]

            (4)

where  λ_n is the wave number that satisfies the equation of tan(λL)=K₀/(EAλ),

, ,.

If the unit rigidity is very small, namely K₀=0, the pile can be considered to be a full friction pile whose bearing capacity is completely supported by lateral friction resistance. On the contrary, if the unit rigidity is very huge, namely K₀→∞, the pile can be regarded as an end-bearing pile whose bearing capacity is supported by the bedrock. Generally, the pile is treated as the friction and end-bearing pile.

3 Influence of viscosity coefficient on stress and displacement

According to the constitutive equation of viscoelastic body, the relationship between stress and strain can be expressed as

                  (5)

Substituting u(x, t) from Eqn.(4) into Eqn.(5), we can obtain the following expression:

                        (6)

Supposing that the basic parameters of pile, such as the length of pile L=15 m, the diameter of pile d=500 mm, the density of pile ρ=2 430 kg/m³, the elastic modulus of E=27.950 GPa, the equivalent viscous damping coefficient β=0.628, and the impulse I=0.5 kN?s, we can investigate the influence of viscous coefficient on stress and displacement in pile (see Figs.1-3). It can be seen that the influence of viscous coefficient on displacement is smaller than that of the stress. With the increase of viscosity coefficient, the amplitudes of displacement and stress decrease correspondingly. At the same time, the maximum value of the stress wave shifts to deeper position of the pile. In other words, the viscosity coefficient behaves as lag effect to stress wave.

Fig.1 Influence of viscosity coefficient on stress and displacement in full friction pile at time of 1 ms: (a) Stress wave; (b) Displace- ment wave

Fig.2 Influence of viscosity coefficient on stress and displacement in end-bearing pile at time of 1 ms: (a) Stress wave; (b) Displace- ment wave

Fig.3 Influence of viscosity coefficient on stress and displacement in friction and end-bearing pile at time of 1 ms; (a) Stress wave;    (b) Displacement wave

4 Influence of viscosity coefficient on velocity response at top of pile

According to the relationship between velocity and displacement, we can obtain the velocity response at the top of pile:

  (7)

Supposing the basic parameters of pile are the same to those of Fig.1, the influence of viscous coefficient on the velocity response is shown in Fig.4, from which it is found that there is a reflected wave every 8.84 ms. For the full friction pile, the incident wave and reflected wave have the same phase. But for the end-bearing pile, anti-phases were observed between the incident wave and the reflected wave. The energy of wave is assumed to be invariable during propagation when the viscous behavior is not considered. With the increase of viscosity coefficient, the attenuation of energy becomes faster, and the frequency becomes lower.

5 Conclusions

1) Based on constitutive theory of viscoelasticity, the mathematic model of vibration of pile is set up under excitation by a transient force.

2) The viscoelastic behavior of concrete pile was investigated. The influence of viscosity coefficient on the stress, displacement and velocity response was discussed in concrete pile. The viscosity coefficient behaves as lag effect to stress wave.

Fig.4 Velocity response in pile: (a) Full friction pile; (b) End-bearing pile; (c) Friction and end-bearing pile (K₀=0.01E); (d) Friction and end-bearing pile (K₀=0.1E)

References

[1] CEDERBAUM G, MOND M. Stability properties of a viscoelastic column under a periodic force [J]. Journal of Applied Mechanics, 1992, 59(3): 16-19.

[2] SUN Qiang. Dynamical stability analysis of pile under the influence of damp [J]. East China Highway, 1996(1): 53-56.

[3] LIU Xue-shan, FENG Zhi-liang, XU Bin. Free vibration analysis for elastic beam on viscoelastic foundation [J]. Shanghai Journal of Mechanics, 1999, 20(4): 470-476.

[4] LI Ting. The analytic solution of stress’s propagation in the viscoelastic pile [J]. Journal of Vibration and Shock, 2000, 19(4): 9-13.

[5] FENG Zhen-yu, WANG Zhong-min, FAN Li-jian. Dynamic stability analysis of viscoelastic pile with point viscoelastic supports [J]. China Journal of Highway and Transport, 2006, 19(1): 67-70.

[6] QUE Ren-bo, WANG Kui-hua. Theory on longitudinal vibration of pile in various damping soil layer considering three-dimensional wave effect and its applications [J]. Chinese Journal of Rock Mechanics and Engineering, 2007, 26(2): 381-390. (in Chinese)

[7] DING Ke, CHEN Zhi-li, XIE Zhong-qiu. Study on transient response test of a viscoelastic pile [J]. Journal of Vibration and Shock, 2007, 26(6): 114-117.

(Edited by LI Xiang-qun)

Received date: 2008-06-25; Accepted date: 2008-08-05

Corresponding author: DING Ke, PhD; Tel: +86-13974831121; E-mail: kding@sina.com