The dynamic response magnitudes of retaining walls under seismic loadings, such as earthquakes, are influenced by their natural frequencies. Resonances can occur when the natural frequency of a wall is close to the loading frequency, which could result in serious damage or collapse. Although field percussion tests are usually used to study the health state of retaining walls, they are complicated and time consuming. A natural frequency equation for retaining walls with tapered wall facings is established in this paper using the transfer matrix method (TMM). The proposed method is validated against the results of numerical simulations and field tests. Results show that fundamental frequencies decrease gradually with wall height; soil elastic modulus exerts a great influence on the fundamental frequency for walls with smaller facing stiffness; fundamental frequencies are smaller for a hinged toe than a fixed toe condition, and this difference is smaller in taller walls.
Retaining walls are widely used in civil engineering (e.g., roads and railways) due to their advantages, such as simple structural forms and convenient constructions. Earthquakes have occurred frequently in recent years, and the dynamic performances of retaining walls under seismic loadings have received increasing attention. Tatsuoka et al. [
As an inherent structural characteristic, natural frequencies have a significant influence on dynamic displacements under seismic loadings [
For retaining walls in practical engineering, tapered concrete facings are often used to increase their external stabilities. Therefore, the above methods [
The natural frequencies of retaining walls have a significant effect on the dynamic responses of structures. Although field percussion tests and existing analytical methods have been carried out and proposed to study the natural frequencies, they are either costly, time consuming, or limited in structures with constant cross section shapes. To predict the natural frequencies of retaining walls with tapered facings which are generally used in civil engineering, a new method is proposed in this paper. Additionally, examples are used to evaluate the accuracy of the proposed method, and a parametric analysis is carried out to analyze the influences of wall facing, wall height, and soil on its natural frequencies.
The retaining wall model used for analysis is shown in Figure
Cross section of a cantilever retaining wall and the mechanical model.
When the bottom width of the concrete wall facing
Calculation models.
In this paper, the TMM is proposed to solve the problem above. Dividing the wall height
The displacement of the
If the displacement,
The boundary conditions of the
The natural frequency of the
Similarly, the
The boundary conditions of the two adjacent beams is expressed by the following equation:
Let
The relationship between each two adjacent beams is established by equation (
In analytical calculations, numerical simulations, and even model tests, ideal bottom constraint conditions, either a fixed boundary or a hinged boundary [
Simplifying the above equation, we obtain
Judging from equations (
As introduced in Section
To evaluate the accuracy of the method proposed in this paper, the results predicted are compared with those given by Scott [
Calculation parameters.
Parameter | Value |
---|---|
|
6/8/10 |
|
26 |
|
1.0 |
|
0.5 |
|
2320 |
|
1.55/1.16/0.93 |
Table
Calculated results.
|
Proposed method | Scott [ |
FEM [ |
---|---|---|---|
6 | 14.6 | 10.6 | 16.0 |
8 | 9.0 | 6.5 | 9.6 |
10 | 6.5 | 3.2 | 6.4 |
A field vibration test was conducted by Klymenkov et al. [
Predicted results using different methods.
Method | Fundamental frequency (Hz) |
---|---|
Field test [ |
8.00 |
Numerical simulation [ |
9.46 |
Scott [ |
10.16 |
Proposed method | 8.83 |
Xu [
Comparison of the fundamental frequency obtained from a full-scale test and the proposed method.
Method | Fundamental frequency (Hz) |
---|---|
Xu [ |
22.41 |
Proposed method | 21.37 |
To study the influence of different parameters on the natural frequencies of retaining walls under the stability precondition, the following basic parameters are defined:
Figure
(a) Effects of wall height on fundamental frequencies; (b) effects of wall height on first four natural frequencies.
Figure
Since retaining walls are usually considered as short-period structures [
Although a larger wall width can result in a larger mass, it also increases the flexural rigidity of beams, so the relationships between wall width and fundamental frequencies are nonlinear. Results in Figure
Effects of wall width on fundamental frequencies for walls with rectangular facings.
The cross section facing shapes of retaining walls are usually trapezoidal. To study the effect of facing shape on the fundamental frequencies of retaining walls, three shapes (i.e., triangular, trapezoidal, and rectangular) are examined in the case of a constant facing cross-sectional area. Figure
Effects of cross section shape on fundamental frequencies.
The relationships between the fundamental frequencies and wall facing modulus are shown in Figure
Effects of wall modulus on fundamental frequencies.
Figure
Effects of soil modulus on fundamental frequencies: (a)
Results in Figure
Effects of boundary condition of wall facing on fundamental frequencies.
The dynamic response of a retaining wall is influenced by the natural frequency of the structure. Although the natural frequency can be obtained by field tests, these percussion tests are complicated and time consuming. Besides, the existing analytical methods are usually limited to structures with constant cross sections. However, the shape of the retaining wall facing is usually trapezoidal in practical engineering. To improve the accuracy and solve the natural frequency of retaining walls with tapered facings, the transfer matrix method (TMM) is employed. The following conclusions are drawn: The results predicted from the proposed method are closer to the FEM results as compared to those obtained using the existing analytical method. Besides, the accuracy of the proposed method is also validated against the results of two full-scale model tests. The fundamental frequencies decrease with wall height. The influence of wall height on fundamental frequencies is more significant on taller walls as compared to the shorter walls. The relationship between fundamental frequencies and wall width is nonlinear because wall mass and flexural rigidity are both influenced by the wall width. In addition, the minimum fundamental frequency is obtained when the value of The fundamental frequencies increase with facing modulus and soil modulus, and the increasing trends are more obvious for shorter walls. The fundamental frequency is smaller for a hinged boundary than a fixed boundary condition, and this difference is negligible with wall height.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.
This work was supported by the project of Science and Technology Research and Development Plan of China Railway Corporation (no. 2014G003-C). The authors are grateful to colleges at the University of Colorado Boulder and University of Birmingham for their assistance. The first author also thanks the CSC for providing the scholarship for occupational training at the University of Colorado Boulder.