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Critical load of slender elastomeric seismic isolators: An experimental perspective
Engineering Structures 40 (2012) 198–204
Contents lists available at SciVerse ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Critical load of slender elastomeric seismic isolators: An experimental perspective Donatello Cardone ⇑, Giuseppe Perrone DiSGG, University of Basilicata, Via Ateneo Lucano 10, 85100 Potenza, Italy
a r t i c l e
i n f o
Article history: Received 27 July 2011 Revised 31 January 2012 Accepted 5 February 2012 Available online 28 March 2012 Keywords: Seismic isolation Elastomeric bearings Critical load Shear strain Buckling
a b s t r a c t One of the most important aspects of the seismic response of elastomeric isolators is their stability under large shear strains. The bearing capacity of elastomeric isolators, indeed, progressively degrades while increasing horizontal displacement. This may greatly influence the design of elastomeric isolators, especially in high seismicity regions, where slender elastomeric isolators subjected to large horizontal displacements are a common practice. In the current design approach the critical load is evaluated based on the Haringx theory, modified to account for large shear strains by approximate correction factors. In this paper the critical behavior of a pair of slender elastomeric devices is experimentally evaluated at different strain amplitudes, ranging from approximately 50% to 150%. The experimental results are then compared to the predictions of a number of semi-empirical and theoretical formulations. The main conclusion of this study is that current design approaches are overly conservative for slender elastomeric seismic isolators, since they underestimate their critical load capacity at moderate-to-large shear strain amplitudes. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction An elastomeric isolation bearing consists of a number of rubber and steel layers mutually vulcanized, to provide high stiffness in the vertical direction together with large deformability in the horizontal direction. The elastomeric isolators work like a filter lengthening the fundamental period of vibration of the structure, thus reducing the seismic effects (interstory drifts, floor accelerations, stresses in the structural members, etc.) generated in the superstructure. However, this reduction is accompanied by large horizontal displacements in the isolators, which may significantly reduce their axial load capacity [1–4]. The earliest theoretical approach for the evaluation of the critical axial load of rubber bearings was introduced by Haringx [5], considering the mechanical characteristics of helical steel springs and rubber rods. Same assumptions have been made later by Gent [6] considering multilayered rubber compression springs. Basically, the Haringx’s theory is based on a linear one-dimensional beam model with shear deformability, within the hypothesis of small displacements. The critical buckling load of elastomeric seismic isolators is expressed as:
Pcr;0 ¼
2 PE sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEIÞeff 1 þ 1 þ 4p2 ðGAs Þeff L2
ð1Þ
⇑ Corresponding author. Tel.: +39 0971205054. E-mail addresses: [email protected] (D. Cardone), giuseppe.perr@ alice.it (G. Perrone). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2012.02.031
in which: ðGAs Þeff and ðEIÞeff are the effective shear rigidity and effective flexural rigidity, respectively, of the elastomeric isolators, computed based on the bending modulus (E) and dynamic shear modulus (Gdyn) of rubber, moment of inertia of the bearing about the axis of bending (I) and bonded rubber area (As); PE is the Euler load for a standard elastic column:
PE ¼
p2 ðEIÞeff
ð2Þ
L2
L is the total height of rubber layers and steel plates excluding top and bottom connecting steel plates. Various authors proposed different relations to evaluate the effective shear and flexural 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:
ðGAs Þeff ¼ Gdyn As ðEIÞeff ¼ Er I
L te
L te
ð3Þ
ð4Þ
where te is the total thickness of the rubber layers and Er is the elastic modulus of the rubber bearing evaluated based on the primary shape factor S1 and rubber Young’s modulus E0 as:
Er ¼ E0 ð1 þ 0:742 S21 Þ
ð5Þ
The primary shape factor S1 is defined as the ratio between the loaded area of the bearing and the area free to bulge of the single rubber layer (S1 D/4ti for circular bearings, where D is the diameter
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of the isolator and ti the thickness of a single rubber layer). The rubber Young’s modulus E0 is usually taken equal to 3.3 Gdyn to 4 Gdyn. The Haringx’s theory has been later applied by Naeim and Kelly [7], with a series of simplified assumptions, for commercial elastomeric seismic isolators. According to Naeim and Kelly [7], the critical buckling load of elastomeric seismic isolators can be expressed in terms of the primary and secondary shape factors S1 and S2, the latter being defined as the ratio between the maximum dimension of the cross section of the isolator and the total height of rubber. For circular elastomeric isolator, for instance, Naeim and Kelly [7] provides:
p
Pcr;0 ¼ pffiffiffi ðGAs Þeff S1 S2 2 2
Pcr;0 ¼ pffiffiffi ðGAs Þeff 2 3
h
u P
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:742 E0 S1 S2 G
(b)
2 PE sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð EI Þ ð EI Þ eff eff 1 þ 1 þ 4p2 ðEAÞ L2 þ ðGA Þ L2 eff
ð8Þ
L te
ð9Þ
In Italy, the current design approach [10] refers to a formulation of the critical load similar to (but more conservative than) that initially proposed by Naeim and Kelly [7] (see Eq. (6)):
Pcr;0 ¼ Gdyn As S1 S2
ð11Þ
For all the above mentioned formulations, the buckling load at the target shear displacement (u) is evaluated as a function of the ratio between the effective area of the inner shim plate (A) and the overlap area of the displaced bearing (Ar) (see Fig. 1):
Pcr ¼ Pcr;0
Ar A
ð12Þ
For circular bearings, for instance, the overlap area at the target displacement is given by:
Ar ¼ ðu sin uÞ with
u ¼ 2 arccos
u D
D2 4
u Fig. 1. (a) Schematic deformed shape of an elastomeric bearing subjected to shear and compression; (b) effective cross section area as a function of shear displacement.
[12,1,3,4,13]. In this paper, the critical load of a couple of slender (low shape factors) elastomeric bearings is experimentally evaluated. Test set-up and experimental program are presented first in detail. Then, the experimental results are compared with the predictions of the theoretical formulations presented in the previous paragraph. 2. Experimental tests
ð10Þ
where Gdyn is the dynamic shear modulus derived from the qualification tests of the elastomeric isolator. More recently, a less conservative variant of the Naeim and Kelly formulation has been adopted in the new European Standard EN11529 [11]:
Pcr;0 ¼ 1:3 Gdyn As S1 S2
Ar
s eff
where ðEAÞeff is the effective axial stiffness of the rubber bearing, evaluated as:
ðEAÞeff ¼ Er A
D
ð7Þ
The secondary shape factor S2 is defined as the ratio between the bearing maximum dimension and the total thickness of all the rubber layers (S2 D/te for circular bearings, where te is the total thickness of all the rubber layers). It is interesting to note that the critical buckling load capacity evaluated considering the expressions (6) and (7), differs by 10–20%, depending on the value (between 3.3G and 4G) assumed for the rubber Young’s modulus. Lanzo [9] modified the Haringx’s expression by taking into account the axial stiffness of the rubber bearing (EA)eff:
Pcr;0 ¼
P F
ð6Þ
Subsequently, Kelly [8] derived a more refined formulation of the buckling load of elastomeric isolator:
p
(a)
ð13Þ
ð14Þ
Several experimental studies of the buckling behavior of elastomeric seismic isolators have been carried out in the past
2.1. Test specimens Test specimens are a couple of 1:2 scaled circular elastomeric bearings with 200 mm diameter and 10 rubber layers with 8 mm thickness. Bearing geometrical properties are summarized in Table 1. The mechanical properties of the specimens have been derived from a number of standard cyclic tests, performed in accordance with the test procedure prescribed in the Italian seismic code [10] for the qualification of elastomeric bearings. The static shear modulus (Gstat), in particular, has been derived from a quasi-static test consisting of five cycles at 0.1 Hz frequency of loading and 100% shear strain amplitude. According to the NTC 2008 [10], Gstat
Table 1 Elastomeric bearings details. Outer diameter Inner diameter Rubber layer thickness Number of rubber layers Steel shim thickness Number of steel shims Total height of rubber Primary shape factor Secondary shape factor
De (mm) D (mm) ti (mm) nti ts (mm) nti te (mm) S1 S2
200 180 8 10 2 9 80 5.63 2.25
200
D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
(a)
(b)
Fig. 2. Hysteresis cycles of the test specimens at 100% shear strain amplitude during (a) quasi-static test for the evaluation of Gstat and (b) dynamic test for the evaluation of Gdyn and neq.
is defined as the slope of the shear–strain curve between 27% and 58% 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 a dynamic test consisting of five cycles at 0.5 Hz frequency of loading and 100% shear strain amplitude with reference to the third cycle of the test (see Fig. 2b). According to the NTC2008 [10], Gdyn is defined as the secant stiffness to the origin of the axis of the cycle of maximum strain amplitude, neq is evaluated as the ratio between the energy dissipated in one complete cycle (equal to the area enclosed in the cycle) and the strain energy stored at the maximum strain amplitude (see Fig. 2b). In the case under consideration, a value of Gdyn = 0.37 MPa and neq = 16% have been obtained. 2.2. Test set-up The specimens have been tested using the uniaxial bearing test facility available at the Laboratory of Structures at the University of Basilicata (see Fig. 3). The test apparatus constitutes a selfbalanced system designed for cyclic testing of a pair of identical counteracting specimens. The apparatus subjects the test specimens to axial loads in the horizontal direction and shear forces and displacements in the vertical direction. The test rig consists of two steel stiff beams connected by three steel columns. Two columns are linked to the base beam through a double pendulum hinge while the third column (on the left in Fig. 3) by a simple hinge. This has been purposely done to preserve the load directions
Fig. 3. Test set-up.
and the co-planarity of the bases of the specimens during the test. The hinge connections accommodate the axial displacement resulting from the axial load being applied to the specimens. The shear displacement is vertically applied by a Schenk hydraulic actuator, connected to the specimens through a Tshaped steel plate. The actuator is served by two pumps, each with 80 l/min maximum capacity, and equipped with two servo-valves with maximum capacity of 63 l/min each. The Shenck actuator can apply 250 kN maximum force and ±125 mm maximum displacement. The force of the actuator is measured by a load cell mounted on the top of the cylinder. The displacement of the actuator is measured and controlled by an internal transducer. The compression load is horizontally applied and kept constant during the tests by an Enerpac double-effect hydraulic jack (see Fig. 3). The hydraulic jack is served by a pump, with 5 l/min capacity, and equipped with an electro-valve with 5 l/min maximum capacity. The hydraulic jack can apply 300 kN maximum force. The force of the hydraulic jack is monitored and measured by a number of load cells of different capacity mounted on the top of the cylinder. The shear displacements of the specimens have been measured by two linear transducers connected to the T-shaped steel plate. The axial displacements of the specimens have been measured by a couple of linear transducers, positioned between the corner of the outer plates of the two specimens, in order to evaluate possible rotations of the specimens around the out-of-plane axis.
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D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
interval of 10 s was allowed to elapse between one step and the next. The test control has been carried out manually in order to prevent, in real time, possible damages to either specimens or test apparatus. A total number of 6 stability test have been performed (see Table 2), while increasing the target shear displacement from 48 to 125 mm, corresponding to six different shear strain amplitude, equal to 60%, 80%, 100%, 120%, 140% and 160%, respectively.
u (mm)
90
45
0
0
The first direct result of the critical load tests is the time history of the shear force F resulting from the application of an increasing axial load P under a given shear displacement u. As an example, Fig. 5 shows the shear force–time 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, the shear force F decreases until it becomes negative, while the shear displacement remains constant. The experimental outcomes have been employed to evaluate the critical buckling load for each imposed shear deformation. It is worth observing that the critical load cannot be simply estimated as the value of the axial load at which the horizontal shear force is zero because this would give a ‘‘constrained’’ evaluation of the critical load, since the critical load is approached while the bearing is locked against further horizontal displacements. In the common practice (e.g. during a seismic event), instead, the horizontal displacement is unconstrained and free to increase, as necessary, when buckling takes place. Reference to the procedure proposed by Nagarajaiah and Ferrell [4], based on equilibrium paths, has been made to determine the unconstrained critical load. Fig. 6 shows the variation of the shear force F with increasing axial load P, for each test performed at a given shear deformation c = 60%, 80%, 100%, 120%, 140% and 160%. The diagram of Fig. 6 has been entered alternatively with a number of F-lines and P-lines to determine a series of equilibrium paths (i.e. a number of smooth curves passing through discrete points) corresponding to given levels of shear force (Fig. 7) and axial load (Fig. 9), respectively. The equilibrium paths are unstable past a limit point, which is the critical load Pcr. This observation can be exploited to estimate the critical load of the rubber bearings under consideration, as explained below. Fig. 7 shows the axial load vs. shear strain curves, corresponding to different levels of shear force in the specimens. The critical load can be identified with the value of the axial load that must be reduced, to maintain constant the shear force in the specimens by further increments of shear displacement. As expected, the critical load decreases while increasing the horizontal displacement, hence shear strain, applied (see Fig. 8). Fig. 9 shows the shear force vs. shear strain curves, corresponding to different values of axial load applied to the specimens. As can be seen, each curve passes through a maximum as the horizontal displacement increases, under constant axial load. The shear
P (KN)
300
2.4. Test results
150
0
0
Fig. 4. Steps of a typical stability test (for more details see Table 2).
The acquisition frequency has been set equal to 10 Hz. The signals recorded during the tests have been filtered, amplified and converted from analog signals to 16 bit digital signals in real time by a proper signal processing software. 2.3. Test procedure The testing procedure followed in this study is similar, to some extent, to that adopted by Buckle et al. [13]. All the stability tests, in particular, were performed on the same couple of specimens, at different levels of shear deformations. Basically, each test consisted in the application of two consecutive ramps of axial load with a ramp of shear displacement between them (see Fig. 4). More precisely, each test can be divided in the following six steps: 1. Applying an initial axial load of 150 kN, corresponding to a design compression stress of 6 MPa. 2. Applying a given target displacement (hence shear deformation). 3. Increasing the axial load until the horizontal force became 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. Both forces and displacements have been applied very slowly during the tests, in order to avoid dynamic effects. Moreover, an
Table 2 Stability test details (see Fig. 4). Test no.
1 2 3 4 5 6
u1 (mm)
48 64 80 96 112 125
c (%) 60 80 100 120 140 160
P1 (kN)
150 150 150 150 150 150
P2 (kN)
280 280 280 280 200 200
Step 1
Step 2
t1 (s)
t2 (s)
t3 (s)
t4 (s)
Step 3 t5 (s)
t6 (s)
Step 4 t7 (s)
t8 (s)
t9 (s)
Step 5 t10 (s)
t11 (s)
Step 6
15 15 15 15 15 15
25 25 25 25 25 25
35 38 42 45 48 51
45 48 52 55 58 61
58 61 65 68 63 66
68 71 75 78 73 76
81 84 88 91 78 81
91 94 98 101 88 91
101 108 114 121 112 117
111 118 124 131 122 127
126 133 139 146 137 142
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D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
125
u (mm)
(a)
100 75 50 25
t (s)
0 0
300
100
150
200
250
300
350
P (KN)
(b)
50
200
100 t (s)
0 0
100
150
200
250
300
350
20
F (KN)
(c)
50
10
t (s)
0 0
50
100
150
200
250
300
350
-10 -20
220 F = 2.2 kN
P (kN)
235
215
200
Axial load (kN)
Fig. 5. Typical results of stability test: shear displacement, axial load and shear force time histories.
195
180 175 P = 162 kN
F=0.2KN F=1.3KN F=1.5KN F=2.2KN F=4.5KN
160
155
140 40
135 0
1
2
3
4
5
6
7 F (kN)
Fig. 6. Axial load vs. shear force variations as a function of the imposed shear strain amplitude.
force and shear strain at which the maximum occurs decrease with increasing axial load. Finally, the horizontal tangent stiffness tends to zero for shear strains lower than the maximum test amplitude; moreover it decreases while increasing axial load and horizontal displacement.
Shear strain (%)
80
120
160
Fig. 7. Axial load vs. shear strain curves as a function of shear force.
2.5. Comparison between analytical and experimental results Fig. 10 compares the experimental values of Pcr derived following the ‘‘equilibrium path’’ procedure described in the last section, to the numerical results provided by the formula proposed by Haringx [5] (Eq. (1)), Naeim and Kelly [7] (Eq. (6)), Kelly [8] (Eq. (7)), Lanzo [9] (Eq. (8)), and those adopted in the Italian seismic code [10] (Eq. (10)) and in the EN15129 European standard [11]
D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
Pcr (kN)
300
experimental values
200
100
Shear strain (%)
0 60
80
100
120
140
160
Fig. 8. Critical load as a function of shear strain amplitude.
Shear force(kN)
5
4
3
2
1 Shear strain(%)
0 0
40
80
120
160
300
Pcr (kN)
Fig. 9. Shear force–strain curves as a function of axial load.
203
results, while the formula proposed by Kelly provides the highest values of critical load. As can be seen, for the slender elastomeric bearings under consideration (with secondary shape factor less than 3 and rubber thickness of 8 mm), the experimental values of Pcr result significantly greater than those predicted by the theoretical formulas, especially at large shear amplitudes (100–150%) where values approximately 2.5–3.5 times greater than expected are found. The relationship adopted in the current Italian seismic code appears to be too conservative for low shape factors elastomeric bearings (S1 < 12, S2 < 3), such as those considered in this study. The formulation of Pcr proposed in the new EN11529 European Standard, being less conservative and fully compatible with many theoretical formulations, seems to be more suitable for slender elastomeric isolators. 3. Conclusions In this paper the critical behavior of a couple of slender elastomeric seismic isolators has been experimentally evaluated at different strain amplitudes, ranging approximately from 50% to 160%. The tests were performed with the specific objectives of finding the effect of shear strain on the critical load of slender (low shape factors) elastomeric seismic isolators and to evaluate the existing design approaches. During the tests, the shear displacement of the specimens was held at a specified value while the axial load was progressively increased until critical load conditions occurred. For the purpose of this experiment, the isolator was considered to be in critical state when the horizontal force became zero or negative. The critical load has been evaluated, for each strain amplitude, based on a series of equilibrium paths derived from the experimental results. Based on the results of this study the following conclusions can be drawn: (i) The critical load decreases with increasing shear strain. (ii) The horizontal stiffness decreases with increasing axial load and horizontal displacement. (iii) Current design procedures seem to be overly conservative for slender (low shape factor) elastomeric isolators because they underestimate the experimentally determined critical load by a factor of approximately 1.5 at low shear strain amplitudes and approximately 3.5 at large shear strain amplitudes. In the near future, it is desirable to extend this experimental investigation considering more specimens, characterized by different shape factors and/or different rubber thickness, in order to evaluate the influence of such parameters on the critical behavior of slender elastomeric seismic isolators.
200
Acknowledgements The authors are grateful to Domenico Nigro (University of Basilicata) for his valuable help in setting up the testing apparatus and executing the experimental tests. This work has been partially funded by MIUR, COFIN 2007.
100
Shear strain(%)
0 60
80
100
120
140
Fig. 10. Comparison between experimental results and critical loads expected based on different theoretical and semi-empirical formulations.
(Eq. (11)). Among the theoretical formulations considered, the formula proposed by Haringx leads to the most conservative
References [1] Buckle IG, Kelly JM. Properties of slender elastomeric isolation bearings during shake table studies of a large-scale model bridge deck. In: Joint sealing and bearing systems for concrete structures (American Concrete Institute), vol. 1. p. 247–69; 1986. [2] Koh CG, Kelly JM. Effects of axial load on elastomeric bearings. Earthquake Engineering Research Center, Rep. UCB/EERC-86/12, Univ. of California, Berkeley, 1986. [3] Buckle IG, Liu H. Experimental determination of critical loads of elastomeric isolators at high shear strain. NCEER Bull 1994;8(3):1–5.
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D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
[4] Nagarajaiah S, Ferrell K. Stability of elastomeric seismic isolation bearings. J Struct Eng 1999;125(9):946–54. [5] Haringx JA. On highly compressive helical springs and rubber rods and their applications for vibration-free mountings. I. Philips Res Rep 1948;3:401–49. [6] Gent AN. Elastic stability of rubber compression springs. J Mech Eng Sci 1964;6(4):318–26. [7] Naeim F, Kelly JM. Design of seismic isolated structures—from theory to practice. New York: John Wiley & Sons; 1999. [8] Kelly TE. Base isolation of structures: design guidelines. Wellington, New Zealand: Holmes Consulting Group Ltd.; 2001.
[9] Lanzo AD. On elastic beam models for stability analysis of multilayered rubber bearings. Int J Solids Struct 2004;41(20):5733–57. [10] NTC 2008 – Norme Tecniche per le Costruzioni, D.M. 14/01/08, Rome, 2008 [in Italian]. [11] EN15129 – European Standard EN 15129: anti-seismic devices, 2009. [12] Aiken D, Kelly JM, Tajirian FF. Mechanics of low shape factor elastomeric seismic isolation bearings. Report no. UCB/EERC-89/13, 1989. [13] Buckle I, Nagarajaiah S, Ferrell K. Stability of elastomeric isolation bearings: experimental study. J Struct Eng 2002;128(1):3–11.
Contents lists available at SciVerse ScienceDirect
Engineering Structures journal homepage: www.elsevier.com/locate/engstruct
Critical load of slender elastomeric seismic isolators: An experimental perspective Donatello Cardone ⇑, Giuseppe Perrone DiSGG, University of Basilicata, Via Ateneo Lucano 10, 85100 Potenza, Italy
a r t i c l e
i n f o
Article history: Received 27 July 2011 Revised 31 January 2012 Accepted 5 February 2012 Available online 28 March 2012 Keywords: Seismic isolation Elastomeric bearings Critical load Shear strain Buckling
a b s t r a c t One of the most important aspects of the seismic response of elastomeric isolators is their stability under large shear strains. The bearing capacity of elastomeric isolators, indeed, progressively degrades while increasing horizontal displacement. This may greatly influence the design of elastomeric isolators, especially in high seismicity regions, where slender elastomeric isolators subjected to large horizontal displacements are a common practice. In the current design approach the critical load is evaluated based on the Haringx theory, modified to account for large shear strains by approximate correction factors. In this paper the critical behavior of a pair of slender elastomeric devices is experimentally evaluated at different strain amplitudes, ranging from approximately 50% to 150%. The experimental results are then compared to the predictions of a number of semi-empirical and theoretical formulations. The main conclusion of this study is that current design approaches are overly conservative for slender elastomeric seismic isolators, since they underestimate their critical load capacity at moderate-to-large shear strain amplitudes. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction An elastomeric isolation bearing consists of a number of rubber and steel layers mutually vulcanized, to provide high stiffness in the vertical direction together with large deformability in the horizontal direction. The elastomeric isolators work like a filter lengthening the fundamental period of vibration of the structure, thus reducing the seismic effects (interstory drifts, floor accelerations, stresses in the structural members, etc.) generated in the superstructure. However, this reduction is accompanied by large horizontal displacements in the isolators, which may significantly reduce their axial load capacity [1–4]. The earliest theoretical approach for the evaluation of the critical axial load of rubber bearings was introduced by Haringx [5], considering the mechanical characteristics of helical steel springs and rubber rods. Same assumptions have been made later by Gent [6] considering multilayered rubber compression springs. Basically, the Haringx’s theory is based on a linear one-dimensional beam model with shear deformability, within the hypothesis of small displacements. The critical buckling load of elastomeric seismic isolators is expressed as:
Pcr;0 ¼
2 PE sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðEIÞeff 1 þ 1 þ 4p2 ðGAs Þeff L2
ð1Þ
⇑ Corresponding author. Tel.: +39 0971205054. E-mail addresses: [email protected] (D. Cardone), giuseppe.perr@ alice.it (G. Perrone). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2012.02.031
in which: ðGAs Þeff and ðEIÞeff are the effective shear rigidity and effective flexural rigidity, respectively, of the elastomeric isolators, computed based on the bending modulus (E) and dynamic shear modulus (Gdyn) of rubber, moment of inertia of the bearing about the axis of bending (I) and bonded rubber area (As); PE is the Euler load for a standard elastic column:
PE ¼
p2 ðEIÞeff
ð2Þ
L2
L is the total height of rubber layers and steel plates excluding top and bottom connecting steel plates. Various authors proposed different relations to evaluate the effective shear and flexural 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:
ðGAs Þeff ¼ Gdyn As ðEIÞeff ¼ Er I
L te
L te
ð3Þ
ð4Þ
where te is the total thickness of the rubber layers and Er is the elastic modulus of the rubber bearing evaluated based on the primary shape factor S1 and rubber Young’s modulus E0 as:
Er ¼ E0 ð1 þ 0:742 S21 Þ
ð5Þ
The primary shape factor S1 is defined as the ratio between the loaded area of the bearing and the area free to bulge of the single rubber layer (S1 D/4ti for circular bearings, where D is the diameter
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D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
of the isolator and ti the thickness of a single rubber layer). The rubber Young’s modulus E0 is usually taken equal to 3.3 Gdyn to 4 Gdyn. The Haringx’s theory has been later applied by Naeim and Kelly [7], with a series of simplified assumptions, for commercial elastomeric seismic isolators. According to Naeim and Kelly [7], the critical buckling load of elastomeric seismic isolators can be expressed in terms of the primary and secondary shape factors S1 and S2, the latter being defined as the ratio between the maximum dimension of the cross section of the isolator and the total height of rubber. For circular elastomeric isolator, for instance, Naeim and Kelly [7] provides:
p
Pcr;0 ¼ pffiffiffi ðGAs Þeff S1 S2 2 2
Pcr;0 ¼ pffiffiffi ðGAs Þeff 2 3
h
u P
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:742 E0 S1 S2 G
(b)
2 PE sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ð EI Þ ð EI Þ eff eff 1 þ 1 þ 4p2 ðEAÞ L2 þ ðGA Þ L2 eff
ð8Þ
L te
ð9Þ
In Italy, the current design approach [10] refers to a formulation of the critical load similar to (but more conservative than) that initially proposed by Naeim and Kelly [7] (see Eq. (6)):
Pcr;0 ¼ Gdyn As S1 S2
ð11Þ
For all the above mentioned formulations, the buckling load at the target shear displacement (u) is evaluated as a function of the ratio between the effective area of the inner shim plate (A) and the overlap area of the displaced bearing (Ar) (see Fig. 1):
Pcr ¼ Pcr;0
Ar A
ð12Þ
For circular bearings, for instance, the overlap area at the target displacement is given by:
Ar ¼ ðu sin uÞ with
u ¼ 2 arccos
u D
D2 4
u Fig. 1. (a) Schematic deformed shape of an elastomeric bearing subjected to shear and compression; (b) effective cross section area as a function of shear displacement.
[12,1,3,4,13]. In this paper, the critical load of a couple of slender (low shape factors) elastomeric bearings is experimentally evaluated. Test set-up and experimental program are presented first in detail. Then, the experimental results are compared with the predictions of the theoretical formulations presented in the previous paragraph. 2. Experimental tests
ð10Þ
where Gdyn is the dynamic shear modulus derived from the qualification tests of the elastomeric isolator. More recently, a less conservative variant of the Naeim and Kelly formulation has been adopted in the new European Standard EN11529 [11]:
Pcr;0 ¼ 1:3 Gdyn As S1 S2
Ar
s eff
where ðEAÞeff is the effective axial stiffness of the rubber bearing, evaluated as:
ðEAÞeff ¼ Er A
D
ð7Þ
The secondary shape factor S2 is defined as the ratio between the bearing maximum dimension and the total thickness of all the rubber layers (S2 D/te for circular bearings, where te is the total thickness of all the rubber layers). It is interesting to note that the critical buckling load capacity evaluated considering the expressions (6) and (7), differs by 10–20%, depending on the value (between 3.3G and 4G) assumed for the rubber Young’s modulus. Lanzo [9] modified the Haringx’s expression by taking into account the axial stiffness of the rubber bearing (EA)eff:
Pcr;0 ¼
P F
ð6Þ
Subsequently, Kelly [8] derived a more refined formulation of the buckling load of elastomeric isolator:
p
(a)
ð13Þ
ð14Þ
Several experimental studies of the buckling behavior of elastomeric seismic isolators have been carried out in the past
2.1. Test specimens Test specimens are a couple of 1:2 scaled circular elastomeric bearings with 200 mm diameter and 10 rubber layers with 8 mm thickness. Bearing geometrical properties are summarized in Table 1. The mechanical properties of the specimens have been derived from a number of standard cyclic tests, performed in accordance with the test procedure prescribed in the Italian seismic code [10] for the qualification of elastomeric bearings. The static shear modulus (Gstat), in particular, has been derived from a quasi-static test consisting of five cycles at 0.1 Hz frequency of loading and 100% shear strain amplitude. According to the NTC 2008 [10], Gstat
Table 1 Elastomeric bearings details. Outer diameter Inner diameter Rubber layer thickness Number of rubber layers Steel shim thickness Number of steel shims Total height of rubber Primary shape factor Secondary shape factor
De (mm) D (mm) ti (mm) nti ts (mm) nti te (mm) S1 S2
200 180 8 10 2 9 80 5.63 2.25
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(a)
(b)
Fig. 2. Hysteresis cycles of the test specimens at 100% shear strain amplitude during (a) quasi-static test for the evaluation of Gstat and (b) dynamic test for the evaluation of Gdyn and neq.
is defined as the slope of the shear–strain curve between 27% and 58% 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 a dynamic test consisting of five cycles at 0.5 Hz frequency of loading and 100% shear strain amplitude with reference to the third cycle of the test (see Fig. 2b). According to the NTC2008 [10], Gdyn is defined as the secant stiffness to the origin of the axis of the cycle of maximum strain amplitude, neq is evaluated as the ratio between the energy dissipated in one complete cycle (equal to the area enclosed in the cycle) and the strain energy stored at the maximum strain amplitude (see Fig. 2b). In the case under consideration, a value of Gdyn = 0.37 MPa and neq = 16% have been obtained. 2.2. Test set-up The specimens have been tested using the uniaxial bearing test facility available at the Laboratory of Structures at the University of Basilicata (see Fig. 3). The test apparatus constitutes a selfbalanced system designed for cyclic testing of a pair of identical counteracting specimens. The apparatus subjects the test specimens to axial loads in the horizontal direction and shear forces and displacements in the vertical direction. The test rig consists of two steel stiff beams connected by three steel columns. Two columns are linked to the base beam through a double pendulum hinge while the third column (on the left in Fig. 3) by a simple hinge. This has been purposely done to preserve the load directions
Fig. 3. Test set-up.
and the co-planarity of the bases of the specimens during the test. The hinge connections accommodate the axial displacement resulting from the axial load being applied to the specimens. The shear displacement is vertically applied by a Schenk hydraulic actuator, connected to the specimens through a Tshaped steel plate. The actuator is served by two pumps, each with 80 l/min maximum capacity, and equipped with two servo-valves with maximum capacity of 63 l/min each. The Shenck actuator can apply 250 kN maximum force and ±125 mm maximum displacement. The force of the actuator is measured by a load cell mounted on the top of the cylinder. The displacement of the actuator is measured and controlled by an internal transducer. The compression load is horizontally applied and kept constant during the tests by an Enerpac double-effect hydraulic jack (see Fig. 3). The hydraulic jack is served by a pump, with 5 l/min capacity, and equipped with an electro-valve with 5 l/min maximum capacity. The hydraulic jack can apply 300 kN maximum force. The force of the hydraulic jack is monitored and measured by a number of load cells of different capacity mounted on the top of the cylinder. The shear displacements of the specimens have been measured by two linear transducers connected to the T-shaped steel plate. The axial displacements of the specimens have been measured by a couple of linear transducers, positioned between the corner of the outer plates of the two specimens, in order to evaluate possible rotations of the specimens around the out-of-plane axis.
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interval of 10 s was allowed to elapse between one step and the next. The test control has been carried out manually in order to prevent, in real time, possible damages to either specimens or test apparatus. A total number of 6 stability test have been performed (see Table 2), while increasing the target shear displacement from 48 to 125 mm, corresponding to six different shear strain amplitude, equal to 60%, 80%, 100%, 120%, 140% and 160%, respectively.
u (mm)
90
45
0
0
The first direct result of the critical load tests is the time history of the shear force F resulting from the application of an increasing axial load P under a given shear displacement u. As an example, Fig. 5 shows the shear force–time 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, the shear force F decreases until it becomes negative, while the shear displacement remains constant. The experimental outcomes have been employed to evaluate the critical buckling load for each imposed shear deformation. It is worth observing that the critical load cannot be simply estimated as the value of the axial load at which the horizontal shear force is zero because this would give a ‘‘constrained’’ evaluation of the critical load, since the critical load is approached while the bearing is locked against further horizontal displacements. In the common practice (e.g. during a seismic event), instead, the horizontal displacement is unconstrained and free to increase, as necessary, when buckling takes place. Reference to the procedure proposed by Nagarajaiah and Ferrell [4], based on equilibrium paths, has been made to determine the unconstrained critical load. Fig. 6 shows the variation of the shear force F with increasing axial load P, for each test performed at a given shear deformation c = 60%, 80%, 100%, 120%, 140% and 160%. The diagram of Fig. 6 has been entered alternatively with a number of F-lines and P-lines to determine a series of equilibrium paths (i.e. a number of smooth curves passing through discrete points) corresponding to given levels of shear force (Fig. 7) and axial load (Fig. 9), respectively. The equilibrium paths are unstable past a limit point, which is the critical load Pcr. This observation can be exploited to estimate the critical load of the rubber bearings under consideration, as explained below. Fig. 7 shows the axial load vs. shear strain curves, corresponding to different levels of shear force in the specimens. The critical load can be identified with the value of the axial load that must be reduced, to maintain constant the shear force in the specimens by further increments of shear displacement. As expected, the critical load decreases while increasing the horizontal displacement, hence shear strain, applied (see Fig. 8). Fig. 9 shows the shear force vs. shear strain curves, corresponding to different values of axial load applied to the specimens. As can be seen, each curve passes through a maximum as the horizontal displacement increases, under constant axial load. The shear
P (KN)
300
2.4. Test results
150
0
0
Fig. 4. Steps of a typical stability test (for more details see Table 2).
The acquisition frequency has been set equal to 10 Hz. The signals recorded during the tests have been filtered, amplified and converted from analog signals to 16 bit digital signals in real time by a proper signal processing software. 2.3. Test procedure The testing procedure followed in this study is similar, to some extent, to that adopted by Buckle et al. [13]. All the stability tests, in particular, were performed on the same couple of specimens, at different levels of shear deformations. Basically, each test consisted in the application of two consecutive ramps of axial load with a ramp of shear displacement between them (see Fig. 4). More precisely, each test can be divided in the following six steps: 1. Applying an initial axial load of 150 kN, corresponding to a design compression stress of 6 MPa. 2. Applying a given target displacement (hence shear deformation). 3. Increasing the axial load until the horizontal force became 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. Both forces and displacements have been applied very slowly during the tests, in order to avoid dynamic effects. Moreover, an
Table 2 Stability test details (see Fig. 4). Test no.
1 2 3 4 5 6
u1 (mm)
48 64 80 96 112 125
c (%) 60 80 100 120 140 160
P1 (kN)
150 150 150 150 150 150
P2 (kN)
280 280 280 280 200 200
Step 1
Step 2
t1 (s)
t2 (s)
t3 (s)
t4 (s)
Step 3 t5 (s)
t6 (s)
Step 4 t7 (s)
t8 (s)
t9 (s)
Step 5 t10 (s)
t11 (s)
Step 6
15 15 15 15 15 15
25 25 25 25 25 25
35 38 42 45 48 51
45 48 52 55 58 61
58 61 65 68 63 66
68 71 75 78 73 76
81 84 88 91 78 81
91 94 98 101 88 91
101 108 114 121 112 117
111 118 124 131 122 127
126 133 139 146 137 142
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125
u (mm)
(a)
100 75 50 25
t (s)
0 0
300
100
150
200
250
300
350
P (KN)
(b)
50
200
100 t (s)
0 0
100
150
200
250
300
350
20
F (KN)
(c)
50
10
t (s)
0 0
50
100
150
200
250
300
350
-10 -20
220 F = 2.2 kN
P (kN)
235
215
200
Axial load (kN)
Fig. 5. Typical results of stability test: shear displacement, axial load and shear force time histories.
195
180 175 P = 162 kN
F=0.2KN F=1.3KN F=1.5KN F=2.2KN F=4.5KN
160
155
140 40
135 0
1
2
3
4
5
6
7 F (kN)
Fig. 6. Axial load vs. shear force variations as a function of the imposed shear strain amplitude.
force and shear strain at which the maximum occurs decrease with increasing axial load. Finally, the horizontal tangent stiffness tends to zero for shear strains lower than the maximum test amplitude; moreover it decreases while increasing axial load and horizontal displacement.
Shear strain (%)
80
120
160
Fig. 7. Axial load vs. shear strain curves as a function of shear force.
2.5. Comparison between analytical and experimental results Fig. 10 compares the experimental values of Pcr derived following the ‘‘equilibrium path’’ procedure described in the last section, to the numerical results provided by the formula proposed by Haringx [5] (Eq. (1)), Naeim and Kelly [7] (Eq. (6)), Kelly [8] (Eq. (7)), Lanzo [9] (Eq. (8)), and those adopted in the Italian seismic code [10] (Eq. (10)) and in the EN15129 European standard [11]
D. Cardone, G. Perrone / Engineering Structures 40 (2012) 198–204
Pcr (kN)
300
experimental values
200
100
Shear strain (%)
0 60
80
100
120
140
160
Fig. 8. Critical load as a function of shear strain amplitude.
Shear force(kN)
5
4
3
2
1 Shear strain(%)
0 0
40
80
120
160
300
Pcr (kN)
Fig. 9. Shear force–strain curves as a function of axial load.
203
results, while the formula proposed by Kelly provides the highest values of critical load. As can be seen, for the slender elastomeric bearings under consideration (with secondary shape factor less than 3 and rubber thickness of 8 mm), the experimental values of Pcr result significantly greater than those predicted by the theoretical formulas, especially at large shear amplitudes (100–150%) where values approximately 2.5–3.5 times greater than expected are found. The relationship adopted in the current Italian seismic code appears to be too conservative for low shape factors elastomeric bearings (S1 < 12, S2 < 3), such as those considered in this study. The formulation of Pcr proposed in the new EN11529 European Standard, being less conservative and fully compatible with many theoretical formulations, seems to be more suitable for slender elastomeric isolators. 3. Conclusions In this paper the critical behavior of a couple of slender elastomeric seismic isolators has been experimentally evaluated at different strain amplitudes, ranging approximately from 50% to 160%. The tests were performed with the specific objectives of finding the effect of shear strain on the critical load of slender (low shape factors) elastomeric seismic isolators and to evaluate the existing design approaches. During the tests, the shear displacement of the specimens was held at a specified value while the axial load was progressively increased until critical load conditions occurred. For the purpose of this experiment, the isolator was considered to be in critical state when the horizontal force became zero or negative. The critical load has been evaluated, for each strain amplitude, based on a series of equilibrium paths derived from the experimental results. Based on the results of this study the following conclusions can be drawn: (i) The critical load decreases with increasing shear strain. (ii) The horizontal stiffness decreases with increasing axial load and horizontal displacement. (iii) Current design procedures seem to be overly conservative for slender (low shape factor) elastomeric isolators because they underestimate the experimentally determined critical load by a factor of approximately 1.5 at low shear strain amplitudes and approximately 3.5 at large shear strain amplitudes. In the near future, it is desirable to extend this experimental investigation considering more specimens, characterized by different shape factors and/or different rubber thickness, in order to evaluate the influence of such parameters on the critical behavior of slender elastomeric seismic isolators.
200
Acknowledgements The authors are grateful to Domenico Nigro (University of Basilicata) for his valuable help in setting up the testing apparatus and executing the experimental tests. This work has been partially funded by MIUR, COFIN 2007.
100
Shear strain(%)
0 60
80
100
120
140
Fig. 10. Comparison between experimental results and critical loads expected based on different theoretical and semi-empirical formulations.
(Eq. (11)). Among the theoretical formulations considered, the formula proposed by Haringx leads to the most conservative
References [1] Buckle IG, Kelly JM. Properties of slender elastomeric isolation bearings during shake table studies of a large-scale model bridge deck. In: Joint sealing and bearing systems for concrete structures (American Concrete Institute), vol. 1. p. 247–69; 1986. [2] Koh CG, Kelly JM. Effects of axial load on elastomeric bearings. Earthquake Engineering Research Center, Rep. UCB/EERC-86/12, Univ. of California, Berkeley, 1986. [3] Buckle IG, Liu H. Experimental determination of critical loads of elastomeric isolators at high shear strain. NCEER Bull 1994;8(3):1–5.
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[4] Nagarajaiah S, Ferrell K. Stability of elastomeric seismic isolation bearings. J Struct Eng 1999;125(9):946–54. [5] Haringx JA. On highly compressive helical springs and rubber rods and their applications for vibration-free mountings. I. Philips Res Rep 1948;3:401–49. [6] Gent AN. Elastic stability of rubber compression springs. J Mech Eng Sci 1964;6(4):318–26. [7] Naeim F, Kelly JM. Design of seismic isolated structures—from theory to practice. New York: John Wiley & Sons; 1999. [8] Kelly TE. Base isolation of structures: design guidelines. Wellington, New Zealand: Holmes Consulting Group Ltd.; 2001.
[9] Lanzo AD. On elastic beam models for stability analysis of multilayered rubber bearings. Int J Solids Struct 2004;41(20):5733–57. [10] NTC 2008 – Norme Tecniche per le Costruzioni, D.M. 14/01/08, Rome, 2008 [in Italian]. [11] EN15129 – European Standard EN 15129: anti-seismic devices, 2009. [12] Aiken D, Kelly JM, Tajirian FF. Mechanics of low shape factor elastomeric seismic isolation bearings. Report no. UCB/EERC-89/13, 1989. [13] Buckle I, Nagarajaiah S, Ferrell K. Stability of elastomeric isolation bearings: experimental study. J Struct Eng 2002;128(1):3–11.