- Email: [email protected]
Variation of the vertical stiffness of strip-shaped fiber-reinforced elastomeric isolators under lateral loading
Accepted Manuscript Variation of the Vertical Stiffness of Strip-Shaped Fiber-Reinforced Elastomeric Isolators under Lateral Loading Peyman M. Osgooei, Dimitrios Konstantinidis, Michael J. Tait PII: DOI: Reference:
S0263-8223(16)00112-4 http://dx.doi.org/10.1016/j.compstruct.2016.01.089 COST 7195
To appear in:
Composite Structures
Please cite this article as: Osgooei, P.M., Konstantinidis, D., Tait, M.J., Variation of the Vertical Stiffness of StripShaped Fiber-Reinforced Elastomeric Isolators under Lateral Loading, Composite Structures (2016), doi: http:// dx.doi.org/10.1016/j.compstruct.2016.01.089
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Variation of the Vertical Stiffness of Strip-Shaped Fiber-Reinforced Elastomeric Isolators under Lateral Loading Peyman M. Osgooei1, Dimitrios Konstantinidis2, Michael J. Tait3
Abstract A fiber-reinforced elastomeric isolator (FREI) is a relatively new type of isolator that utilizes fiber material for the reinforcing layers. FREIs can be installed in a bonded or unbonded application. In this study, finite element analysis (FEA) is carried out on bonded and unbonded strip-shaped FREIs to investigate the variation in vertical stiffness as the isolator undergoes lateral displacement. The vertical stiffness of the isolators under pure compression obtained by FEA was in good agreement with the predictions of two available closed-form solutions. As the lateral displacement increases, it was observed that for bonded FREIs the vertical stiffness decreases monotonically; whereas, for unbonded FREIs, the vertical stiffness decreased up to 175% shear deformation, where an increase in vertical stiffness was observed. FEA results confirmed that the effective overlap area method provides reasonable estimates for the vertical stiffness of bonded FREIs. It is observed that as the applied vertical stress increases, the vertical stiffness of bonded and unbonded FREIs increases. Finally, the paper shows that under large lateral displacements, bonded FREIs develop large tensile stresses in the regions outside the overlapping areas, while the tensile stresses that develop in unbonded FREIs are very low and confined in a small region. Keywords: fiber-reinforced elastomeric isolators; laminated rubber bearings; seismic isolation; combined lateral and vertical loading; vertical response; finite element analysis 1
PhD Candidate, Dept. of Civil Engineering, McMaster University Assistant Professor, Dept. of Civil Engineering, McMaster University, 1280 Main St W, Hamilton, Ontario L8S 4L8, Canada; E-mail: [email protected] (corresponding author) 3 Professor, Dept. of Civil Engineering, McMaster University 2
1
1. Introduction A steel-reinforced elastomeric isolator (SREI) is the most common type of seismic isolator in use. This type of isolator is composed of layers of elastomer interleaved with steel reinforcement shims. Thick steel end plates are typically bonded to the top and bottom surfaces of the bearing for connecting purposes. The application of SREIs in North America is limited to high importance structures (e.g., historical buildings, emergency centers), or buildings with sensitive or valuable contents [1]. Currently it is not economically feasible to employ SREIs for residential or ordinary importance structures. Kelly [2, 3] suggested that by reducing the costs and weight of SREIs, the application of elastomeric bearings (and in a more general case, seismic isolation) can be extended to housing and commercial buildings. Replacing the steel reinforcing shims with a lighter material (for example, fiber reinforced polymer) can reduce the weight of SREIs. Unlike SREIs, which require a hot vulcanization process, a cold vulcanization process can be utilized to construct fiber-reinforced elastomeric isolators (FREIs) [4], which could potentially reduce the high manufacturing costs associated with SREIs. In addition, FREIs can be manufactured in large pads where individual isolators can be cut to the desired size, leading to further potential reductions in cost and turnaround time. FREIs can be installed in both bonded (Fig. 1-a) and unbonded (Fig. 1-b) applications. Unbonded FREIs do not have end plates. Due to the flexibility of the fiber reinforcing layers, unbonded FREIs undergo a unique rollover deformation (Fig. 1-b). This rollover has been shown to increase the seismic isolating efficiency by reducing the effective lateral stiffness of the isolators [5]. In addition, the unbonded application results in a reduced stress demand on the
2
elastomer and fiber reinforcement layers [5], which makes it possible to use a simpler manufacturing process for unbonded FREIs. Figure 2 compares the lateral load-displacement relationship of bonded and unbonded FREIs. While the effective (secant) stiffness of the bonded FREI is unaffected by the change in the displacement, the effective stiffness of the isolator in an unbonded application varies with lateral displacement. Note that in this discussion, geometric effects that can reduce the effective lateral stiffness of a bonded isolator [6–8], especially for slender configurations, are not considered. Under small displacements, the effective stiffness of the unbonded FREI is constant. As the displacement increases, due to the rollover effect, the effective stiffness of the unbonded FREI decreases, until a point where an increase in the stiffness is observed. This increase in the effective stiffness in unbonded FREIs is due to the contact of the originally vertical faces of the isolator with the loading surfaces and can limit the isolation displacement under large (i.e., maximum-considered-earthquake) ground motions [4, 9] . The vertical response of FREIs has been investigated in a number of analytical [10–13] and numerical [14–18] studies. Furthermore, experimental studies involving vertical compression and lateral cyclic tests have shown that FREIs achieve both adequate vertical stiffness and the required lateral flexibility to render them suitable for seismic isolation [2–4, 19–23]; However, in these studies, FREIs were investigated under pure compression, and the effect of lateral displacement on the compression response characteristics of the isolators was neglected. As such, the variation in the vertical stiffness of unbonded FREIs subjected to lateral loading requires investigation.
3
In this paper, finite element analysis (FEA) is employed to study the vertical response of FREIs, in both bonded and unbonded applications, under different lateral displacement amplitudes. Two strip-shaped isolators with the same overall dimensions but different shape factors, (defined for a single elastomer layer as the ratio of the total loaded area to the load-free area), are modeled and analyzed using MSC Marc [24], a general-purpose finite element program. First, results from the FEA under pure compression are compared against two available closed-form solutions. Then, the vertical stiffness of FREIs is characterized at various levels of lateral displacement and under different values of average vertical stress. The accuracy of the effective overlap area method [25], which is widely used for estimating the vertical stiffness of bonded SREIs under lateral displacement, is evaluated for bonded FREIs by comparing its predictions against FEA results. The paper examines how the downward vertical displacement of the top support of a FREI varies with increasing lateral displacement. Finally, the stress distribution in the overlapping and non-overlapping regions of bonded and unbonded FREIs is investigated.
2. Finite Element Analysis Two strip-shaped isolators are considered in this study. Table 1 shows the dimensions of the isolators. Both isolators had a width of = 300 mm and a width-to-height aspect ratio of 3.0. A shear modulus of = 0.8 MPa and a bulk modulus of = 2000 MPa, corresponding to an
elastic modulus of = 2.4 MPa and a Poisson’s ratio of = 0.4998, were used for the elastomer. The elastic modulus and the Poisson’s ratio of the fiber reinforcement material were selected as = 20 GPa and = 0.2, respectively. 4
FEA of FREIs is challenging due to near-incompressibility of the elastomer material, large deformations, large strains, and complex contact. In the literature, a number of FEA studies have been carried out on FREIs subjected to lateral loading [26, 5, 27, 28, 15, 23, 29, 30]. In this study, 2D FEA were carried out using MSC Marc (2013) [24]. As seen in Osgooei et al. [16], the number of elements along the height of the isolator influences modeling error. In this study, a total of 84 and 104 four-node linear plane-strain elements were used along the height of isolators S11 and S22, respectively. In order to consider the interaction of the originally vertical faces of the unbonded FREI with the loading surfaces, a touching contact was defined between the mid-elastomer layers and the loading surfaces. In this contact model, the nodal points of the elastomer layers were allowed to detach from the loading supports when the tensile stress values were reached. Under static loading conditions, an isotropic hyperelastic material model can be used for rubbers. For an isotropic hyperelastic model, the strain energy function, , is expressed in terms of the three invariants of the Cauchy-Green deformation tensor (either the left, B, or the right, C), defined by [31] = + + (1)
= + + (2) = (3)
where , , and are the eigenvalues of B or C, also known as the principal stretches. For an incompressible material, the third invariant is equal to 1 and the strain energy function can be expressed by [31]
= ( , ) (4) 5
In this study, the neo-Hookean constitutive model is used for the strain energy function, defined by [31] = ( − 3) (5)
where is the material constant and is related to the shear modulus of the elastomer by =
(6) 2
Using the strain energy function, the stress-strain relationship for each deformation state can be calculated. For example, assuming a uniaxial tension deformation state where = and assuming an incompressible material ( = 1), we have = =
1
√
(7)
Using the neo-Hookean strain energy density function, the strain energy density function becomes = # + $ − 3% (8)
The fiber-reinforcement layers were modeled using a linear-elastic isotropic material model. Quadrilateral plane-strain elements were used to model the elastomer and fiber-reinforcement layers. The top and bottom loading surfaces were modeled using rigid wire elements. Figure 3 shows the FEA model of the isolator with = 11. The FREI models were subjected to an average vertical stress (i.e., the vertical load divided by the loaded area) of &̅ = 2 MPa and were subsequently loaded to lateral displacements of (/*+ = 0.25, 0.50, 0.75, 1.00, 1.25, 1.50 and 1.75. At each displacement amplitude, the applied
lateral displacement was held constant, while the vertical load (,) on the isolator was subjected to a ±20% change, and the variation in the vertical displacement (/0 ) was monitored. The 6
vertical stiffness of the isolator at lateral displacement (, 01 , was calculated by dividing the
difference between the maximum and minimum values of the vertical force (,234 and ,256 , respectively) by the difference between the maximum and minimum values of the vertical displacement (/0,234 and /0,256 , respectively) 01 =
,234 − ,256 (9) /0,234 − /0,256
The notation 0 used in the following section denotes the vertical stiffness of the isolator at zero lateral displacement (i.e., under pure compression). It is noted that although slipping is possible in unbonded elastomeric bearings [32–34], in the FEA the contact between the elastomer is modelled with a high friction coefficient, as the effect of slip is beyond the scope of this study.
3. Results and Discussion 3.1 Vertical Stiffness under Pure Compression Table 2 shows the values of the vertical stiffness of the isolators under pure compression load, obtained from FEA results. The results are compared against two closed-form solutions for FREIs, the Pressure Solution (PS) presented by Kelly and Calabrese [15], and the Pressure Approach (PA) by Tsai [35]. The effective compression modulus ( 7 ) of an infinitely long strip elastomeric pad perfectly bonded to flexible reinforcement layers using the PS method is [15]: 7 =
8 tanh? :1 − @ (10) 8+9 ?
7
and 8, 9 and ? are defined in Eqs. (11), (12) and (13), respectively. 8=
9=
12 (11)
12* (12) *
? = 8 + 9 (13)
where * is the thickness of the elastomer pad, and and * are the elastic modulus and thickness of the reinforcement layers, respectively. With the assumption that all elastomer layers have the
same compression modulus (neglecting the effect of boundary conditions), the vertical stiffness of an isolator with A layers of elastomer can be calculated by 0 =
7 B (14) A*
Tsai [35] derived the effective compression modulus of the Cth elastomer layer 7
(D)
in a
FREI bonded to rigid end plates using the PA method: 7 = 2E + F1 − (D)
? − I tanh?D ?D − I tanh?D GH D KL (15) J + H J + 2E ?D ?D ?D − ?D ?D − ?D
where E and are Lame’s constants and
1 ) K (16) ?D = GI + ID + ID − M(I + ID + ID − 4I ID 2 1 ) K (17) ?D = GI + ID + ID + M(I + ID + ID − 4I ID 2 1 12E I = N (18) * + 2E 8
12EO1 − P I = N (19) * * ID =
I (−QDR + 2QD − QDS ) (20) 12QD
ID =
I (QD + QDS ) (21) 2QD
C C QD = 4 :1 − @ (22) A A The total vertical stiffness of the isolator can be calculated using 0 =
B
* ∑UDV
1 (D) 7
(23)
It can be observed from Table 2 that the predictions of the PA assuming rigid end condition are in better agreement with the FEA results. As expected, the error between the predictions of the analytical solutions and the FEA results reduces as the shape factor of the isolator increases. These findings are in agreement with those in Osgooei et al. [16].
3.2 Vertical Stiffness under Lateral Displacement Table 3 shows the FEA results of the vertical stiffness of each isolator in both bonded and unbonded applications obtained at different lateral displacement amplitudes. The vertical stiffness values, 01 , are normalized with respect 0 . For both S11 and S22 and up to (/*+ = 1.50, the reduction in the vertical stiffness of the isolator is greater in the unbonded application. At (/*+ = 1.75 however, an increase in the vertical stiffness of unbonded FREI is observed,
9
which is due to the contact of the middle elastomer layers with the loading surfaces. Figure 4 compares the deformed shape of S11 at (/*+ = 1.75 in bonded (left) and unbonded (right) applications. For the unbonded isolator, three of the mid-elastomer layers are in contact with the bottom loading surface and thus participate in the vertical load bearing mechanism of the isolator. Further increase in the vertical stiffness of the isolator can be expected at larger lateral displacement amplitudes. It can be observed from the values in Table 3 that for the bonded FREI, the change in the vertical stiffness at different displacement amplitudes is almost unaffected by the shape factor of the isolator. Figure 5 plots the normalized vertical stiffness of the isolators at the displacement amplitudes considered in the analyses. For the bonded isolators, the reduction in the vertical stiffness varies almost linearly with lateral displacement. As suggested in [25] for bonded SREIs, the vertical stiffness under lateral displacement can be estimated by considering an effective overlapping area under compression, as shown in Fig. 6, and neglecting the contribution of the two triangle-shaped areas in the vertical stiffness of the isolator. Extending the method to bonded FREIs, the vertical stiffness of a strip isolated under lateral displacement u can be estimated from 01 = 0 #1 −
( % (24) 2
Table 4 compares the values of the vertical stiffness of S11 and S22 in a bonded application, calculated using Eq. (24), as well as the corresponding error with respect to the FEA results. For both isolators, the predictions of Eq. (24) are lower than the FEA results for (/*+ < 1.00, and higher than the FEA results for (/*+ > 1.00, although the maximum difference is less than 12%.
10
Equation (24) predicts that the vertical stiffness of the isolator becomes zero at ( = 2. For bonded isolators, designers consider a limit on the minimum effective area [36]. Assuming a 40% limit on the effective overlapping area and using Eq. (24), the maximum allowable isolation displacement for S11 and S22 will be (/*+ = 1.92 and (/*+ = 2.07, respectively. For unbonded isolators S11 and S22, the vertical stiffness of the isolator reaches a minimum value of 01 ⁄ 0 = 0.47 at (/*+ = 1.50, and increases with an increase in the lateral displacements. This feature of unbonded FREI allows larger lateral displacement amplitudes to be considered for an unbonded isolator, as they have been tested up to (/*+ = 3.00 [37] while maintaining sufficient vertical load bearing capacity.
3.3 Stress Distribution Figures 7 and 8 show the contours of normalized Z (i.e., Z /&̅, where Z is the normal stress in the vertical direction, and &̅ = 2 MPa) in S11 and S22 isolators, respectively. The stress
contours are shown for (/*+ = 0 (i.e. under pure compression), 1.00, and 1.75. As the isolators undergo lateral displacement, stress concentration is observed at the corners of the isolators. For the bonded isolators, an increase in the lateral displacement results in a decrease in the effective overlapping area, thus increasing the peak values of Z in the center of the isolator. Also, for the bonded isolators under lateral displacement, the regions outside the effective overlapping areas experience tensile Z stresses, which develop in order to equilibrate the unbalanced moment that is generated when the isolator is sheared [38]. For unbonded FREI under lateral displacement, however, the regions that experience tensile Z stresses are very small, and the values of these tensile stresses are significantly smaller than for the bonded case. In an unbonded FREI, a larger
11
portion of the isolator experiences compressive stresses, compared to a bonded FREI. As a result, the increase in the peak Z stress values in the center of the unbonded FREI as the isolator is displaced is considerably lower than those in the bonded FREI.
3.4 Vertical Displacement Figures 9 and 10 plot the variations in the normalized downward vertical displacement (/0 /*+ ) of S11 and S22, respectively, under different lateral displacement amplitudes (/*+ . Note that the
value of /0 /*+ at (/*+ = 0 is due to pure compression under an average vertical stress of &̅ = 2 MPa. For the bonded isolators, the downward vertical displacement increases as the applied lateral displacement of the isolator increases. The vertical displacement follows a parabolic trend as a function of lateral displacement. For S11 for example, the vertical displacement at lateral displacements of (/*+ = 1.00 and (/*+ = 1.75 compared with the vertical displacement under pure compression shows 90% and 372% increase, respectively. For the unbonded FREI, the geometry change of the isolator due to the rollover deformation tends to decrease the downward vertical displacement as the isolator undergoes lateral displacement. Thus, the variation in the vertical displacement of isolators under different lateral displacement amplitudes is considerably lower compared to the bonded isolators. For S11 and S22, the maximum variation in the vertical displacement of the isolator at different lateral displacement amplitudes compared to the vertical displacement under pure compression load is 23% and 49%, respectively.
3.5 Influence of Applied Vertical Stress Figure 11 compares the normalized effective stiffness of S11 calculated under average vertical stress values of &̅ = 2, 4 and 6 MPa for bonded [Figure 11(a)] and unbonded [Figure 11(b)] 12
applications. The vertical stiffness values are normalized with respect to the effective stiffness value under pure compression ( 0 ) predicted using the PA method (Table 2). It can be observed that as the applied vertical stress increases, the vertical stiffness of the bearings increases. This stiffening behavior is induced by the geometric nonlinearity caused by larger bulging of elastomer layers as well as the shortening of the elastomer layers as the vertical load increases. Under different vertical compression loads, the same overall trend in the vertical stiffness (reduction in bonded isolators and reduction and then increase in unbonded isolators) is observed as the isolator undergoes lateral displacement. As the lateral displacement increases, the overall reduction in the vertical stiffness of the isolator decreases with an increase in the vertical compression load. At (/*+ = 1.00, for example, the vertical stiffness of the bonded isolator shows 29%, 24% and 16% reduction compared to the vertical stiffness under pure compression load, for &̅ = 2, 4 and 6 MPa, respectively. For unbonded application, the reduction in the vertical stiffness is obtained 39%, 35% and 31%, for &̅ = 2, 4 and 6 MPa, respectively.
4. Conclusions In this paper, FEA is carried out on two strip-shaped FREIs to investigate the variation in the vertical stiffness of the isolators under different lateral displacement amplitudes. Two shape factor values of = 11 (S11) and = 22 (S22) were considered for the isolators and their vertical compression response was studied in both bonded and unbonded applications. Under pure compression load, the FEA results were compared with the values from two analytical solutions: the pressure solution, presented by Kelly and Calabrese [15], and the pressure approach, presented by Tsai [35]. It was found that the predictions of the PA method are in better
13
agreement with the FEA results. The error between the predictions of the analytical solutions and the FEA results decreases as the shape factor of the isolator increases. In general, the FEA results were in good agreement with the results from analytical solutions, with a maximum discrepancy of 12.1% for S11 and 6.1% for S22. FEA results showed that for bonded isolators, the vertical stiffness decreases as the lateral displacement increases. This reduction in the vertical stiffness of bonded FREIs can be predicted using the effective overlapping area method. A similar trend is observed for FREIs in unbonded application up to 150% shear deformation. For (/*+ > 1.50, an increase is observed in the vertical stiffness of unbonded isolators. This increase is due to the contact of mid-elastomer layers with the loading support surfaces due to the rollover, and is beneficial as it allows considering larger isolation displacement values for design. For the bonded isolators, the increase in the lateral displacement results in an increase the peak values of Z in the center of the isolators, and the regions outside the effective overlapping
areas experience appreciable tensile Z stresses. However, for an unbonded FREI under lateral displacement, the tensile Z stresses that develop are very low and confined in a small region.
The increase in peak compressive Z stress in the central region of the unbonded isolator due to increasing lateral displacement is considerably lower than in the bonded isolators. Under a constant vertical load, as the applied lateral displacement increases, the vertical displacement of the bonded isolator increases with a parabolic trend. The variation in the vertical displacement under different lateral displacement amplitudes is considerably lower in the unbonded application relative to the bonded application. This is attributed to the fact that under
14
rollover deformation, the isolator tends to push the loading surfaces against the direction of the applied vertical load. The response of the isolator with the shape factor of = 11 was investigated under three
vertical load values, corresponding to average vertical pressure values of &̅ = 2, 4 and 6 MPa. It was observed that as the applied vertical compression load increases, the vertical stiffness of the isolator in both bonded and unbonded applications increase. As the lateral displacement increases, the overall reduction in the vertical stiffness of the isolator decreases with an increase in the vertical compression load.
The present study used 2D FEA to investigate the variation of the vertical properties of FREIs subjected to lateral loading. The characterization of the vertical-lateral response interaction in unbonded and bonded FREIs of various geometries requires further investigation using 3D FEA and experimental testing.
Acknowledgments This research was carried out as part of the mandate of the Centre for Effective Design of Structures (CEDS) at McMaster University and is partially funded by the Ontario Ministry of Economic Development and Innovation and by the Natural Sciences and Engineering Research Council of Canada (NSERC).
References 1.
Kelly JM, Konstantinidis D. Mechanics of Rubber Bearings for Seismic and Vibration Isolation. Chichester, UK: Wiley; 2011.
15
2.
Kelly JM. Analysis of fiber-reinforced elastomeric isolators. Journal of Seismology and Earthquake Engineering, 1999; 2(1): 19–34.
3.
Kelly JM. Seismic isolation systems for developing countries. Earthquake Spectra 2002; 18(3): 385–406.
4.
Toopchi-Nezhad H, Tait MJ, Drysdale RG. Lateral response evaluation of fiber-reinforced neoprene seismic isolators utilized in an unbonded application. Journal of Structural Engineering, 2008; 134(10): 1627–37.
5.
Toopchi-Nezhad H, Tait MJ, Drysdale RG. Bonded versus unbonded strip fiber reinforced elastomeric isolators: finite element analysis. Composite Structures, 2011; 93(2): 850–9.
6.
Koh CG, Kelly JM. A simple mechanical model for elastomeric bearings used in base isolation. International Journal of Mechanical Sciences, 1988; 30(12): 933–43.
7.
Buckle IG, Nagarajaiah S, Ferrell K. Stability of elastomeric isolation bearings: experimental study. Journal of Structural Engineering, 2002; 128(1): 3–11.
8.
Han X, Warn G. Mechanistic model for simulating critical behavior in elastomeric bearings. Journal of Structural Engineering, 2015; 141(5): 04014140.
9.
Van Engelen NC, Konstantinidis D, Tait MJ. Structural and non-structural performance of a seismically isolated building using stable unbonded fiber-reinforced elastomeric isolators. Earthquake Engineering and Structural Dynamics, 2015 (in press).
10. Tsai HC, Kelly JM. Bending stiffness of fiber-reinforced circular seismic isolators. Journal of Engineering Mechanics, 2002; 128(11): 1150–7. 11. Pinarbasi S, Akyuz U, Mengi Y. A new formulation for the analysis of elastic layers bonded to rigid surfaces. International Journal of Solids Structures, 2006; 43(14): 4271–96.
16
12. Pinarbasi S, Mengi Y. Elastic layers bonded to flexible reinforcements. International Journal of Solids Structures, 2008; 45(3): 794–820. 13. Pinarbasi S, Okay F. Compression of hollow-circular fiber-reinforced rubber bearings. Structural Engineering Mechanics, 2011; 38(3): 361–84. 14. Toopchi-Nezhad H, Tait MJ, Drysdale RG. Influence of thickness of individual elastomer layers (first shape factor) on the response of unbonded fiber-reinforced elastomeric bearings. Journal of Composite Materials, 2013; 47(27): 3433-50. 15. Kelly JM, Calabrese A. Mechanics of Fiber Reinforced Bearings. PEER Report 2012/101, Pacific Earthquake Engineering Research Center, University of California, Berkeley; 2012. 16. Osgooei PM, Tait MJ, Konstantinidis D. Three-dimensional finite element analysis of circular fiber-reinforced elastomeric bearings under compression. Composite Structures, 2014; 108: 191–204. 17. Van Engelen NC, Osgooei PM, Tait MJ, Konstantinidis D. Experimental and finite element study on the compression properties of modified rectangular fiber-reinforced elastomeric isolators (MR-FREIs). Engineering Structures, 2014; 74: 52–64. 18. Al-Anany YM, Tait MJ. A numerical study on the compressive and rotational behaviour of fiber reinforced elastomeric isolators (FREI). Composite Structures, 2015 (in press, doi:10.1016/j.compstruct.2015.07.042). 19. Kelly JM, Takhirov SM. Analytical and Experimental Study of Fiber-Reinforced Elastomeric Isolators. PEER Report 2001/11, Pacific Earthquake Engineering Research Center, University of California, Berkeley; 2001.
17
20. Moon BY, Kang GJ, Kang BS, Kelly JM. Design and manufacturing of fiber-reinforced elastomeric isolator for seismic isolation. Journal of Material Processing and Technology, 2002; 130–131:145–50. 21. Toopchi‐Nezhad H, Tait MJ, Drysdale RG. Testing and modeling of square carbon fiber‐reinforced elastomeric seismic isolators. Structural Control and Health Monitoring, 2008; 15(6): 876-900. 22. Russo G, Pauletta M, Cortesia A, Dal Binanco A. Experimental behavior of carbon fiber reinforced isolators. In: Proc. 2008 Seismic Engineering International Conference Commemorating the 1908 Messina and Reggio Calabria Earthquake, Reggio Calabria, Italy; 2008. 23. Hedayati Dezfuli F, Shahria Alam M. Sensitivity analysis of carbon fiber-reinforced elastomeric isolators based on experimental tests and finite element simulations. Bulletin of Earthquake Engineering, 2014; 12(2): 1025–43. 24. MSC Marc. Theory and user information, vol. A. Santa Ana: MSC Software Corporation; 2013. 25. Warn GP, Whittaker AS, Constantinou MC. Vertical stiffness of elastomeric and lead– rubber seismic isolation bearings. Journal of Structural Engineering, 2007; 133(9): 1227– 36. 26. Mordini A, Strauss A. An innovative earthquake isolation system using fibre reinforced rubber bearings. Engineering Structures, 2008; 30(10): 2739–51. 27. Kelly JM, Marsico MR. Stability and post-buckling behavior in nonbolted elastomeric isolators. Seismic Isolation and Protection Systems, 2010; 1(1): 41–54.
18
28. Gerhaher U, Strauss A, Bergmeister K. Numerical modeling of elastomeric bearings in structural engineering. Advances in Materials Science, 2011; 11(3): 51–63. 29. Osgooei PM, Tait MJ, Konstantinidis D. Finite element analysis of unbonded square fiberreinforced elastomeric isolators (FREIs) under lateral loading in different directions. Composite Structures, 2014; 113: 164–73. 30. Osgooei PM, Van Engelen NC, Konstantinidis D, Tait MJ. Experimental and finite element study on the lateral response of modified rectangular fiber-reinforced elastomeric isolators (MR-FREIs). Engineering Structures, 2015; 85: 293–303. 31. Bonet J, Wood RD. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge, UK: Cambridge University Press; 1997. 32. Kelly JM, Konstantinidis D Effect of friction on unbonded elastomeric bearings. Journal of Engineering Mechanics, 2009; 135(9): 953–60. 33. Russo G, Pauletta M. Sliding instability of fiber-reinforced elastomeric isolators in unbonded applications. Engineering Structures, 2013; 48: 70–80. 34. Van Engelen NC, Osgooei PM, Tait, MJ, Konstantinidis D. Partially bonded fiber‐reinforced elastomeric isolators (PB‐FREIs). Structural Control and Health Monitoring, 2015; 22(3): 417–32. 35. Tsai HC. Compression stiffness of infinite-strip bearings of laminated elastic material interleaving with flexible reinforcements. International Journal of Solids and Structures, 2004; 41: 6647–60. 36. Christopoulos C, Filiatrault A. Principles of Passive Supplemental Damping and Seismic Isolation. Pavia, Italy; IUSS Press; 2006.
19
37. Tait MJ, de Raaf MGP, Toopchi-Nezhad H. Stability of fiber-reinforced elastomeric bearings in an unbonded application. Journal of Composite Materials, 2011; 45(18): 1873– 84. 38. Konstantinidis D, Kelly JM, Makris N. Experimental Investigation on the Seismic Response of Bridge Bearings. Report No. EERC-2008/02. Earthquake Engineering Research Center, University of California, Berkeley, CA; 2008.
20
Table 1. Characteristics of the isolators considered in this study Bearing
b (mm)
S
t (mm)
tf (mm)
n
tr (mm)
h (mm)
S11
300
11.2
13.4
1
7
93.8
99.8
Aspect ratio 3.0
S22
300
22.4
6.7
1
13
87.1
99.1
3.0
Table 2. Vertical stiffness of isolators under pure compression, 0 (N/mm) FEA
PS
Difference
PA
Difference
S11
936
822
12.1
877
6.3
S22
2,280
2,140
6.1
2,254
1.1
Table 3. Normalized vertical stiffness of isolators, 01 / 0 S11
S22
(/*+
Bonded
Unbonded
Bonded
Unbonded
0.25
0.98
0.90
0.97
0.94
0.50
0.92
0.83
0.90
0.84
0.75
0.82
0.69
0.80
0.77
1.00
0.71
0.61
0.71
0.66
1.25
0.59
0.54
0.61
0.57
1.50
0.52
0.47
0.52
0.47
1.75
0.42
0.56
0.44
0.69
21
Table 4. Predicted normalized values for vertical stiffness of bonded FREIs using the effective overlapping area, compared with FEA results. S11
(/*+
01 / 0
S22
Difference with FEA (%)
01 / 0
Difference with FEA (%)
0.25
0.92
- 5.9
0.93
- 4.4
0.50
0.84
- 8.3
0.85
- 5.0
0.75
0.77
- 6.6
0.78
- 2.2
1.00
0.69
- 3.2
0.71
0.0
1.25
0.61
3.2
0.64
4.4
1.50
0.53
2.1
0.56
8.6
1.75
0.45
7.8
0.49
11.8
22
(a) (b) Fig. 1. Deformed shape of (a) bonded and (b) unbonded FREI.
Lateral Force
Bonded Application Applicaon
Unonded Applicaon Unbonded Application
Lateral Displacement Fig. 2. Lateral load-displacement relationships of bonded and unbonded FREI.
Fig. 3. Finite element model of S11.
23
(a)
(b)
Fig. 4. Deformed shapes of (a) bonded and (b) unbonded S11 at (/*+ = 1.75.
1.2
_]^(∕ _]
1.0 0.8 0.6
S11-Bonded S22-Bonded
0.4
S11-Unbonded
0.2
S22-Unbonded
0.0 0.0
0.5
1.0
1.5
(∕*_`
2.0
Fig. 5. Normalized effective vertical stiffness of isolators at different lateral displacement amplitudes.
2b-u
u
effective area
h
2b
Fig. 6. The effective overlapping area of bonded isolators.
24
u / t r = 1.00
u / t r = 1.75
Unbonded
Bonded
u / tr = 0
-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
-2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
-3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
Fig. 7. Contours of normalized Z (i.e., Z /&̅) stress distribution in S11 at (/*+ = 0, 1.00 and 1.75.
u / t r = 1.00
u / t r = 1.75
Unbonded
Bonded
u / tr = 0
-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
-2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
-3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
Fig. 8. Contours of normalized Z (i.e., Z /&̅) stress distribution in S22 at (/*+ = 0, 1.00 and 1.75.
25
0.04
0.04
0.03
0.03
0.02
〖/" " 〗_ ]/*_`
〖/" " 〗_ ]/*_`
0.01
0.01
0.02
0.00
0.00 0.0
0.5
(/*_`
1.0
1.5
2.0
0.0
0.5
(a)
(/*_`
1.0
1.5
2.0
(b)
Fig. 9. Variation of downward vertical displacement of S11 in (a) bonded and (b) unbonded application at different lateral displacement amplitudes.
0.010
0.010
〖/" " 〗_ ]/*_`
0.015
〖/" " 〗_ ]/*_`
0.015
0.005
0.005
0.000
0.000 0.0
0.5
(/*_`
1.0
1.5
2.0
0.0
(a)
0.5
(/*_`
1.0
1.5
2.0
(b)
Fig. 10. Variation of downward vertical displacement of S22 in (a) bonded and (b) unbonded application at different lateral displacement amplitudes.
26
1.4
1.2
1.2
1.0
1.0
〖 " " 〗_]/ _]^(( =0)
〖 " " 〗_]/ _]^(( =0)
1.4
0.8
0.8
0.6
0.6
2 MPa 4 MPa 6 MPa
0.4 0.2
0.4
4 MPa
0.2
0.0
2 MPa 6 MPa
0.0 0.0
0.5
1.0
(/*_`
1.5
2.0
0.0
(a)
0.5
1.0
(/*_`
1.5
2.0
(b)
Fig. 11. Variation of normalized effective stiffness of S11 in (a) bonded and (b) unbonded application under different average compression stress values.
27
S0263-8223(16)00112-4 http://dx.doi.org/10.1016/j.compstruct.2016.01.089 COST 7195
To appear in:
Composite Structures
Please cite this article as: Osgooei, P.M., Konstantinidis, D., Tait, M.J., Variation of the Vertical Stiffness of StripShaped Fiber-Reinforced Elastomeric Isolators under Lateral Loading, Composite Structures (2016), doi: http:// dx.doi.org/10.1016/j.compstruct.2016.01.089
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Variation of the Vertical Stiffness of Strip-Shaped Fiber-Reinforced Elastomeric Isolators under Lateral Loading Peyman M. Osgooei1, Dimitrios Konstantinidis2, Michael J. Tait3
Abstract A fiber-reinforced elastomeric isolator (FREI) is a relatively new type of isolator that utilizes fiber material for the reinforcing layers. FREIs can be installed in a bonded or unbonded application. In this study, finite element analysis (FEA) is carried out on bonded and unbonded strip-shaped FREIs to investigate the variation in vertical stiffness as the isolator undergoes lateral displacement. The vertical stiffness of the isolators under pure compression obtained by FEA was in good agreement with the predictions of two available closed-form solutions. As the lateral displacement increases, it was observed that for bonded FREIs the vertical stiffness decreases monotonically; whereas, for unbonded FREIs, the vertical stiffness decreased up to 175% shear deformation, where an increase in vertical stiffness was observed. FEA results confirmed that the effective overlap area method provides reasonable estimates for the vertical stiffness of bonded FREIs. It is observed that as the applied vertical stress increases, the vertical stiffness of bonded and unbonded FREIs increases. Finally, the paper shows that under large lateral displacements, bonded FREIs develop large tensile stresses in the regions outside the overlapping areas, while the tensile stresses that develop in unbonded FREIs are very low and confined in a small region. Keywords: fiber-reinforced elastomeric isolators; laminated rubber bearings; seismic isolation; combined lateral and vertical loading; vertical response; finite element analysis 1
PhD Candidate, Dept. of Civil Engineering, McMaster University Assistant Professor, Dept. of Civil Engineering, McMaster University, 1280 Main St W, Hamilton, Ontario L8S 4L8, Canada; E-mail: [email protected] (corresponding author) 3 Professor, Dept. of Civil Engineering, McMaster University 2
1
1. Introduction A steel-reinforced elastomeric isolator (SREI) is the most common type of seismic isolator in use. This type of isolator is composed of layers of elastomer interleaved with steel reinforcement shims. Thick steel end plates are typically bonded to the top and bottom surfaces of the bearing for connecting purposes. The application of SREIs in North America is limited to high importance structures (e.g., historical buildings, emergency centers), or buildings with sensitive or valuable contents [1]. Currently it is not economically feasible to employ SREIs for residential or ordinary importance structures. Kelly [2, 3] suggested that by reducing the costs and weight of SREIs, the application of elastomeric bearings (and in a more general case, seismic isolation) can be extended to housing and commercial buildings. Replacing the steel reinforcing shims with a lighter material (for example, fiber reinforced polymer) can reduce the weight of SREIs. Unlike SREIs, which require a hot vulcanization process, a cold vulcanization process can be utilized to construct fiber-reinforced elastomeric isolators (FREIs) [4], which could potentially reduce the high manufacturing costs associated with SREIs. In addition, FREIs can be manufactured in large pads where individual isolators can be cut to the desired size, leading to further potential reductions in cost and turnaround time. FREIs can be installed in both bonded (Fig. 1-a) and unbonded (Fig. 1-b) applications. Unbonded FREIs do not have end plates. Due to the flexibility of the fiber reinforcing layers, unbonded FREIs undergo a unique rollover deformation (Fig. 1-b). This rollover has been shown to increase the seismic isolating efficiency by reducing the effective lateral stiffness of the isolators [5]. In addition, the unbonded application results in a reduced stress demand on the
2
elastomer and fiber reinforcement layers [5], which makes it possible to use a simpler manufacturing process for unbonded FREIs. Figure 2 compares the lateral load-displacement relationship of bonded and unbonded FREIs. While the effective (secant) stiffness of the bonded FREI is unaffected by the change in the displacement, the effective stiffness of the isolator in an unbonded application varies with lateral displacement. Note that in this discussion, geometric effects that can reduce the effective lateral stiffness of a bonded isolator [6–8], especially for slender configurations, are not considered. Under small displacements, the effective stiffness of the unbonded FREI is constant. As the displacement increases, due to the rollover effect, the effective stiffness of the unbonded FREI decreases, until a point where an increase in the stiffness is observed. This increase in the effective stiffness in unbonded FREIs is due to the contact of the originally vertical faces of the isolator with the loading surfaces and can limit the isolation displacement under large (i.e., maximum-considered-earthquake) ground motions [4, 9] . The vertical response of FREIs has been investigated in a number of analytical [10–13] and numerical [14–18] studies. Furthermore, experimental studies involving vertical compression and lateral cyclic tests have shown that FREIs achieve both adequate vertical stiffness and the required lateral flexibility to render them suitable for seismic isolation [2–4, 19–23]; However, in these studies, FREIs were investigated under pure compression, and the effect of lateral displacement on the compression response characteristics of the isolators was neglected. As such, the variation in the vertical stiffness of unbonded FREIs subjected to lateral loading requires investigation.
3
In this paper, finite element analysis (FEA) is employed to study the vertical response of FREIs, in both bonded and unbonded applications, under different lateral displacement amplitudes. Two strip-shaped isolators with the same overall dimensions but different shape factors, (defined for a single elastomer layer as the ratio of the total loaded area to the load-free area), are modeled and analyzed using MSC Marc [24], a general-purpose finite element program. First, results from the FEA under pure compression are compared against two available closed-form solutions. Then, the vertical stiffness of FREIs is characterized at various levels of lateral displacement and under different values of average vertical stress. The accuracy of the effective overlap area method [25], which is widely used for estimating the vertical stiffness of bonded SREIs under lateral displacement, is evaluated for bonded FREIs by comparing its predictions against FEA results. The paper examines how the downward vertical displacement of the top support of a FREI varies with increasing lateral displacement. Finally, the stress distribution in the overlapping and non-overlapping regions of bonded and unbonded FREIs is investigated.
2. Finite Element Analysis Two strip-shaped isolators are considered in this study. Table 1 shows the dimensions of the isolators. Both isolators had a width of = 300 mm and a width-to-height aspect ratio of 3.0. A shear modulus of = 0.8 MPa and a bulk modulus of = 2000 MPa, corresponding to an
elastic modulus of = 2.4 MPa and a Poisson’s ratio of = 0.4998, were used for the elastomer. The elastic modulus and the Poisson’s ratio of the fiber reinforcement material were selected as = 20 GPa and = 0.2, respectively. 4
FEA of FREIs is challenging due to near-incompressibility of the elastomer material, large deformations, large strains, and complex contact. In the literature, a number of FEA studies have been carried out on FREIs subjected to lateral loading [26, 5, 27, 28, 15, 23, 29, 30]. In this study, 2D FEA were carried out using MSC Marc (2013) [24]. As seen in Osgooei et al. [16], the number of elements along the height of the isolator influences modeling error. In this study, a total of 84 and 104 four-node linear plane-strain elements were used along the height of isolators S11 and S22, respectively. In order to consider the interaction of the originally vertical faces of the unbonded FREI with the loading surfaces, a touching contact was defined between the mid-elastomer layers and the loading surfaces. In this contact model, the nodal points of the elastomer layers were allowed to detach from the loading supports when the tensile stress values were reached. Under static loading conditions, an isotropic hyperelastic material model can be used for rubbers. For an isotropic hyperelastic model, the strain energy function, , is expressed in terms of the three invariants of the Cauchy-Green deformation tensor (either the left, B, or the right, C), defined by [31] = + + (1)
= + + (2) = (3)
where , , and are the eigenvalues of B or C, also known as the principal stretches. For an incompressible material, the third invariant is equal to 1 and the strain energy function can be expressed by [31]
= ( , ) (4) 5
In this study, the neo-Hookean constitutive model is used for the strain energy function, defined by [31] = ( − 3) (5)
where is the material constant and is related to the shear modulus of the elastomer by =
(6) 2
Using the strain energy function, the stress-strain relationship for each deformation state can be calculated. For example, assuming a uniaxial tension deformation state where = and assuming an incompressible material ( = 1), we have = =
1
√
(7)
Using the neo-Hookean strain energy density function, the strain energy density function becomes = # + $ − 3% (8)
The fiber-reinforcement layers were modeled using a linear-elastic isotropic material model. Quadrilateral plane-strain elements were used to model the elastomer and fiber-reinforcement layers. The top and bottom loading surfaces were modeled using rigid wire elements. Figure 3 shows the FEA model of the isolator with = 11. The FREI models were subjected to an average vertical stress (i.e., the vertical load divided by the loaded area) of &̅ = 2 MPa and were subsequently loaded to lateral displacements of (/*+ = 0.25, 0.50, 0.75, 1.00, 1.25, 1.50 and 1.75. At each displacement amplitude, the applied
lateral displacement was held constant, while the vertical load (,) on the isolator was subjected to a ±20% change, and the variation in the vertical displacement (/0 ) was monitored. The 6
vertical stiffness of the isolator at lateral displacement (, 01 , was calculated by dividing the
difference between the maximum and minimum values of the vertical force (,234 and ,256 , respectively) by the difference between the maximum and minimum values of the vertical displacement (/0,234 and /0,256 , respectively) 01 =
,234 − ,256 (9) /0,234 − /0,256
The notation 0 used in the following section denotes the vertical stiffness of the isolator at zero lateral displacement (i.e., under pure compression). It is noted that although slipping is possible in unbonded elastomeric bearings [32–34], in the FEA the contact between the elastomer is modelled with a high friction coefficient, as the effect of slip is beyond the scope of this study.
3. Results and Discussion 3.1 Vertical Stiffness under Pure Compression Table 2 shows the values of the vertical stiffness of the isolators under pure compression load, obtained from FEA results. The results are compared against two closed-form solutions for FREIs, the Pressure Solution (PS) presented by Kelly and Calabrese [15], and the Pressure Approach (PA) by Tsai [35]. The effective compression modulus ( 7 ) of an infinitely long strip elastomeric pad perfectly bonded to flexible reinforcement layers using the PS method is [15]: 7 =
8 tanh? :1 − @ (10) 8+9 ?
7
and 8, 9 and ? are defined in Eqs. (11), (12) and (13), respectively. 8=
9=
12 (11)
12* (12) *
? = 8 + 9 (13)
where * is the thickness of the elastomer pad, and and * are the elastic modulus and thickness of the reinforcement layers, respectively. With the assumption that all elastomer layers have the
same compression modulus (neglecting the effect of boundary conditions), the vertical stiffness of an isolator with A layers of elastomer can be calculated by 0 =
7 B (14) A*
Tsai [35] derived the effective compression modulus of the Cth elastomer layer 7
(D)
in a
FREI bonded to rigid end plates using the PA method: 7 = 2E + F1 − (D)
? − I tanh?D ?D − I tanh?D GH D KL (15) J + H J + 2E ?D ?D ?D − ?D ?D − ?D
where E and are Lame’s constants and
1 ) K (16) ?D = GI + ID + ID − M(I + ID + ID − 4I ID 2 1 ) K (17) ?D = GI + ID + ID + M(I + ID + ID − 4I ID 2 1 12E I = N (18) * + 2E 8
12EO1 − P I = N (19) * * ID =
I (−QDR + 2QD − QDS ) (20) 12QD
ID =
I (QD + QDS ) (21) 2QD
C C QD = 4 :1 − @ (22) A A The total vertical stiffness of the isolator can be calculated using 0 =
B
* ∑UDV
1 (D) 7
(23)
It can be observed from Table 2 that the predictions of the PA assuming rigid end condition are in better agreement with the FEA results. As expected, the error between the predictions of the analytical solutions and the FEA results reduces as the shape factor of the isolator increases. These findings are in agreement with those in Osgooei et al. [16].
3.2 Vertical Stiffness under Lateral Displacement Table 3 shows the FEA results of the vertical stiffness of each isolator in both bonded and unbonded applications obtained at different lateral displacement amplitudes. The vertical stiffness values, 01 , are normalized with respect 0 . For both S11 and S22 and up to (/*+ = 1.50, the reduction in the vertical stiffness of the isolator is greater in the unbonded application. At (/*+ = 1.75 however, an increase in the vertical stiffness of unbonded FREI is observed,
9
which is due to the contact of the middle elastomer layers with the loading surfaces. Figure 4 compares the deformed shape of S11 at (/*+ = 1.75 in bonded (left) and unbonded (right) applications. For the unbonded isolator, three of the mid-elastomer layers are in contact with the bottom loading surface and thus participate in the vertical load bearing mechanism of the isolator. Further increase in the vertical stiffness of the isolator can be expected at larger lateral displacement amplitudes. It can be observed from the values in Table 3 that for the bonded FREI, the change in the vertical stiffness at different displacement amplitudes is almost unaffected by the shape factor of the isolator. Figure 5 plots the normalized vertical stiffness of the isolators at the displacement amplitudes considered in the analyses. For the bonded isolators, the reduction in the vertical stiffness varies almost linearly with lateral displacement. As suggested in [25] for bonded SREIs, the vertical stiffness under lateral displacement can be estimated by considering an effective overlapping area under compression, as shown in Fig. 6, and neglecting the contribution of the two triangle-shaped areas in the vertical stiffness of the isolator. Extending the method to bonded FREIs, the vertical stiffness of a strip isolated under lateral displacement u can be estimated from 01 = 0 #1 −
( % (24) 2
Table 4 compares the values of the vertical stiffness of S11 and S22 in a bonded application, calculated using Eq. (24), as well as the corresponding error with respect to the FEA results. For both isolators, the predictions of Eq. (24) are lower than the FEA results for (/*+ < 1.00, and higher than the FEA results for (/*+ > 1.00, although the maximum difference is less than 12%.
10
Equation (24) predicts that the vertical stiffness of the isolator becomes zero at ( = 2. For bonded isolators, designers consider a limit on the minimum effective area [36]. Assuming a 40% limit on the effective overlapping area and using Eq. (24), the maximum allowable isolation displacement for S11 and S22 will be (/*+ = 1.92 and (/*+ = 2.07, respectively. For unbonded isolators S11 and S22, the vertical stiffness of the isolator reaches a minimum value of 01 ⁄ 0 = 0.47 at (/*+ = 1.50, and increases with an increase in the lateral displacements. This feature of unbonded FREI allows larger lateral displacement amplitudes to be considered for an unbonded isolator, as they have been tested up to (/*+ = 3.00 [37] while maintaining sufficient vertical load bearing capacity.
3.3 Stress Distribution Figures 7 and 8 show the contours of normalized Z (i.e., Z /&̅, where Z is the normal stress in the vertical direction, and &̅ = 2 MPa) in S11 and S22 isolators, respectively. The stress
contours are shown for (/*+ = 0 (i.e. under pure compression), 1.00, and 1.75. As the isolators undergo lateral displacement, stress concentration is observed at the corners of the isolators. For the bonded isolators, an increase in the lateral displacement results in a decrease in the effective overlapping area, thus increasing the peak values of Z in the center of the isolator. Also, for the bonded isolators under lateral displacement, the regions outside the effective overlapping areas experience tensile Z stresses, which develop in order to equilibrate the unbalanced moment that is generated when the isolator is sheared [38]. For unbonded FREI under lateral displacement, however, the regions that experience tensile Z stresses are very small, and the values of these tensile stresses are significantly smaller than for the bonded case. In an unbonded FREI, a larger
11
portion of the isolator experiences compressive stresses, compared to a bonded FREI. As a result, the increase in the peak Z stress values in the center of the unbonded FREI as the isolator is displaced is considerably lower than those in the bonded FREI.
3.4 Vertical Displacement Figures 9 and 10 plot the variations in the normalized downward vertical displacement (/0 /*+ ) of S11 and S22, respectively, under different lateral displacement amplitudes (/*+ . Note that the
value of /0 /*+ at (/*+ = 0 is due to pure compression under an average vertical stress of &̅ = 2 MPa. For the bonded isolators, the downward vertical displacement increases as the applied lateral displacement of the isolator increases. The vertical displacement follows a parabolic trend as a function of lateral displacement. For S11 for example, the vertical displacement at lateral displacements of (/*+ = 1.00 and (/*+ = 1.75 compared with the vertical displacement under pure compression shows 90% and 372% increase, respectively. For the unbonded FREI, the geometry change of the isolator due to the rollover deformation tends to decrease the downward vertical displacement as the isolator undergoes lateral displacement. Thus, the variation in the vertical displacement of isolators under different lateral displacement amplitudes is considerably lower compared to the bonded isolators. For S11 and S22, the maximum variation in the vertical displacement of the isolator at different lateral displacement amplitudes compared to the vertical displacement under pure compression load is 23% and 49%, respectively.
3.5 Influence of Applied Vertical Stress Figure 11 compares the normalized effective stiffness of S11 calculated under average vertical stress values of &̅ = 2, 4 and 6 MPa for bonded [Figure 11(a)] and unbonded [Figure 11(b)] 12
applications. The vertical stiffness values are normalized with respect to the effective stiffness value under pure compression ( 0 ) predicted using the PA method (Table 2). It can be observed that as the applied vertical stress increases, the vertical stiffness of the bearings increases. This stiffening behavior is induced by the geometric nonlinearity caused by larger bulging of elastomer layers as well as the shortening of the elastomer layers as the vertical load increases. Under different vertical compression loads, the same overall trend in the vertical stiffness (reduction in bonded isolators and reduction and then increase in unbonded isolators) is observed as the isolator undergoes lateral displacement. As the lateral displacement increases, the overall reduction in the vertical stiffness of the isolator decreases with an increase in the vertical compression load. At (/*+ = 1.00, for example, the vertical stiffness of the bonded isolator shows 29%, 24% and 16% reduction compared to the vertical stiffness under pure compression load, for &̅ = 2, 4 and 6 MPa, respectively. For unbonded application, the reduction in the vertical stiffness is obtained 39%, 35% and 31%, for &̅ = 2, 4 and 6 MPa, respectively.
4. Conclusions In this paper, FEA is carried out on two strip-shaped FREIs to investigate the variation in the vertical stiffness of the isolators under different lateral displacement amplitudes. Two shape factor values of = 11 (S11) and = 22 (S22) were considered for the isolators and their vertical compression response was studied in both bonded and unbonded applications. Under pure compression load, the FEA results were compared with the values from two analytical solutions: the pressure solution, presented by Kelly and Calabrese [15], and the pressure approach, presented by Tsai [35]. It was found that the predictions of the PA method are in better
13
agreement with the FEA results. The error between the predictions of the analytical solutions and the FEA results decreases as the shape factor of the isolator increases. In general, the FEA results were in good agreement with the results from analytical solutions, with a maximum discrepancy of 12.1% for S11 and 6.1% for S22. FEA results showed that for bonded isolators, the vertical stiffness decreases as the lateral displacement increases. This reduction in the vertical stiffness of bonded FREIs can be predicted using the effective overlapping area method. A similar trend is observed for FREIs in unbonded application up to 150% shear deformation. For (/*+ > 1.50, an increase is observed in the vertical stiffness of unbonded isolators. This increase is due to the contact of mid-elastomer layers with the loading support surfaces due to the rollover, and is beneficial as it allows considering larger isolation displacement values for design. For the bonded isolators, the increase in the lateral displacement results in an increase the peak values of Z in the center of the isolators, and the regions outside the effective overlapping
areas experience appreciable tensile Z stresses. However, for an unbonded FREI under lateral displacement, the tensile Z stresses that develop are very low and confined in a small region.
The increase in peak compressive Z stress in the central region of the unbonded isolator due to increasing lateral displacement is considerably lower than in the bonded isolators. Under a constant vertical load, as the applied lateral displacement increases, the vertical displacement of the bonded isolator increases with a parabolic trend. The variation in the vertical displacement under different lateral displacement amplitudes is considerably lower in the unbonded application relative to the bonded application. This is attributed to the fact that under
14
rollover deformation, the isolator tends to push the loading surfaces against the direction of the applied vertical load. The response of the isolator with the shape factor of = 11 was investigated under three
vertical load values, corresponding to average vertical pressure values of &̅ = 2, 4 and 6 MPa. It was observed that as the applied vertical compression load increases, the vertical stiffness of the isolator in both bonded and unbonded applications increase. As the lateral displacement increases, the overall reduction in the vertical stiffness of the isolator decreases with an increase in the vertical compression load.
The present study used 2D FEA to investigate the variation of the vertical properties of FREIs subjected to lateral loading. The characterization of the vertical-lateral response interaction in unbonded and bonded FREIs of various geometries requires further investigation using 3D FEA and experimental testing.
Acknowledgments This research was carried out as part of the mandate of the Centre for Effective Design of Structures (CEDS) at McMaster University and is partially funded by the Ontario Ministry of Economic Development and Innovation and by the Natural Sciences and Engineering Research Council of Canada (NSERC).
References 1.
Kelly JM, Konstantinidis D. Mechanics of Rubber Bearings for Seismic and Vibration Isolation. Chichester, UK: Wiley; 2011.
15
2.
Kelly JM. Analysis of fiber-reinforced elastomeric isolators. Journal of Seismology and Earthquake Engineering, 1999; 2(1): 19–34.
3.
Kelly JM. Seismic isolation systems for developing countries. Earthquake Spectra 2002; 18(3): 385–406.
4.
Toopchi-Nezhad H, Tait MJ, Drysdale RG. Lateral response evaluation of fiber-reinforced neoprene seismic isolators utilized in an unbonded application. Journal of Structural Engineering, 2008; 134(10): 1627–37.
5.
Toopchi-Nezhad H, Tait MJ, Drysdale RG. Bonded versus unbonded strip fiber reinforced elastomeric isolators: finite element analysis. Composite Structures, 2011; 93(2): 850–9.
6.
Koh CG, Kelly JM. A simple mechanical model for elastomeric bearings used in base isolation. International Journal of Mechanical Sciences, 1988; 30(12): 933–43.
7.
Buckle IG, Nagarajaiah S, Ferrell K. Stability of elastomeric isolation bearings: experimental study. Journal of Structural Engineering, 2002; 128(1): 3–11.
8.
Han X, Warn G. Mechanistic model for simulating critical behavior in elastomeric bearings. Journal of Structural Engineering, 2015; 141(5): 04014140.
9.
Van Engelen NC, Konstantinidis D, Tait MJ. Structural and non-structural performance of a seismically isolated building using stable unbonded fiber-reinforced elastomeric isolators. Earthquake Engineering and Structural Dynamics, 2015 (in press).
10. Tsai HC, Kelly JM. Bending stiffness of fiber-reinforced circular seismic isolators. Journal of Engineering Mechanics, 2002; 128(11): 1150–7. 11. Pinarbasi S, Akyuz U, Mengi Y. A new formulation for the analysis of elastic layers bonded to rigid surfaces. International Journal of Solids Structures, 2006; 43(14): 4271–96.
16
12. Pinarbasi S, Mengi Y. Elastic layers bonded to flexible reinforcements. International Journal of Solids Structures, 2008; 45(3): 794–820. 13. Pinarbasi S, Okay F. Compression of hollow-circular fiber-reinforced rubber bearings. Structural Engineering Mechanics, 2011; 38(3): 361–84. 14. Toopchi-Nezhad H, Tait MJ, Drysdale RG. Influence of thickness of individual elastomer layers (first shape factor) on the response of unbonded fiber-reinforced elastomeric bearings. Journal of Composite Materials, 2013; 47(27): 3433-50. 15. Kelly JM, Calabrese A. Mechanics of Fiber Reinforced Bearings. PEER Report 2012/101, Pacific Earthquake Engineering Research Center, University of California, Berkeley; 2012. 16. Osgooei PM, Tait MJ, Konstantinidis D. Three-dimensional finite element analysis of circular fiber-reinforced elastomeric bearings under compression. Composite Structures, 2014; 108: 191–204. 17. Van Engelen NC, Osgooei PM, Tait MJ, Konstantinidis D. Experimental and finite element study on the compression properties of modified rectangular fiber-reinforced elastomeric isolators (MR-FREIs). Engineering Structures, 2014; 74: 52–64. 18. Al-Anany YM, Tait MJ. A numerical study on the compressive and rotational behaviour of fiber reinforced elastomeric isolators (FREI). Composite Structures, 2015 (in press, doi:10.1016/j.compstruct.2015.07.042). 19. Kelly JM, Takhirov SM. Analytical and Experimental Study of Fiber-Reinforced Elastomeric Isolators. PEER Report 2001/11, Pacific Earthquake Engineering Research Center, University of California, Berkeley; 2001.
17
20. Moon BY, Kang GJ, Kang BS, Kelly JM. Design and manufacturing of fiber-reinforced elastomeric isolator for seismic isolation. Journal of Material Processing and Technology, 2002; 130–131:145–50. 21. Toopchi‐Nezhad H, Tait MJ, Drysdale RG. Testing and modeling of square carbon fiber‐reinforced elastomeric seismic isolators. Structural Control and Health Monitoring, 2008; 15(6): 876-900. 22. Russo G, Pauletta M, Cortesia A, Dal Binanco A. Experimental behavior of carbon fiber reinforced isolators. In: Proc. 2008 Seismic Engineering International Conference Commemorating the 1908 Messina and Reggio Calabria Earthquake, Reggio Calabria, Italy; 2008. 23. Hedayati Dezfuli F, Shahria Alam M. Sensitivity analysis of carbon fiber-reinforced elastomeric isolators based on experimental tests and finite element simulations. Bulletin of Earthquake Engineering, 2014; 12(2): 1025–43. 24. MSC Marc. Theory and user information, vol. A. Santa Ana: MSC Software Corporation; 2013. 25. Warn GP, Whittaker AS, Constantinou MC. Vertical stiffness of elastomeric and lead– rubber seismic isolation bearings. Journal of Structural Engineering, 2007; 133(9): 1227– 36. 26. Mordini A, Strauss A. An innovative earthquake isolation system using fibre reinforced rubber bearings. Engineering Structures, 2008; 30(10): 2739–51. 27. Kelly JM, Marsico MR. Stability and post-buckling behavior in nonbolted elastomeric isolators. Seismic Isolation and Protection Systems, 2010; 1(1): 41–54.
18
28. Gerhaher U, Strauss A, Bergmeister K. Numerical modeling of elastomeric bearings in structural engineering. Advances in Materials Science, 2011; 11(3): 51–63. 29. Osgooei PM, Tait MJ, Konstantinidis D. Finite element analysis of unbonded square fiberreinforced elastomeric isolators (FREIs) under lateral loading in different directions. Composite Structures, 2014; 113: 164–73. 30. Osgooei PM, Van Engelen NC, Konstantinidis D, Tait MJ. Experimental and finite element study on the lateral response of modified rectangular fiber-reinforced elastomeric isolators (MR-FREIs). Engineering Structures, 2015; 85: 293–303. 31. Bonet J, Wood RD. Nonlinear Continuum Mechanics for Finite Element Analysis. Cambridge, UK: Cambridge University Press; 1997. 32. Kelly JM, Konstantinidis D Effect of friction on unbonded elastomeric bearings. Journal of Engineering Mechanics, 2009; 135(9): 953–60. 33. Russo G, Pauletta M. Sliding instability of fiber-reinforced elastomeric isolators in unbonded applications. Engineering Structures, 2013; 48: 70–80. 34. Van Engelen NC, Osgooei PM, Tait, MJ, Konstantinidis D. Partially bonded fiber‐reinforced elastomeric isolators (PB‐FREIs). Structural Control and Health Monitoring, 2015; 22(3): 417–32. 35. Tsai HC. Compression stiffness of infinite-strip bearings of laminated elastic material interleaving with flexible reinforcements. International Journal of Solids and Structures, 2004; 41: 6647–60. 36. Christopoulos C, Filiatrault A. Principles of Passive Supplemental Damping and Seismic Isolation. Pavia, Italy; IUSS Press; 2006.
19
37. Tait MJ, de Raaf MGP, Toopchi-Nezhad H. Stability of fiber-reinforced elastomeric bearings in an unbonded application. Journal of Composite Materials, 2011; 45(18): 1873– 84. 38. Konstantinidis D, Kelly JM, Makris N. Experimental Investigation on the Seismic Response of Bridge Bearings. Report No. EERC-2008/02. Earthquake Engineering Research Center, University of California, Berkeley, CA; 2008.
20
Table 1. Characteristics of the isolators considered in this study Bearing
b (mm)
S
t (mm)
tf (mm)
n
tr (mm)
h (mm)
S11
300
11.2
13.4
1
7
93.8
99.8
Aspect ratio 3.0
S22
300
22.4
6.7
1
13
87.1
99.1
3.0
Table 2. Vertical stiffness of isolators under pure compression, 0 (N/mm) FEA
PS
Difference
PA
Difference
S11
936
822
12.1
877
6.3
S22
2,280
2,140
6.1
2,254
1.1
Table 3. Normalized vertical stiffness of isolators, 01 / 0 S11
S22
(/*+
Bonded
Unbonded
Bonded
Unbonded
0.25
0.98
0.90
0.97
0.94
0.50
0.92
0.83
0.90
0.84
0.75
0.82
0.69
0.80
0.77
1.00
0.71
0.61
0.71
0.66
1.25
0.59
0.54
0.61
0.57
1.50
0.52
0.47
0.52
0.47
1.75
0.42
0.56
0.44
0.69
21
Table 4. Predicted normalized values for vertical stiffness of bonded FREIs using the effective overlapping area, compared with FEA results. S11
(/*+
01 / 0
S22
Difference with FEA (%)
01 / 0
Difference with FEA (%)
0.25
0.92
- 5.9
0.93
- 4.4
0.50
0.84
- 8.3
0.85
- 5.0
0.75
0.77
- 6.6
0.78
- 2.2
1.00
0.69
- 3.2
0.71
0.0
1.25
0.61
3.2
0.64
4.4
1.50
0.53
2.1
0.56
8.6
1.75
0.45
7.8
0.49
11.8
22
(a) (b) Fig. 1. Deformed shape of (a) bonded and (b) unbonded FREI.
Lateral Force
Bonded Application Applicaon
Unonded Applicaon Unbonded Application
Lateral Displacement Fig. 2. Lateral load-displacement relationships of bonded and unbonded FREI.
Fig. 3. Finite element model of S11.
23
(a)
(b)
Fig. 4. Deformed shapes of (a) bonded and (b) unbonded S11 at (/*+ = 1.75.
1.2
_]^(∕ _]
1.0 0.8 0.6
S11-Bonded S22-Bonded
0.4
S11-Unbonded
0.2
S22-Unbonded
0.0 0.0
0.5
1.0
1.5
(∕*_`
2.0
Fig. 5. Normalized effective vertical stiffness of isolators at different lateral displacement amplitudes.
2b-u
u
effective area
h
2b
Fig. 6. The effective overlapping area of bonded isolators.
24
u / t r = 1.00
u / t r = 1.75
Unbonded
Bonded
u / tr = 0
-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
-2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
-3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
Fig. 7. Contours of normalized Z (i.e., Z /&̅) stress distribution in S11 at (/*+ = 0, 1.00 and 1.75.
u / t r = 1.00
u / t r = 1.75
Unbonded
Bonded
u / tr = 0
-1.50 -1.25 -1.00 -0.75 -0.50 -0.25 0.00
-2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
-3.00 -2.50 -2.00 -1.50 -1.00 -0.50 0.00 0.50 1.00
Fig. 8. Contours of normalized Z (i.e., Z /&̅) stress distribution in S22 at (/*+ = 0, 1.00 and 1.75.
25
0.04
0.04
0.03
0.03
0.02
〖/" " 〗_ ]/*_`
〖/" " 〗_ ]/*_`
0.01
0.01
0.02
0.00
0.00 0.0
0.5
(/*_`
1.0
1.5
2.0
0.0
0.5
(a)
(/*_`
1.0
1.5
2.0
(b)
Fig. 9. Variation of downward vertical displacement of S11 in (a) bonded and (b) unbonded application at different lateral displacement amplitudes.
0.010
0.010
〖/" " 〗_ ]/*_`
0.015
〖/" " 〗_ ]/*_`
0.015
0.005
0.005
0.000
0.000 0.0
0.5
(/*_`
1.0
1.5
2.0
0.0
(a)
0.5
(/*_`
1.0
1.5
2.0
(b)
Fig. 10. Variation of downward vertical displacement of S22 in (a) bonded and (b) unbonded application at different lateral displacement amplitudes.
26
1.4
1.2
1.2
1.0
1.0
〖 " " 〗_]/ _]^(( =0)
〖 " " 〗_]/ _]^(( =0)
1.4
0.8
0.8
0.6
0.6
2 MPa 4 MPa 6 MPa
0.4 0.2
0.4
4 MPa
0.2
0.0
2 MPa 6 MPa
0.0 0.0
0.5
1.0
(/*_`
1.5
2.0
0.0
(a)
0.5
1.0
(/*_`
1.5
2.0
(b)
Fig. 11. Variation of normalized effective stiffness of S11 in (a) bonded and (b) unbonded application under different average compression stress values.
27