Design criteria or Fiber Reinorced Rubber Bearings J. M. Kelly Earthquake Engineering Research Center University o Caliornia, Berkeley A. Calabrese & G. Serino Department o Structural Engineering University o Naples Federico II SUMMARY: This paper contains the indings o a study on the mechanical behaviour o unbonded Fiber-Reinorced Bearings (FRB). Typical FRBs consist o several layers o rubber that are bonded to iber reinorcing sheets. The purpose o the reinorcement is to prevent the rubber rom bulging laterally under compressive load. The most important aspects o these bearings are (i) they do not have thick end plates; (ii) they are not bonded to the top and bottom support suraces; and (iii) their reinorcements are very lexible. These aspects may seem to be design deiciencies, but they have the advantage o eliminating the presence o tensile stresses in the bearing by allowing it to roll o the supports when it is sheared. This reduces the typical bonding requirements. The weight and the cost o isolators is reduced by using iber reinorcing, no end-plates and no bonding to the support suraces, oering a low-cost lightweight isolation system. The paper introduces simple theories, valid as design criteria, or the determination o the tensile stresses in the reinorcement, the vertical stiness o the bearing, the ultimate lateral displacement. A good benchmark to test the theories is proposed in this work using the output o Finite Element Analysis (FEA). Keywords: Low-Cost Fiber Reinorced Isolators, Design Criteria, FEA 1. INTRODUCTION Modern seismic protection systems have shown their good perormance in past earthquakes, or example in the 2011 o the Paciic coast o Tohoku Earthquake on March 11, 2011. Among the dierent available techniques, seismic isolation seems to be the most applicable or a large range o structures. Generally, an isolation bearing consists o thin sheets o rubber bonded to steel plates. The system has enough vertical rigidity to sustain gravitational load and horizontal lexibility to shit the undamental requency o the building away rom the dominant requency range o most earthquakes. This technique is generally applied to expensive and strategically relevant buildings. Consequently, or this kind o applications, devices are large, expensive and heavy. In order to ensure the diusion o base isolation to common residential buildings, especially in earthquake prone areas in developing countries, a reduction o the cost and weight o the devices is required. The possibility o replacing the steel reinorcing plates in the bearing with iber reinorcement is presented in this paper. A iber reinorcement is not only lighter than steel but it allows a less labour-intensive manuacturing process that would reduce production cost. This paper ocuses on the behaviour o Fiber Reinorced Bearings (FRBs) under pure compression and shear introducing reerences or sizing low cost devices. 2. BEHAVIOUR UNDER COMPRESSION 2.1. Vertical stiness and stress in the reinorcement The essential characteristic o an elastomeric isolator is the very large ratio o the vertical stiness to the horizontal stiness. This ratio can be in the order o magnitude o hundreds. In the case o FRBs, this large ratio is due to the iber sheets which allow the rubber to shear reely while preventing its
lateral bulging. For design purposes, it is particularly important to predict the vertical stiness and the collapse condition o FRBs under compression. Under compression, collapse o the bearing can occur or global ailure due to buckling o the device, local ruptures o the reinorcement or the detachment o the rubber rom the iber sheets. Thereore, an accurate knowledge o the global characteristics o the device and o the stress distributions at the rubber iber interaces and in the iber reinorcement is necessary. To predict the compression stiness o thin elastomeric pad a linear elastic theory is used. The irst analyses o the compression stiness or rigid reinorcement were done by Rocard (1973), Gent and Lindley (1959), and Gent and Meinecke (1970). Further developments in the study o traditional elastomeric bearings were made by Kelly and Takhirov (2002) using a simpliied version o the previous theories. Tsai and Kelly (2001) also extended the theoretical results to the analysis o iber-reinorced elastomeric isolators developing an approximate closed-orm solution (the pressure solution) based on several assumptions. The kinematic assumptions are as ollows: (i) points on a vertical line beore deormation lie on a parabola ater loading, (ii) horizontal planes remain horizontal, and (iii) the reinorcement stretching produces a constant displacement trough the thickness. Furthermore, rubber is assumed incompressible, and the pressure is assumed to be the dominant stress component. For simplicity we consider the solution or a long rectangular strip pad o thickness t in which a rectangular Cartesian coordinate system, (x,y,z), is located in the middle surace o the pad, as shown in Fig.2.1. Figure 2.1. Ininitely long rectangular pad showing dimensions In the previous hypothesis, the pressure solution gives the stress in the reinorcement, eective compressive modulus, E c : F x, and the F x cosh x / b Et 1 t cosh (2.1) Et Ec t In which tanh 1 (2.2)
2 2 12Gb E t t (2.3) where: - is the vertical displacement, - E is the elastic modulus o the reinorcement, - t is the thickness o the reinorcement, - G is the shear modulus o the rubber, - B 2b is the base length o the bearing. 2 When 0, i.e., E, imposing S b 2t r, E c becomes Ec 4GS. This result is the same that can be derived applying the pressure solution to bearings with rigid reinorcement. The ormula 2 also shows that Ec 4GS or all the values o E. Once E c, the instantaneous compression modulus o iber-rubber composite under a speciied level o vertical load, is known the vertical stiness o a bearing can be easily computed as KV EcA tr where A is the area o the bearing and t r is the total thickness o rubber in the device. The pressure solution can be applied even or compressible material. In case strip type bearings, the solution o the problem is Ec K 2 tanh 1 2 2 (2.4) where K is the bulk modulus o the rubber, 2 2 2 12Gb Kt and 2 2 2. 2.2. Theoretical results According to the previous solution Eqns. 2.1-2.4 we can determine the vertical stiness o strip type FRB bearings, K V,and the tensile stress in the reinorcement, x. In Table 2.1 some calculation examples are reported or bearings with dierent shape actors, S. The dierent values o shape actor are obtained by increasing the values o the base o the device (B= 250; 300; 350; 400; 450; 500mm). Results are reerred to bearings with a longitudinal dimension o 750mm. As shown in Fig.2.1, each device is made o twenty-eight rubber layers with twenty-nine interlea iber sheets. Each rubber layer is t r = 6.36 mm thick, and each iber sheet is t = 0.07 mm thick. Table 2.1. Applications o the pressure solution. GEOMETRICAL AND MECHANICAL PROPERTIES B [mm] 250 300 350 400 450 500 H [mm] 180 L [mm] 750 t [mm] 0.07 n [-] 29 iber layers tr [mm] 6.37 E [MPa] 14000 G [MPa] 0.70 COMPRESSION OF PAD WITH RIGID REINFORCEMENT E c [Mpa] 1082.93 1559.42 2122.54 2772.30 3508.69 4331.72 K v [N/mm] 1140919 1971509 3130683 4673206 6653842 9127356
COMPRESSION STIFFNESS WITH COMPRESSIBILITY OF THE ELASTOMER (K=2000MPa) β [-] 1.27 1.53 1.78 2.04 2.29 2.55 E c [MPa] 658.29 809.58 940.60 1051.90 1145.77 1224.92 K v [N/mm] 693539 1023520 1387352 1773163 2172814 2581018 COMPRESSION STIFFNESS WITH FLEXIBLE REINFORCEMENT α [-] 4.59 5.51 6.43 7.34 8.26 9.18 E c [Mpa] 120.60 126.19 130.19 133.19 135.52 137.39 K v [N/mm] 127059 159541 192028 224515 257003 289491 FLEXIBLE REINFORCEMENT AND COMPRESSIBILITY(K=2000MPa) λ [-] 4.76 5.72 6.67 7.62 8.58 9.53 E c [Mpa] 113.10 118.11 121.69 124.37 126.45 128.12 K v [N/mm] 119160 149319 179481 209643 239806 269969 2.3. FEA results Figure 2.1 Cross section o a bearing The goal o this section is to veriy the validity o the results provided by the pressure solution by comparing them with results rom FEA. A series o FEA o rectangular elastomeric bearings were conducted using the general-purpose inite element program MSC.Marc 2005 [MSC.Sotware 2004]. The analyses were perormed using a twodimensional model under the plane strain assumption assuming specialized element types that automatically address numerical issues to get accurate solutions to large strain problems, change o contact conditions, sliding and near-incompressibility o the rubber. The results o analysis presented in this paper are based on the widely popular mixed method proposed by Herrmann (1965). A restricted case o the general Hellinger-Reissner variational principle is used to derive the stiness equations. The inite element discretization consists o square our-node elements with side length o 2 mm and is denser at the contact interace. The iber reinorcement o the bearing is modeled using the rebar element. It is a tension element o a liner elastic isotropic material with Young s modulus E = 14000MPa, and thickness t = 0.07mm. The rubber is modelled by a single-parameter Mooney-Rivlin material (i.e., Neo-Hookean) with strain energy unction that is described by the shear modulus G = 0.7 MPa, and the bulk modulus κ = 2000 MPa. The top and bottom support suraces are modeled as rigid lines. The contact between the rubber and the support suraces is modeled by Coulomb riction with μ = 0.9. As mentioned above, the analysis is carried out under the plane strain assumption. Moreover, large strain theory is employed. The kinematics o deormation is described ollowing the Updated Lagrangian ormulation (i.e., the Lagrangian rame o reerence is redeined at the last completed iteration o the current increment). Furthermore, Full Newton-Raphson solution method is used. The bearings are loaded in two steps: 1) Pure compression [0-3.45MPa] 2) Horizontal displacement is increased till collapse (= u ). Figures 2.2 and 2.3 show the contour maps o the equivalent stress obtained rom analysis with Marc or bearings 250 and 500 under pure compression
Figure 2.2. Von Mises stress contours [MPa] at peak vertical orce in a bearing o base B=250mm Figure 2.3. Von Mises stress contours [MPa] at peak vertical orce in a bearing o base B=500mm The contours show a stress concentration in the core o the bearings. For a given compressive orce P (average pressure = PA=3.45 MPa) in the considered range o bases, the maximum equivalent stress in the core o the bearing is unaected by the change o the dimension o the device. Thereore, the stress in the core o the bearing is the same when the base dimension is modiied. However, as expected, a new arrangement o stress distribution along the base length can be observed when the dimensions are changed. As a result, or bigger bearings as we move towards the ree edges, the stress drops less aggressively than or smaller ones. It is worth noting that due to the rictional restraint o the supports, the iber layers closest to the supports are in compression. Fig.2.4 is a plot o the vertical stiness as a unction o the shape actor or the six bearings o dierent geometry.
3.5 x 105 Vertical test results (SET2 t =0.07mm) vertical stiness [N/mm] 3 2.5 2 1.5 FR PS FRC PS FR FEA 1 20 22 24 26 28 30 32 34 36 38 40 42 shape actor [-] Figure 2.4. Vertical stiness Vs. shape actor FR PS and FRC PS are the theoretical vertical stiness results o iber reinorced bearings or incompressible and compressible material respectively. The FEA outputs are plotted as FR FEA. FEA and pressure solution output result in good agreement. FEA gives higher values o vertical stiness because o a stiening contribution o the quadratic mesh. Fig. 2.5 shows the plot o tensile stress as a unction o the dimensionless length o the device, x B, or the mid-height iber layer, where the stresses are the largest. In the igure, the solid line represents the PS result and the dotted line represent the FEA result. 120 Stress distribution in the reinorcement (B=250mm) PS t=0.07 100 FEA t=0.07 stress in the iber layer [MPa] 80 60 40 20 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 x/b [-] Figure 2.5. Stress distribution in the reinorcement (FEA VS. Pressure Solution) The pressure solution gives accurate results or the description o the stress o the iber in the bearing s core. Towards the ree edges o the bearings, or a length o 10% o the base, the FE and the PS results are dierent. This dierence is very low or higher shape actor but is signiicant as the shape actor decreases.
3. STABILITY OF HORIZONTAL DISPLACEMENT 3.1. Design criterion The basic premise or the analysis o these bearings is that the regions o the bearing that have rolled o the rigid supports are ree o all stress and that the volume under the contact area has constant shear stress. Under this assumption, the active area that produces the orce o resistance F to displacement, is B and thus the orce (per unit width o the bearing) is F G B, but / t r, thus F G B / t r, and consequently the orce displacement curve has zero slope when df G B 2 0, i.e., when B / 2 d t r (3.1) The implication o this result is that the bearing remains stable in the sense o positive tangential orce-displacement relationship so long as the displacement is less than hal the length in the direction o the displacement. As a result o the limiting displacement analysis, it is possible to determine a simple design criterion or this type o bearing. We need only to determine a maximum required design displacement which normally would depend on the site, the anticipated isolation period and damping. I we denote this by then the requirement or positive incremental horizontal stiness requires that the width B o the bearing in the direction o the displacement be at least twice the displacement, i.e. B 2. 3.2. FEA Shear Results In the second part o the analysis, a constant vertical load is applied, and then the horizontal displacement is increased in order to investigate the ultimate behavior o the bearings. Figures 3.1 is a Von Mises stress contour map or B=250 at the displacement that corresponds to the peak orce in the load-displacement curve. Figure 3.1 Von Mises stress contours [MPa] at peak horizontal orce in a bearing o base B=250mm The aorementioned igure clearly shows the avourable response o an isolator which is not bonded to the top or bottom supports. This is due to the elimination o tension in the elastomer. In a bonded bearing under the simultaneous action o shear and compression, the presence o an unbalanced moment at both top and bottom suraces produces a distribution o tensile stresses in the triangular
region outside the overlap between top and bottom. The compression load is carried through the overlap area, and the triangular regions created by the shear displacement provide the tensile stresses to balance the moment. These tensile stresses must be sustained by the elastomer and also by the bonding between the elastomer and the steel reinorcement plates. The provision o these bonding requirements is the main reason or the high cost o current designs o isolator bearings or buildings. With the elimination o these tension stresses, the bonding requirements or this type o bearing are reduced. Figures 3.2 is the stress contour in the reinorcement at peak horizontal orce. Figure 3.2 Tension contours [MPa] in the iber reinorcement at maximum shear (B=250) In Fig.3.3 the horizontal load is plotted as a unction o the horizontal displacement or the six bearings under a vertical pressure o 3.45 MPa. horizontal load [kn] 200 150 100 50 250 300 350 400 450 500 Horizontal test results at 3.45MPa vertical pressure 0 0 0.05 0.1 0.15 0.2 0.25 horizontal displacement [m] Figure 3.3 Force-displacement curves Fig.3.4 shows the horizontal load VS the dimensionless horizontal displacement. These curves are straight lines up to the value o the lateral load or which roll-o o the right end starts to occur. A progressive reduction o the lateral tangent stiness is then observed with urther
increase o the lateral load. The large deormation that FRBs experience can be modeled accurately in Marc. However, the inite element mesh distorts so heavily that the analysis becomes grossly inaccurate or stops due to individual mesh elements turning inside out and pre-speciied convergence criteria not being satisied. The values that describe the ultimate behavior are summarized in Table 3.1. horizontal load [kn] 200 150 100 50 250 300 350 400 450 500 Horizontal test results at 3.45MPa vertical pressure 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 horizontal displacement/base [-] Figure 3.4 Force-displacement/base curves Table 3.1 Ultimate perormances o the bearings under horizontal load. B S F h,u τ u Δ u Δ u /B γ u [mm] [-] [kn] [MPa] [mm] [-] [%] NAME 500 43.48 170.5 0.45 204 0.4 1.1 450 39.13 139.9 0.41 195 0.4 1.1 400 34.78 104.1 0.35 190 0.5 1.1 350 26.09 77.17 0.29 167 0.5 0.9 300 21.74 52.13 0.23 136 0.5 0.8 250 43.48 32.92 0.18 104 0.4 0.6 4. CONCLUSIONS In this study, the ollowing observations and conclusions can be made. The most important aspects o FRB are: they do not have thick end plates, they are not bonded to the top and bottom support suraces, and their reinorcements are lexible. These eatures at irst sight might seem to be deiciencies o their design, but they have the advantage to eliminate the presence o tensile stresses in the bearing by allowing it to roll o the supports. This reduces the costly stringent bonding requirements that are typical or conventional bearings. The validity o the pressure solution, a method developed to describe the global behavior and the stress distribution in rubber bearings, and the validity o an ultimate analysis to investigate the peak horizontal displacement are discussed by comparing the theoretical results with a FEA. Bearings with dierent shape actors were modelled and loaded under pure compression and shear. For both load conditions, the bearings exhibit a linear behavior under a wide range o displacement. For the theoretical and the numerical solution, in the considered range o geometries, the variation o the vertical stiness as a unction o the shape actor is linear. The pressure solution result, except near the ree edges, is particularly valid to determine the stress distribution in the reinorcement. Moreover, the resultant maximum stress in the reinorcement is in good agreement to the FE outputs.
In the considered range o dimensions, bearings with a dierent shape actor exhibit at peak lateral displacement, under the same vertical pressure, the same maximum stress in the reinorcement. This value is approximately three times larger than when the bearing is under pure compression. The shear tests conducted in Marc or the dierent bearings show that the peak horizontal displacement is approximately hal o the base length. The theoretical model, which was useul to determine the peak horizontal displacement, gives good results even when compared against the FEA with non-zero vertical load. It was also observed that or a ixed axial load, the shear load goes through a maximum as the shear delection is urther increased. The displacement at which this maximum occurs decreases with increasing axial load. Thereore, it can be concluded that urther FEA are necessary to precisely evaluate the inluence o the vertical load on the horizontal behavior o the bearings. Moreover, this study was conducted considering the shape actor S as the only variable. Analytical results showed that the shape actor is an important geometrical parameter that characterizes the mechanics o the bearings; however, there are other parameters, such as the slenderness o the bearing, that have to be taken into account to ully describe the global behavior o the device. The comparison between the theoretical and the FEA outputs shows potential to validate urther theoretical results. Results o the theoretical model, such as the vertical stiness, the maximum stress in the reinorcement, and the peak horizontal displacement o the bearing, are in good agreement with the results rom FEA. REFERENCES Gent, A.N., Lindley, P.B. (1959). Compression o bounded rubber blocks, Proceeding Institution o Mechanical Engineers, 173(3), 111-122. Gent, A.N., Meinecke, E.A. (1970). Compression, bending and shear o bonded rubber blocks, Polymer Engineering and Science, 10 (1), 48-53. Herrmann, L.R. (1965). Elasticity equations or incompressible and nearly incompressible materials by a variational theorem, AIAA J. 3, pp. 1896 1900. Kelly, J.M., Takhirov, S.M. (2002). Analytical and experimental study o iber-reinorced bearings, PEER 2002/11. Rocard, Y. (1937). Note sur le calcul des proprietes elastique des supports en caoutchouc adherent, Journal de Physique et de Radium, 8, 197. Tsai, HC., Kelly, J.M. (2001). Stiness analysis o iber-reinorced elastomeric isolators, PEER 2001/05.