Keywords

1 Introduction

The karst areas in the south of China are mainly composed of soluble carbonate rocks. Under the action of dissolution, weathered, denuded and red soil, the karst area is formed with various terrain. The representative soil layers or strata in karst areas are: carbonate rock layer, calcareous mud stone layer and red clay layer, the geological structure of karst area determines the activity and its nature of earthquake [1, 2], for some small magnitude earthquake can cause considerable damage in karst area, As a result, free vibration analysis is really essential in karst area. So far, among the study on the influence of karst cave in karst area, boundary is often simplified as a fixed end constraint, viscoelastic artificial boundary was seldom used. In order to study the influence of karst cave and boundary condition on free field vibration analysis in karst area, a three-dimensional karst area finite element model of 1000 m × 1000 m × 500 m was established by ANSYS finite element software, and the effects of fixed end constraint and artificial boundary constraint on natural vibration frequency of karst free field were studied, the influence of karst cave on the displacement, velocity and acceleration of free field, as well as the influence of karst cave on spectral curve of center node on surface of free field was researched, which can provide reference for engineering construction in karst area.

2 Artificial Boundary of Karst Foundation

The purpose of artificial boundary is to convert an infinite model into a finite model which is easy for implementation [3]. At present, the widely used artificial boundary in engineering mainly includes viscous boundary, transmission boundary and viscoelastic boundary [4, 5]. The earliest adhesion boundary aims to set up the damper on the artificial boundary to absorb the external wave whose concept is simple and easy to implement. However, it ignores the resilience of elastic boundaries. The transmission boundary proposed by the academician Liao Zhenpeng is the process of using artificial boundary to simulate the boundary crossing, and it establishes the inner point to simulate the displacement of the boundary node at the edge of the boundary at the same time, however, when the input time is long and the waveform is highly irregular, the seismic wave will cause instability [6].

The viscoelastic boundary is the spring unit which is set up in parallel with the damping element on the viscous boundary, which possesses a good stability [7]. It is a local artificial boundary to use the viscoelastic boundary for the boundary effect of the distance simulation of karst foundation. It overcomes the low frequency drift caused by the viscous boundary, as well as it can simulate the elastic recovery performance of the outer half of the artificial boundary and can be excellently combined with the finite element software to meet engineering requirements [8]. The viscoelastic boundary is the stress boundary, and the elastic boundary element parameter is on the principle of dissipating energy by using viscous dampers, i.e. the vibration energy is dissipated by the viscous damping force produced by the damper which is set up at the boundary. In this paper, we adopt the concentrated viscoelastic artificial boundary. The specific parameters are as follows:

$$ {\text{The}}\,{\text{normal}}\,{\text{elastic}}\,{\text{stiffness}}\,{\text{is:}}\;K_{N} = \alpha_{N} \frac{G}{R}\sum\limits_{i = 1}^{n} {A_{i} } $$
(1)
$$ {\text{The}}\,{\text{tangential}}\,{\text{elastic}}\,{\text{stiffness}}\,{\text{is:}}\,K_{T} = \alpha_{T} \frac{G}{R}\sum\limits_{i = 1}^{n} {A_{i} } $$
(2)
$$ {\text{The}}\,{\text{normal}}\,{\text{damping}}\,{\text{coefficient}}\,{\text{is:}}\,C_{N} = \rho c_{p} \sum\limits_{i = 1}^{n} {A_{i} } $$
(3)
$$ {\text{The}}\,{\text{tangential}}\,{\text{damping}}\,{\text{coefficient}}\,{\text{is:}}\,C_{T} = \rho c_{s} \sum\limits_{i = 1}^{n} {A_{i} } $$
(4)

\( R \): distance from source to artificial boundary, \( c_{p} \) and \( c_{s} \): wave velocity of \( P \) wave and \( S \) wave, \( G \): shear modulus of the foundation, \( \rho \): mass density of foundation, \( \alpha_{N} \) and \( \alpha_{T} \): correction coefficients of spring stiffness for normal and tangential in viscoelastic artificial boundary, based on parameter robustness and a large number of literatures [9,10,11], correction factor can be set as 1.33 and 0.67, formulas of wave velocity were shown as follows:

$$ c_{p} = \sqrt {\frac{\lambda + 2G}{\rho }} = \sqrt {\frac{(1 - \upsilon )E}{(1 + \upsilon )(1 - 2\upsilon )\rho }} $$
(5)
$$ c_{s} = \sqrt {\frac{G}{\rho }} = \sqrt {\frac{E}{2(1 + \upsilon )\rho }} $$
(6)
$$ m = \rho r,\;c = \rho c_{p} ,\;k = \frac{4G}{r} $$
(7)

The viscoelastic boundary is a stress boundary, which is the actual state of infinite domain, it is a function associated with the acceleration, velocity and displacement of boundary nodes. The force on the artificial boundary node is applied to the free space according to the equilibrium of the boundary node force:

Normal node force: \( R_{N} = K_{N} \cdot w_{N} + C \cdot \dot{w}_{N} + f_{N} \)

Tangential force: \( R_{T} = K_{T} \cdot w_{T} + C \cdot \dot{w}_{T} + f_{T} \)

These parameters are obtained by an infinite domain linear elastic medium:

$$ f_{N} = - A\sigma = A\rho c_{p} \dot{w}_{N} $$
(8)
$$ f_{T} = - A\tau = A\rho c_{s} \dot{w}_{T} $$
(9)

3 Overview of Karst Free Field Finite Model

In order to analyze the vibration of free field in karst area, a three-dimensional finite element model of free site in karst area was established by ANSYS finite element software. The model size is 1000 m × 1000 m × 500 m, the field length direction was set as X axis direction, the width direction was set as Y axis direction, and the vertical direction was set as Z axis direction. Solid 45 element was used to simulate soil, element grid size is 25 m × 25 m × 25 m, whose precision meets engineering requirements, viscoelastic artificial boundary condition was simulated by Combin14 unit. For the shape of actual cave is complicated and various, it was simplified as a cube of 50 m × 50 m × 25 m, which was located at 37.5 m below the surface of free ground in karst area. Sketch map of cave position with a red mark is shown in Fig. 1. Sketch map of viscoelastic artificial boundary condition is shown in Fig. 2.

Fig. 1.
figure 1

Sketch map of karst cave position

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Fig. 2.
figure 2

Sketch map of viscoelastic artificial boundary condition

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Material characteristics were as follows, elasticity modulus: 8000 MPa, deformation modulus: 4000 MPa, density: 2680 kg/m3, Poisson’s ratio: 0.25, damping ratio: 0.05, Longitudinal wave velocity: 0.60 m/s, Shear wavevelocity: 11937500 m/s.

4 The Influence of Boundary Condition on Natural Vibration Characteristics of Free Field

In order to study the influence of boundary condition on natural vibration characteristics of free field in karst area, the boundary condition of finite model was divided into three kinds: the first kind of boundary condition refers to only applying fixed end constraint at the bottom of finite element model, the second kind of boundary condition refers to applying fixed end constraint at the bottom and four sides of finite element model, the third boundary condition refers to the application of viscoelastic artificial boundary condition at the bottom and four sides of finite element model. Finite element model of the three boundary conditions was analyzed respectively, the results were shown in Table 1.

Table 1. Influence of boundary conditions on natural vibration characteristics of free field.
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Table 1 shows that, for free field model, compared with the 1st and the 2nd boundary condition, after viscoelastic artificial boundary condition was applied in finite element model, natural frequencies of free field model in karst area were reduced at different degrees, and vibration modes were also changed, which is more consistent with actual situation.

5 The Influence of Karst Cave on Displacement, Velocity and Acceleration of Karst Free Field

In order to study the influence of karst cave on the displacement, velocity and acceleration of karst free field. Viscoelastic artificial boundary condition was adopted in the finite element, EI Centro seismic wave was applied in the finite element, finite element model with karst cave and without karst cave were both analyzed. The maximum displacement, maximum speed and maximum acceleration of each node between free field with karst cave and free filed without karst cave were subtracted. The difference contour line of maximum displacement, maximum velocity maximum acceleration on each node was shown in Figs. 3, 4 and 5 respectively.

Fig. 3.
figure 3

Difference contour line between the maximum displacement of each node (m)

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Fig. 4.
figure 4

Difference contour line between the maximum speed of each node (m/s)

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Fig. 5.
figure 5

Difference contour line between the maximum acceleration of each node (m/s2)

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Based on Figs. 3, 4 and 5, after viscoelastic artificial boundary was adopted in finite element and EI Centro earthquake wave was applied in finite element, the response of the maximum displacement, the maximum velocity and the maximum acceleration in the free field with karst cave in karst area compared with them of free filed without karst cave were spread around the karst cave, and the significant change of the maximum value of the displacement, velocity and the acceleration appeared at the top of karst cave.

6 Influence of Karst Cave on Spectrum Curve for Surface Center of Free Field

In order to study the influence of karst cave on the spectral curve of the center surface in free field, free field finite element model was divided into two working conditions: free field with karst cave and free field without karst cave, at the same time viscoelastic artificial boundary conditions and EI Centro seismic wave were applied in the finite element. Then the acceleration epoch curve was extracted from the center of earth surface in free field finite element model, and Fourier transform was carried out. In karst area, the variation trend of spectrum curve for the center node of earth surface was consistent, peak frequency were mainly located between 0 Hz and 5 Hz. In order to study the influence of karst cave on karst field better, the spectral curve values for center node of earth surface in free field with karst cave were subtracted by them without karst cave, the results were shown in Fig. 6.

Fig. 6.
figure 6

Amplitude difference of acceleration for free field with and without karst cave

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It can be seen from Fig. 6 that, for the surface center node of karst free field, the amplitude difference of acceleration between free field with karst and free field without karst is significant in low order frequency, while it is the opposite in high order frequency, and the influence of karst cave on vertical direction is bigger than horizontal direction.

7 Conclusions

In this paper, a three-dimensional finite element model of free field in karst area was established, and the influence of boundary condition and karst cave on the analysis of free field vibration in karst area was studied respectively. Main results and conclusions are as follows:

  1. (1)

    For the free field finite element model in karst area, compared with the fixed boundary condition, the natural frequency of karst free field model was reduced to a certain extent by using viscoelastic artificial boundary condition.

  2. (2)

    After artificial boundary was used in the free field finite element model in karst areas and EI Centro earthquake wave was applied in free field finite element, compared with the response of free field without karst cave, the change trend of maximum displacement, maximum velocity and maximum acceleration were spread around the karst cave, and the change of maximum displacement, maximum velocity and maximum acceleration is the biggest on the top of karst cave.

  3. (3)

    After artificial boundary was used in free field finite element model in karst areas and EI Centro earthquake wave was applied in free field finite element, compared with spectrum curve of free field without karst cave, the amplitude difference between free field with karst and free field without karst is significant in low order frequency, while it is the opposite in high order frequency, and the influence of karst cave on vertical direction is bigger than horizontal direction, which can provide reference for construction of karst region.