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<ul><li><p>ic</p><p>asparinacegiopradieh, ras ohis</p><p> 2012 Elsevier Ltd. All rights reserved.</p><p>onsistd, to parge dlators wbrationory dretc.) gs accoors, w</p><p>displacements. The critical buckling load of elastomeric seismicisolators is expressed as:</p><p>Pcr;0 2 PE1</p><p>1 4p2 EIeffGAseff L2</p><p>s 1EIeff ErI </p><p>Lte</p><p>4</p><p>where te is the total thickness of the rubber layers and Er is the elas-tic modulus of the rubber bearing evaluated based on the primaryshape factor S1 and rubber Youngs modulus E0 as:</p><p>Er E01 0:742 S21 5The primary shape factor S1 is dened as the ratio between theloaded area of the bearing and the area free to bulge of the singlerubber layer (S1 D/4ti for circular bearings, whereD is the diameter</p><p> Corresponding author. Tel.: +39 0971205054.E-mail addresses: donatello.cardone@unibas.it (D. Cardone), giuseppe.perr@</p><p>Engineering Structures 40 (2012) 198204</p><p>Contents lists available at</p><p>g</p><p>lsealice.it (G. Perrone).reduce their axial load capacity [14].The earliest theoretical approach for the evaluation of the critical</p><p>axial load of rubber bearingswas introduced by Haringx [5], consid-ering themechanical characteristics of helical steel springs and rub-ber rods. Same assumptions have been made later by Gent [6]considering multilayered rubber compression springs. Basically,the Haringxs theory is based on a linear one-dimensional beammodel with shear deformability, within the hypothesis of small</p><p>and bottom connecting steel plates.Various authors proposed different relations to evaluate the</p><p>effective shear and exural rigidity of laminated rubber bearings.In this paper, reference to the formula derived by Buckle and Kelly[1], Koh and Kelly [2] has been made:</p><p>GAseff Gdyn As Lte</p><p>31. Introduction</p><p>An elastomeric isolation bearing cand steel layers mutually vulcanizethe vertical direction together with lizontal direction. The elastomeric isoening the fundamental period of vireducing the seismic effects (interststresses in the structural members,structure. However, this reduction izontal displacements in the isolat0141-0296/$ - see front matter 2012 Elsevier Ltd. Adoi:10.1016/j.engstruct.2012.02.031s of a number of rubberrovide high stiffness ineformability in the hor-ork like a lter length-of the structure, thus</p><p>ifts, oor accelerations,enerated in the super-mpanied by large hori-hich may signicantly</p><p>in which: GAseff and EIeff are the effective shear rigidity and effec-tive exural rigidity, respectively, of the elastomeric isolators, com-puted based on the bending modulus (E) and dynamic shearmodulus (Gdyn) of rubber, moment of inertia of the bearing aboutthe axis of bending (I) and bonded rubber area (As);</p><p>PE is the Euler load for a standard elastic column:</p><p>PE p2 EIeff</p><p>L22</p><p>L is the total height of rubber layers and steel plates excluding topBuckling elastomeric seismic isolators, since they underestimate their critical load capacity at moderate-to-largeshear strain amplitudes.Critical load of slender elastomeric seism</p><p>Donatello Cardone , Giuseppe PerroneDiSGG, University of Basilicata, Via Ateneo Lucano 10, 85100 Potenza, Italy</p><p>a r t i c l e i n f o</p><p>Article history:Received 27 July 2011Revised 31 January 2012Accepted 5 February 2012Available online 28 March 2012</p><p>Keywords:Seismic isolationElastomeric bearingsCritical loadShear strain</p><p>a b s t r a c t</p><p>One of the most importantlarge shear strains. The beincreasing horizontal displcially in high seismicity replacements are a commonon the Haringx theory, moIn this paper the critical b</p><p>different strain amplitudescompared to the predictionThe main conclusion of t</p><p>Engineerin</p><p>journal homepage: www.ell rights reserved.isolators: An experimental perspective</p><p>ects of the seismic response of elastomeric isolators is their stability underg capacity of elastomeric isolators, indeed, progressively degrades whilement. This may greatly inuence the design of elastomeric isolators, espe-ns, where slender elastomeric isolators subjected to large horizontal dis-ctice. In the current design approach the critical load is evaluated baseded to account for large shear strains by approximate correction factors.avior of a pair of slender elastomeric devices is experimentally evaluated atnging from approximately 50% to 150%. The experimental results are thenf a number of semi-empirical and theoretical formulations.study is that current design approaches are overly conservative for slender</p><p>SciVerse ScienceDirect</p><p>Structures</p><p>vier .com/locate /engstruct</p></li><li><p>[12,1,3,4,13]. In this paper, the critical load of a couple of slender(low shape factors) elastomeric bearings is experimentally evalu-ated. Test set-up and experimental program are presented rst indetail. Then, the experimental results are compared with the pre-dictions of the theoretical formulations presented in the previousparagraph.</p><p>2. Experimental tests</p><p>2.1. Test specimens</p><p>Test specimens are a couple of 1:2 scaled circular elastomericbearings with 200 mm diameter and 10 rubber layers with 8 mmthickness. Bearing geometrical properties are summarized inTable 1.</p><p>The mechanical properties of the specimens have been derivedfrom a number of standard cyclic tests, performed in accordancewith the test procedure prescribed in the Italian seismic code[10] for the qualication of elastomeric bearings. The static shearmodulus (Gstat), in particular, has been derived from a quasi-statictest consisting of ve cycles at 0.1 Hz frequency of loading and100% shear strain amplitude. According to the NTC 2008 [10], Gstat</p><p>h</p><p>PF</p><p>(a) </p><p>Table 1Elastomeric bearings details.</p><p>Outer diameter De (mm) 200Inner diameter D (mm) 180Rubber layer thickness ti (mm) 8Number of rubber layers nti 10Steel shim thickness ts (mm) 2Number of steel shims nti 9Total height of rubber t (mm) 80</p><p>ring Structures 40 (2012) 198204 199of the isolator and ti the thickness of a single rubber layer). The rub-ber Youngs modulus E0 is usually taken equal to 3.3 Gdyn to 4 Gdyn.</p><p>The Haringxs theory has been later applied by Naeim and Kelly[7], with a series of simplied assumptions, for commercial elasto-meric seismic isolators. According toNaeimandKelly [7], the criticalbuckling load of elastomeric seismic isolators can be expressed intermsof theprimary and secondary shape factors S1 and S2, the latterbeing dened as the ratio between the maximum dimension of thecross section of the isolator and the total height of rubber. For circu-lar elastomeric isolator, for instance, Naeim and Kelly [7] provides:</p><p>Pcr;0 p2</p><p>2</p><p>p GAseff S1 S2 6</p><p>Subsequently, Kelly [8] derived a more rened formulation of thebuckling load of elastomeric isolator:</p><p>Pcr;0 p2</p><p>3</p><p>p GAseff 0:742 E0</p><p>G</p><p>r S1 S2 7</p><p>The secondary shape factor S2 is dened as the ratio between thebearing maximum dimension and the total thickness of all the rub-ber layers (S2 D/te for circular bearings, where te is the total thick-ness of all the rubber layers). It is interesting to note that the criticalbuckling load capacity evaluated considering the expressions (6)and (7), differs by 1020%, depending on the value (between 3.3Gand 4G) assumed for the rubber Youngs modulus.</p><p>Lanzo [9] modied the Haringxs expression by taking into ac-count the axial stiffness of the rubber bearing (EA)eff:</p><p>Pcr;0 2 PE</p><p>11 4p2 EI eff</p><p>EA eff L2 EI eff</p><p>GAs eff L2</p><p> s 8</p><p>where EAeff is the effective axial stiffness of the rubber bearing,evaluated as:</p><p>EAeff Er A Lte</p><p>9</p><p>In Italy, the current design approach [10] refers to a formulation ofthe critical load similar to (but more conservative than) that ini-tially proposed by Naeim and Kelly [7] (see Eq. (6)):</p><p>Pcr;0 Gdyn As S1 S2 10where Gdyn is the dynamic shear modulus derived from the quali-cation tests of the elastomeric isolator.</p><p>More recently, a less conservative variant of the Naeim andKelly formulation has been adopted in the new European StandardEN11529 [11]:</p><p>Pcr;0 1:3 Gdyn As S1 S2 11For all the above mentioned formulations, the buckling load at thetarget shear displacement (u) is evaluated as a function of the ratiobetween the effective area of the inner shim plate (A) and the over-lap area of the displaced bearing (Ar) (see Fig. 1):</p><p>Pcr Pcr;0 ArA 12</p><p>For circular bearings, for instance, the overlap area at the targetdisplacement is given by:</p><p>Ar u sinu D2</p><p>413</p><p>with</p><p>u 2arccos u </p><p>14</p><p>D. Cardone, G. Perrone / EngineeDSeveral experimental studies of the buckling behavior of elasto-meric seismic isolators have been carried out in the pastu</p><p>Fig. 1. (a) Schematic deformed shape of an elastomeric bearing subjected to shearand compression; (b) effective cross section area as a function of sheardisplacement.u</p><p>P</p><p>Ar</p><p>D(b) e</p><p>Primary shape factor S1 5.63Secondary shape factor S2 2.25</p></li><li><p>ring(a)</p><p>(b)</p><p>200 D. Cardone, G. Perrone / Engineeis dened as the slope of the shearstrain curve between 27% and58% shear deformation with reference to the third cycle of the test(see Fig. 2a). In the case under consideration, a value of Gstat =0.113 MPa has been obtained. The dynamic shear modulus (Gdyn)and the equivalent damping (neq) have been derived from adynamic test consisting of ve cycles at 0.5 Hz frequency of loadingand 100% shear strain amplitude with reference to the third cycleof the test (see Fig. 2b). According to the NTC2008 [10], Gdyn isdened as the secant stiffness to the origin of the axis of the cycleof maximum strain amplitude, neq is evaluated as the ratio betweenthe energy dissipated in one complete cycle (equal to the areaenclosed in the cycle) and the strain energy stored at the maximumstrain amplitude (see Fig. 2b). In the case under consideration, avalue of Gdyn = 0.37 MPa and neq = 16% have been obtained.</p><p>2.2. Test set-up</p><p>The specimens have been tested using the uniaxial bearing testfacility available at the Laboratory of Structures at the University ofBasilicata (see Fig. 3). The test apparatus constitutes a self-balanced system designed for cyclic testing of a pair of identicalcounteracting specimens. The apparatus subjects the test speci-mens to axial loads in the horizontal direction and shear forcesand displacements in the vertical direction. The test rig consistsof two steel stiff beams connected by three steel columns. Two col-umns are linked to the base beam through a double pendulumhinge while the third column (on the left in Fig. 3) by a simplehinge. This has been purposely done to preserve the load directions</p><p>Fig. 2. Hysteresis cycles of the test specimens at 100% shear strain amplitudeduring (a) quasi-static test for the evaluation of Gstat and (b) dynamic test for theevaluation of Gdyn and neq.Structures 40 (2012) 198204and the co-planarity of the bases of the specimens during the test.The hinge connections accommodate the axial displacementresulting from the axial load being applied to the specimens.</p><p>The shear displacement is vertically applied by a Schenkhydraulic actuator, connected to the specimens through a T-shaped steel plate. The actuator is served by two pumps, each with80 l/min maximum capacity, and equipped with two servo-valveswith maximum capacity of 63 l/min each. The Shenck actuatorcan apply 250 kN maximum force and 125 mm maximum dis-placement. The force of the actuator is measured by a load cellmounted on the top of the cylinder. The displacement of the actu-ator is measured and controlled by an internal transducer.</p><p>The compression load is horizontally applied and kept constantduring the tests by an Enerpac double-effect hydraulic jack (seeFig. 3). The hydraulic jack is served by a pump, with 5 l/min capac-ity, and equipped with an electro-valve with 5 l/min maximumcapacity. The hydraulic jack can apply 300 kN maximum force.The force of the hydraulic jack is monitored and measured by anumber of load cells of different capacity mounted on the top ofthe cylinder.</p><p>The shear displacements of the specimens have been measuredby two linear transducers connected to the T-shaped steel plate.The axial displacements of the specimens have been measuredby a couple of linear transducers, positioned between the cornerof the outer plates of the two specimens, in order to evaluate pos-sible rotations of the specimens around the out-of-plane axis.</p><p>Fig. 3. Test set-up.</p></li><li><p>interval of 10 s was allowed to elapse between one step and thenext. The test control has been carried out manually in order toprevent, in real time, possible damages to either specimens or testapparatus.</p><p>A total number of 6 stability test have been performed (seeTable 2), while increasing the target shear displacement from 48to 125 mm, corresponding to six different shear strain amplitude,equal to 60%, 80%, 100%, 120%, 140% and 160%, respectively.</p><p>45</p><p>90</p><p>u (m</p><p>m)</p><p>D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198204 2012.3. Test procedure</p><p>The testing procedure followed in this study is similar, to someextent, to that adopted by Buckle et al. [13]. All the stability tests,in particular, were performed on the same couple of specimens, atThe acquisition frequency has been set equal to 10 Hz. The sig-nals recorded during the tests have been ltered, amplied andconverted from analog signals to 16 bit digital signals in real timeby a proper signal processing software.</p><p>00</p><p>0</p><p>150</p><p>300</p><p>0</p><p>P (K</p><p>N)</p><p>Fig. 4. Steps of a typical stability test (for more details see Table 2).different levels of shear deformations.Basically, each test consisted in the application of two consecu-</p><p>tive ramps of axial load with a ramp of shear displacementbetween them (see Fig. 4). More precisely, each test can be dividedin the following six steps:</p><p>1. Applying an initial axial load of 150 kN, corresponding to adesign compression stress of 6 MPa.</p><p>2. Applying a given target displacement (hence shear deformation).3. Increasing the axial load until the horizontal force became</p><p>negative.4. Decreasing the axial load to the initial value of 150 kN.5. Decreasing the initial displacement to 0.6. Decreasing the axial load to 0.</p><p>Both forces and displacements have been applied very slowlyduring the tests, in order to avoid dynamic effects. Moreover, an</p><p>Table 2Stability test details (see Fig. 4).</p><p>Test no. u1 (mm) c (%) P1 (kN) P2 (kN) Step 1 Step 2</p><p>t1 (s) t2 (s) t3 (s)</p><p>1 48 60 150 280 15 25 352 64 80 150 280 15 25 383 80 100 150 280 15 25 424 96 120 150 280 15 25 455 112 140 150 200 15 25 486 125 160 150 200 15 25 51Step 3 Step 4 Step 5 Step 6</p><p>t4 (s) t5 (s) t6 (s) t7 (s) t8 (s) t9 (s) t10 (s) t11 (s)</p><p>45 58 68 81 91 101 111 12648 61 71 84 94 108 118 13352 65 75 88 98 114 124 13955 68 78 91 101 121 131 1462.4. Test results</p><p>The rst direct result of the critical load tests is the time historyof the shear force F resulting from the application of an increasingaxial load P under a given shear displacement u. As an example,Fig. 5 shows the shear forcetime history derived from Test 3(see Table 2), featuring an imposed shear deformation of 100%.From Fig. 5 it is evident that as the axial force P is increased, theshear force F decreases until it becomes negative, while the sheardisplacement remains constant.</p><p>The experimental outcomes have been employed to evaluatethe critical buckling load for each imposed shear deformation. Itis worth obs...</p></li></ul>

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