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Stabilizing pile is widely used in the landslide controlling projects and shows excellent seismic performance under the action of earthquake. Therefore, in order to improve seismic design theory, it is of importance to study the seismic response characteristics of stabilizing pile based on elastic-plastic analysis. In view of this, elastic-plastic constitutive model was established to deduce the plastic zone of stabilizing pile. Based on elastic-plastic analysis, the seismic response characteristics and the influence of different section sizes, material strengths, and peak ground motion acceleration (PGA) were analyzed by ANSYS 3D. Resultantly, the elastic-plastic fourth-order tensor

Stabilizing pile, as a flexible retaining structure, has advantages of convenient construction, flexible arrangement, and strong stabilizing ability, which is widely used in the treatment of geological hazards in recent years [

Nowadays, many researches about stabilizing pile have been carried out [

The above researches mainly focus on the structure design, the pile-soil interaction, and the seismic response characteristics of reinforced soil. However, the research on elastic-plastic analysis of stabilizing pile under the action of earthquake is still rare [

During the process of the structural design with elastic theory, the stabilizing pile fails to meet the material damage conditions completely so that the design is conservative and the masonry is wasteful. Therefore, it is necessary to establish the elastic-plastic constitutive model, which is advantageous to make full use of materials and maximize their performance.

The elastic constitutive relationship is not applicable when the stress state of stabilizing pile in particular point meets yield condition and goes into plastic deformation stage. Therefore, it is necessary to establish stress-strain relationship for describing the elastic-plastic behavior of that point.

The stress state can be simplified down to plane stress state because the length is much greater than the width of stabilizing pile. On account of von Mises yield criterion, stabilizing pile can be viewed as isotropic material in the elastic stage and it will be viewed as isotropic hardening material in the plastic stage. On the basis, the elastic-plastic stress and strain can be calculated according to the external loading of stabilizing pile, and the size of plastic zone can be calculated in the case of plastic hinge. Finally, a numerical simulation with ANSYS was used to verify the correctness of elastic-plastic constitutive model.

From the elastic-plastic theory, it can be seen that elastic increment plus plastic increment is strain increment [

In (

Combining (

Based on the relationship between stress increment and strain increment,

In von Mises yield criterion, the yield surface is cylindrical surface in principal stress space with the average principal stress axes. The intersecting line among yield plane,

The von Mises yield criterion in

In addition, the equation of the ellipse is shown in (

When the elastic-plastic deformation occurs, the boundary of elastic zone is constantly changing, and plastic state will inevitably follow the change. The yield functions are shown in (

Therefore, the elastic-plastic constitutive tensor of stabilizing pile with isotropic hardening on the plane is shown in (

Slip mass behind cantilever section of stabilizing pile is simplified as known external force. In order to facilitate the mechanical analysis for embedded segment, stress analysis coordinate system of stabilizing pile is established when the top of embedded segment is taken as the origin [

Internal force calculation of stabilizing pile.

Based on the balance of microsection, force relationship is given in (

The stress of stabilizing pile at any position is shown in Figure

Internal force of pile. (a) Shear stress; (b) normal stress.

When the initial yield happens, the yield function equals 0, as shown in (

Therefore, in the plastic zone, the stress relationships are given in (

The elastic-plastic zone of simply supported beam under uniform distributed loading was analyzed by ANSYS, and then the correctness of elastic-plastic constitutive relationship and plastic zone of isotropic hardening material satisfying von Mises yield condition was verified according to the results of numerical simulation. The material is supposed to be elastic-plastic and bilinear isotropic hardening material, which meets von Mises yield criterion. The sectional dimension was taken as

The numerical model and material property in ANSYS.

According to the stress analysis, it is known that the stress of cross-section in the middle of beam is the maximum. Therefore, elastic-plastic limited state of the cross-section should be considered firstly. When

In (

Through (

Elastic limited state analyzed by ANSYS.

Plastic limited state analyzed by ANSYS.

The comparison of equivalent stress and yield strength of the material itself under the external loading can be used as a basis for judging whether the structure shows plastic zone. This means that when the two values are equal at a certain point, the pile reaches the critical point of yielding at that point. Therefore, the range of plastic zone can be obtained according to the position of the critical point, and the subsequent yield equivalent stress of each part of plastic zone can be obtained on the basis of plastic hardening criterion.

Through (

Plastic zone of simply supported beam.

When

The plastic zone analyzed by ANSYS.

When

Elastic strain and plastic strain analyzed by ANSYS.

The numerical simulation was used to analyze the simply supported beam subjected to uniformly distributed loading to verify the correctness of the elastic-plastic constitutive model and plastic zone mentioned above. The results show that the derived plastic zone and plastic strain are consistent with that of numerical simulation; namely, the formulas of elastic-plastic constitutive model and plastic zone proposed in this paper can be applied to the elastic-plastic analysis of isotropic hardening materials which meet the von Mises yield criterion.

However, the design of stabilizing pile is carried out by static method in Chinese code for seismic design of railway [

A model was established by ANASYS 3D to analyze the seismic response characteristics of stabilizing pile and the piles were made of C20 reinforced concrete, as shown in Figure

The calculation parameters of stabilizing piles.

Materials | Elastic modulus [MPa] | Poisson ratio | Cohesion [kPa] | Internal friction angle [°] | Yield strength [MPa] | Density [kg/m^{3}] |
---|---|---|---|---|---|---|

Sliding mass | 50 | 0.3 | 15 | 28 | — | 1900 |

Bedrock | | 0.25 | 35 | 30 | — | 2500 |

stabilizing pile 1 (C20) | | 0.2 | — | — | 13.4 | 2400 |

stabilizing pile 2 (C25) | | 0.2 | — | — | 16.7 | 2400 |

stabilizing pile 3 (C30) | | 0.2 | — | — | 20.1 | 2400 |

Structural design and landslide characteristics of the model. (a) The landslide characteristics; (b) the design parameters of stabilizing pile.

When the numerical calculation was performed with ANSYS and LS-DYNA, the first was implicit static analysis by ANSYS, and then the results of static analysis would be input into LS-DYNA as the initial conditions. Finally, the Wenchuan Wolong seismic wave was input for explicit dynamic analysis. In the simulation, block element was chosen to simulate slide mass, bedrock, and stabilizing pile, where SOLID 185 block element is in implicit analysis with ANSYS and SOLID 164 block element is in explicit analysis with LS-DYNA. In addition, the D-P constitutive model was adopted in the numerical model. The no reflection boundary conditions and Rayleigh damping were selected for dynamic calculation. The mesh generation is shown in Figure

The calculated model.

Wenchuan wolong seismic wave, with time 0~15 s, was input after baseline correction and the direction was horizontal direction which was perpendicular to stabilizing pile [

The acceleration and acceleration spectral curve after modification. (a) Acceleration time history curve; (b) Fourier spectral (EW) curve.

In the simulation, the revised Wenchuan Wolong seismic wave was input along the stress direction of piles at the bottom of the model. In addition, PGA was selected 0.8 g to damage those piles. On the basis, the seismic response characteristics of displacement and equivalent stress during the process of damage were studied.

In the time history response analysis of stabilizing pile, the time and position of maximum displacement and maximum stress were considered. The displacement time history curve of the pile top and the equivalent stress time history curve of embedded section are shown in Figure

Displacement and equivalent stress time history curve of stabilizing piles. (a) The time history curve of displacement; (b) the time history curve of equivalent stress.

Some conclusions can be seen from Figure

The plastic strain cloud diagram of stabilizing pile at different vibration times.

According to the time history response analysis of stabilizing pile, the displacement and equivalent stress increase with the increase of the vibration time. Considering the seismic performance, the 13.63 s was selected as the most severe time of response and the worst time of seismic design, and then the overall displacement and equivalent stress response characteristics of stabilizing pile (that is to the meaning of “seismic response characteristics” in this paper) were studied [

The dynamic response curve of stabilizing pile. (a) Horizontal displacement; (b) equivalent stress.

As is indicted from Figure

In addition, under the action of static state, the maximum of equivalent stress is 3.73 MPa which has not reached the yield strength of the material, so it is still in elastic state. However, the maximum is 13.4 MPa under the action of earthquake, which exceeds the yield strength, so it leads to the development of connected plastic zone. Finally, stabilizing pile starts to damage and loses its working performance gradually. The equivalent stress and plastic strain cloud diagram are shown in Figure

The plastic strain cloud diagram of stabilizing pile under the action of seismic force. (a) Equivalent stress; (b) plastic strain.

The influence of material strength, section size, and PGA on the seismic response characteristics of stabilizing pile was also analyzed by controlling variable method, where the material strength included C20, C25, and C30 of reinforced concrete, the design dimensions included

The displacement response curves of stabilizing pile in different section sizes, PGAs, and material strengths are shown in Figure

The displacement response curves of stabilizing pile in different working conditions. (a) Different section sizes; (b) different PGAs; (c) different material strengths.

The plastic strain cloud diagram of stabilizing pile in different working conditions. (a) Different section sizes; (b) different PGAs; (c) different material strengths.

It can be seen from Figures

Similarly, when the material strength is larger, the displacement response is smaller, whereas the displacement difference of the stabilizing pile under static and seismic states is not obvious with the same structural material strength. Additionally, the plastic threshold of the material is improved by increasing the strength. Therefore, if the material strength changes, its plastic yield condition and bearing capacity can be enhanced availably.

The equivalent stress response curves of stabilizing pile in different section sizes, PGAs, and material strengths are shown in Figure

The equivalent stress curve of stabilizing pile in different working conditions. (a) Different section sizes; (b) different PGAs; (c) different material strengths.

The equivalent stress cloud diagram of stabilizing pile in different working conditions. (a) Different section sizes; (b) different PGAs; (c) different material strengths.

As is indicated from Figures

The variation of section size can efficiently reduce the equivalent stress of stabilizing pile. To be specific, under the action of earthquake, the maximum of equivalent stress near the embedded section is 16.7 MPa, 13.6 MPa, and 11.3 MPa when the section size is

Additionally, the plastic threshold of the material is improved by increasing the strength. Therefore, if the material strength of stabilizing pile increases, its plastic yield condition and bearing capacity can be enhanced availably.

In order to analyze the seismic response characteristics of stabilizing pile based on elastic-plastic analysis, the constitutive relationship with isotropic hardening was deduced according to von Mises yield criterion, and then a numerical model was established by ANSYS. In view of this, the seismic response characteristics and the influence of different design parameters on seismic performance were analyzed. The following conclusions can be obtained.

(1) The elastic-plastic constitutive relationship of isotropic hardening for stabilizing pile was obtained, and the elastic-plastic fourth-order tensor

(2) As for the piles of same size and same property, the ultimate stress based on elastic-plastic analysis is greater than that of static method. Therefore, the material of stabilizing pile working with elastic-plastic state will be decreased under the action of earthquake.

(3) Under the action of earthquake, stabilizing pile is in the elastic stage at the beginning. With the increase of ground motion time, the section starts to exhibit elastic-plastic state and then the plastic zone expands gradually. Finally, the plastic zone runs through the whole section so that the stabilizing pile loses its working performance.

(4) Under the different design parameters, stabilizing pile shows different seismic response characteristics. Specifically, with the increase of section size, the displacement and equivalent stress decrease and so does the plastic zone. However, the increase of material strength improves the plastic threshold of pile. In addition, the stabilizing pile in high intensity seismic area may exhibit connected plastic zone so that losing its service capacity.

(5) According to the analysis of numerical simulation, it also confirms the feasibly that the elastic-plastic constitutive model deduced in this paper can study those isotropic hardening materials which meet von Mises yield criterion.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.

This study is supported by the National Natural Science Foundation of China under Grant no. 41602332 and the Key Technology Research Project of Prevention and Control for Major Work Safety Accident under Grant no. 2014-3189.