A novel costeffective isolator reinforced by engineering plastics has been designed and manufactured for seismic protection for lowrise buildings in less developed areas. The reinforcement is flexible in tension, which is similar to fiberreinforced isolators. However, available solutions for fiberreinforced isolators are not applicable, because the Poisson effect of engineering plastics cannot be neglected, which is done for fiber reinforcement. In this paper, analytical solutions for compression and bending stiffness for rectangular isolators reinforced by engineering plastics are proposed, with both the Poisson effect of the reinforcement and the effect of rubber compressibility taken into consideration. Then, the simplified solutions are also derived, which can greatly improve calculation efficiency. To validate the solutions, finite element analysis is conducted on a set of isolators with different reinforcement stiffnesses. The results show the superiority of the proposed solutions to the previous solutions for fiberreinforced isolators. A series of experimental tests of the isolators are also carried out to verify the solutions. Both the analytical and the simplified solutions match well with the experimental results.
Recent earthquakes in China such as Wenchuan and Yushu earthquakes caused numerous loss of lives, which has emphasized that a greater lose may happen when earthquakes occur in rural areas and less developed areas. Ge et al. [
Zhou [
One of the earliest theoretical analyses of bonded rubber was conducted by Rocard [
Fiber reinforcement is flexible in tension, so the problem becomes much more complex. A widely acceptable method to simplify the problem is the pressure solution method introduced by Kelly [
In engineering practice, even though the value of Poisson’s ratio of rubber may be very close to 0.5, the effect of compressibility is considerable to isolator stiffness and should not be neglected. For engineering plastics, the shear stress and Poisson effect cannot be neglected either. In this paper, the solutions for compression and bending stiffness of rectangular isolators reinforced by engineering plastics considering both rubber compressibility and reinforcement Poisson effect is presented. The derived solutions are expressed in terms of Fourier series which are complex and to the disadvantage of popularizing isolation technology. Thus, the weighted residual approach is introduced to solve the partial differential equation of the stress, and a semianalytical solution in a much simpler form is achieved. To validate the solutions, experimental studies are conducted, and finite element analysis is also carried out. The proposed solutions are proved to have enough accuracy.
Fiberreinforced plastic plate is usually employed as the reinforcement in the novel rectangular isolator, which is a common kind of engineering plastic and belongs to composite material. It was made up of unsaturated polyester matrix and glass fiber cloth in previous research [
Fiberreinforced plastic plate isolator.
Cross section of the isolator.
The isolator is much cheaper and lighter than conventional steelreinforced isolators of the same size, and the processing of plastic is much easier than steel. The isolator promotes the displacement ability of fiberreinforced isolators. Unlike fiberreinforced isolators, the reinforcement of this isolator is composite material made up of plastic matrix and embedded glass fiber. So, the outofplane stiffness of the reinforcement is much higher than fiber sheets. As a result, the rollover phenomenon is limited and the vertical surface of the isolator does not contact with the loading plane even at very large horizontal deformation. It is to be noted that the stresses in two orthogonal directions in this isolator cannot be decoupled as they are in fiberreinforced isolators because of the plastic matrix. So, the analytical deviations should consider the Poisson effect of the reinforcement.
A fiberreinforced plastic plate isolator is constituted of multiple layers. Assuming the stress components of each layer along the height of the isolator is uniform, a layer of rubber and two layers of plastic plates are extracted as an analytical unit. Considering the analytical unit subjected to a vertical compression load
Coordinate system and deformation under compression.
The displacements of the rubber are assumed to obey the kinematic assumption as followed by Kelly and Calabrese [
The pressure solution method [
According to the above equations and the constitutive equations of linear elastic material, there are relationships in rubber derived as follows [
The compressibility of rubber layer is considered by
In a single layer of plastic plate, several layers of glass fiber cloth are embedded, of which the directions are centrosymmetric. The base material of unsaturated polyester is isotropic. Thus, the plastic plate is assumed to be an isotropic material with a modulus of
Stress components on the reinforcement.
The equilibrium equations in
The constitutive equations considering Poisson ratio of the plates are as follows:
The inplane shear strain component of the rubber is as follows:
Substituting equations (
Equation (
Substituting equations (
It is to be noted that the rubber and plastic plate layers are free in the
Integrating equations (
Combining equations (
Differentiating equations (
The Poisson ratio of the reinforcement and the bulk modulus of rubber are both included in the equilibrium equation (
The method of separation of variables is used here combining with Fourier transformation to solve equation (
Considering the vertical normal stress of the plate, there is a relationship as follows:
The normalized compression modulus
Parameter analysis of
To simplify the solution of the square isolator, a numerical method of weighted residuals (MWR) introduced by Xu [
Replacing
Using collocation method to eliminate the residuals by
For a special case of a square isolator with
Simplified solution for
A special case of square cross section with typical geometry parameters is considered. Six models with different elastic moduli of the reinforcement
Parameters of the models.
Model 




P_{1}  0.5  2.61  3.88 
P_{2}  1  1.85  2.87 
P_{3}  2  1.31  2.20 
P_{4}  5  0.83  1.67 
P_{5}  10  0.58  1.45 
P_{6}  20  0.41  1.33 
The Ogden model was used to describe the hyperelastic behavior of rubber. Reinforcement material was modeled by linear elastic isotropic material. The outofplane translational degrees of freedom and the rotational degrees of freedom of the reinforcement layers were restrained by reference points to keep the layer plain. The inplane translational degrees of freedom were released to allow the inplain deformation. The meshing is shown in Figure
FEA model and meshing.
Comparison of
It is observed that the proposed analytical and simplified solutions always present better evaluation of
For an isolator with
The simplified form is calculated using equation (
Applying moment
Deformation under bending.
The displacement functions of rubber in the
The equilibrium of stress in rubber taking into account of rubber compressibility in a pure bending condition is derived by Angeli et al. [
In addition, the expression of
This equilibrium equation of
Equation (
The applied moment
Combining with
The normalized bending modulus
Parameter analysis of
The MWR method is also used here to achieve a simplified solution of bending modulus. After analyzing the shape of the function of
This function automatically satisfies the boundary condition that the stresses on the boundary equal zero. Substituting it into equation (
For a square case, it can be further simplified as follows:
Simplified solution for
The models established in section
Comparison of
The validity of the proposed solutions is confirmed by the curves. Both the analytical and simplified solutions are obviously closer to the FEA results. As
The effect of the bending modulus of a single layer on the horizontal stiffness of an isolator is considered based on the theory of Haringx [
The analytical solution of
The simplified solution for
The analytical and simplified solutions for horizontal stiffness can be obtained by substituting equations (
Eight representative specimens were tested to verify the proposed solutions. The specimens have been designed based on commonly used materials and dimensions of fiberreinforced plastic plate isolators. The detailed parameters are summarized in Table
Characteristics of tested specimens.
No.  Aspect ratio 


2 






B1  1  19400  11  230  5  3  0.4  11.5  1.34 
B2  17400  11  230  5  3  0.4  11.5  1.36  
B3  17400  12  230  4.25  3  0.4  13.53  1.56  
B4  17400  12  230  4.25  3  0.6  13.53  1.91  
B5  3000  12  230  4  3  0.4  14.38  2.10  
B6  2  3000  6  90  5  4  0.4  6.67  0.72 
B7  17400  12  230  4.25  3  0.4  18.04  1.56  
B8  17400  12  230  4.25  3  0.6  18.04  1.91 
All the bearings were tested on the compressionshear testing machine in the Earthquake Engineering Research and Test Center of Guangzhou University, China. The vertical actuator can impose a maximum compression force of 10,000 kN, and the horizontal actuator is capable of a maximum force of ±500 kN.
Standard cyclic vertical and horizontal tests as shown in Figure
Experiment setup. (a) Vertical test. (b) Horizontal test.
Samples of the results of vertical tests of
Vertical test results. (a) Specimens B3 and B5. (b) Specimens B7 and B8.
A sample of the results of horizontal tests of B7 is shown in Figures
Horizontal test results of specimen B7. (a) Forcedisplacement in
The test results and analytical and simplified solutions are listed in Table
Test results of specimens with
Specimen  Vertical stiffness (kN/mm)  Horizontal stiffness (kN/mm)  







 
B1  271  255 (−5.8%)  249 (−8.3%)  12  0.367  0.337 
0.341 (−7.1%) 
B2  205  208 (1.5%)  202 (−1.4%)  15  0.163  0.159 (−2.5%)  0.166 (1.8%) 
B3  373  350 (−6.1%)  343 (−8.1%)  10  0.371  0.371 (0.0%)  0.373 (0.5%) 
B4  432  436 (0.9%)  406 (−6.0%)  15  0.514  0.552 (7.4%)  0.555 
B5  304  278 
268 
5  0.461  0.469 (1.7%)  0.470 (2.0%) 
Test results of specimens with
Specimen  Vertical stiffness (kN/mm)  Horizontal stiffness (kN/mm)  





 
B6  450  474 
3  1.000  0.980 (−2.0%) 
B7  796  817 (2.6%)  15  0.697  0.670 (−3.9%) 
B8  1032  1014 (−1.8%)  15  0.994  1.130 
As can be seen from Tables
The analytical solutions for compression and bending modulus of rectangular fiberreinforced plastic plate isolators are proposed, and simplified solutions are also derived for a special case of square shape. The solutions show advantages to the existing solutions and agree well with experimental tests.
Based on the analytical analysis, finite element analysis, and experimental results, the following conclusions can be made:
The effects of the Poisson ratio of reinforcement and compressibility of rubber cannot be neglected for both compression and bending modulus. The proposed solutions of compression and bending modulus can be used to calculate the vertical and horizontal stiffness of this type of isolators.
Simplified solutions for compression and bending modulus are derived by a method of weighted residuals. The accuracy is controlled by the selection of trial function of pressure
Finite element analysis results show that neglecting the compressibility of rubber overestimates both compression and bending modulus and neglecting the effect of the Poisson ratio underestimates both the moduli. The proposed solutions include both of the effects and have obvious advantage over the solutions by previous papers.
The validity of the proposed solutions is verified by a series of experimental tests. The error of analytical solution for vertical stiffness is less than 8.7% and for horizontal stiffness is less than 11.7%. The error of simplified solution for vertical stiffness is less than 11.7% and for horizontal stiffness is less than 8.0%.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by National Key R&D Program of China under Grant no. 2017YFC0703600, the Program for Changjiang Scholars under Grant no. IRT13057, in part by program for Innovation Team of Department of Education of Guangdong Province under Grant no. 2016KCXTD016, in part by the Program for Ram City Scholar under Grant no. 1201541630, and Guangdong Special Program under Grant no. 2014TX01C141.