RE-CENTERING CAPABILITY EVALUATION OF SEISMIC ISOLATION SYSTEMS BASED ON ENERGY CONCEPTS

Transcription

1 3 th Worl Conference on Earthquake Engineering Vancouver, B.C., Canaa August -6, 004 Paper No. 306 RE-CENTERING CAPABILITY EVALUATION OF SEISMIC ISOLATION SYSTEMS BASED ON ENERGY CONCEPTS Renzo MEDEOT SUMMARY It was not until the most recent years that self-centering capability (sometimes referre to as restoring force) was ientifie as a funamental function of an isolation system. This tary occurrence can perhaps be explaine by the fact that, historically, the first seismic isolators were conventional laminate rubber bearings which are enowe with an optimal self-centering capability. With the introuction in the market of other types of isolators, that are generally not fitte with an intrinsic self-centering capability, the problem of proviing this function has re-asserte its vital role. The purpose of the self-centering capability requirement is not so much that of limiting resiual isplacement at the en of a seismic attack, as instea that of preventing cumulative isplacements uring the seismic event. This type of efect assumes particular importance in cases involving isolators comprising PTFE sliing elements (sometimes referre to as sliers). Moreover, uring the last quarter of the past century energy issipation has increasingly gaine the favor of the esign engineers to mitigate the isastrous effects of a seismic attack. However, energy issipation an self-centering capability are two antithetic functions. Self-centering assumes particular importance in structures locate in close proximity to a fault, where earthquakes characterize by highly asymmetric accelerograms are expecte (Near Fiel or Fling effect). Notwithstaning, self-centering capability was never pai sufficient attention by seismic engineering experts, to the point that the formulation of a criterion to quantify it was only acknowlege for the first time in 99 by the AASHTO Guie Specification for Seismic Isolation Design. Then other criteria were evelope, but none of them is base upon soli theoretical funamentals, but rather make reference to an empirical approach, vali for only one class of evices. In conclusion, present Norms o not furnish an acceptable approach of general valiity to evaluate the self-centering capability of seismic isolation systems. This author evelope a theoretical approach to this problem, suggesting an energy-base criterion for its quantification. The scope of this paper is precisely that of introucing the newly propose criterion. To correctly formulate the problem, some elementary cases are examine that serve to illustrate as well as interpret the requirements aopte to ate by the Norms on this subject. INTRODUCTION Chairman of Technical Committee CEN-TC 340 in charge for the rafting of the European Stanar on Anti-seismic Devices. meeot@iol.it

2 The iea of protecting structures through ecoupling them from the isastrous groun motion generate uring seismic attacks is certainly an ol one. A first example in recore history is that of the temple of Diana in Ephesus, Asia Minor, where a layer of san extene between the founations an the elevate structure was utilize. However, in orer to witness the first practical applications of seismic isolation it was necessary to wait until the last quarter of the 0th century. Although the causes of such elay are multiple, they can be essentially subsume uner lack of: i) aequate software (both Norms an calculation methos), an ii) reliable harware (anti-seismic evices). On the one han, the acaemic worl an the most renowne seismic engineers have eneavore to fill this gap through the evelopment of theoretical frameworks an calculation methoologies whilst, on the other han, research laboratories an specialize sectors of inustry have invente an perfecte numerous mechanical evices apt to satisfy both the theoretical an practical requirements set forth by esign specifications. It goes without saying that, besies transmitting vertical loas, a seismic isolation system must permit free relative movements on the horizontal plane between founation an superstructure, precisely to ensure ecoupling between the soil an the preominant structural mass (e.g.: the brige eck in cases concerning brige structures). Even though the principles of Physics that govern the effects of energy issipation on the control of ynamic phenomena were stuie more than two an a half centuries ago (D Alembert, Traité e ynamique, 743), it took some time before energy issipation came to be ientifie as the most important instrument in the hans of the esign engineer to aequately control seismic response in terms of forces an isplacements between super- an substructure, as well as have it liste amongst the funamental functions of a seismic isolation system. Furthermore, it was not until the most recent years that a fourth funamental function, self-centering capability, was ientifie. This tary occurrence can perhaps be explaine by the fact that, historically, the first seismic isolators were conventional laminate rubber bearings which are enowe with an optimal self-centering capability owing to the elastic restoring force evelope when the same unergo shear eformation. With the introuction in the market of other types of anti-seismic evices that are not fitte with an intrinsic self-centering capability (i.e.: lea rubber bearings, sliing isolators with steel hysteretic elements, friction evices, etc.), the problem of proviing this function has assume a key role (Meeot, [], Braun[]). Notwithstaning, to the latter was never pai sufficient attention by seismic engineering experts, to the point that the formulation of a criterion to quantify it in a Stanar was only acknowlege for the first time in 99 by the AASHTO Guie Specification for Seismic Isolation Design, expressly requiring the following: The Isolation System shall be configure to prouce a lateral restoring force such that the lateral force at the Design Displacement is at least 0.05 W greater than the lateral force at 50 percent of the Design Displacement. The first version of Eurocoe 8, Part: Briges has also acknowlege the same criterion, even though it oes not expressly cite self-centering capability. However, it appears as though the above criterion :

3 i) has no scientific founation, an, most importantly, ii) cannot offer any inication of the actual re-centering capability of the isolation system in question. Let s consier the Figure below, which is a graphical representation of the above cite criterion. F B C F 0, 05 W A 0,5 Figure : Graphical representation of the re-centering capability of an isolation system in accorance with AASHTO Guie Specification for Seismic Isolation Design (version 99) Both characteristic curves which, by the way, are representative of the very type of evices in toay's market fully comply with the requirement F 0, 05 W. However, it shoul be note that they possess istinctly ifferent re-centering capabilities between them. The revision of the aforesai AASHTO Guie Specifications, publishe in 999, as a new requirement, but still maintains the ol one. However, curiously enough, it became much less restrictive: the restoring force at i shall be greater than the restoring force at 0.5 i by not less than W/80. [Note: W/80 = 0,05 W, that is the half of the value state in 99]. The new requirement states the following: The isolation system shall be configure to prouce a lateral restoring force such that the perio corresponing to its tangent stiffness at any isplacement up to its esign isplacement, shall be less than 6 secons Neither of the two above criteria is base upon soli theoretical funamentals, but rather make reference to an empirical approach, vali for only one class of evices, that is to say, those in which the restoring force increases with isplacement in other wors, that which uses a spring-like restoring force as it is asserte in the Commentary. So much so, that it was eeme necessary to a a thir self-centering capability verification criterion for those systems with constant restoring force. In this category the AASHTO Guie Specifications cite the compressible flui springs with preloa an sliing bearings with conical surface. Regaring the cases with constant restoring force, the Guie Specifications prescribe the following:

4 Isolation systems with constant restoring force nee not satisfy the requirements above. In these cases, the combine constant restoring force of the isolation system shall be at least equal to.05 times the characteristic strength of the isolation system uner service conitions. It can be reaily seen that specifying a 5% tolerance on the friction forces is not all that conservative from static point of view. In fact, friction forces largely epen on several uncontrollable parameters an physical conitions of the contact surfaces. It shoul also be note that just the uncertainty of the loas transmitte by the isolator to which the friction force is proportional is usually much greater than 5% (Meeot,[3]). Conversely from ynamic (or energy) point of view this requirement is extremely severe, as it will be emonstrate further in this paper. The scope of this ocument is that of attempting a first theoretical approach to the self-centering problem, suggesting a criterion for its quantification, without pretening to exhaust the subject nonetheless. FORMULATION OF THE PROBLEM As mentione before, the four funamental functions of a seismic isolation system are the following: - Transmission of vertical loas - Lateral flexibility on the horizontal plane - Dissipation of substantial quantities of energy - Self-centering capability Each function can be performe by a single element or one element can perform more functions. For example, in the case of suspene briges, the hanging links can perform the first three functions an, in orer to create a vali seismic isolation system, it is necessary to resort to energy issipation evices such as hyraulic ampers inserte at strategic points in the structure. In the rubber bearings mentione earlier, especially those in the High Damping Rubber Bearings (HDRBs) version, the four functions are performe by a single element, i.e. the rubber. Those evices that can ensure all four functions internally are calle isolators. Amongst them, the Lea Rubber Bearing, Friction Penulum, Friction Sliers, etc. are mentione besies the HDRBs. It shoul be note that energy issipation an self-centering capability (sometimes referre to as restoring force) are two antithetic functions an their relative importance epens primarily on the case uner examination (Meeot, [4]). The term restoring force is misleaing inasmuch as it woul seem to suggest that the evaluation of the self-centering capability of a Seismic Isolation System must be conucte through a comparison of forces, something that is conceptually erroneous. Actually, comparisons must be mae between the Isolation System s capability to elastically (or, better sai, reversibly) store an irreversibly issipate the earthquake energy input. Self-centering assumes particular importance in structures locate in close proximity to a fault, where earthquakes characterize by highly asymmetric accelerograms are expecte (Near Fiel effect). The purpose of the self-centering capability requirement is not so much that of limiting resiual isplacement at the en of a seismic attack, as instea that of preventing cumulative isplacements uring the seismic event, as inicate in Figures an 3, as well as to remey isolator installation imperfections such as their being out of level. This type of efect assumes particular importance in cases involving isolators comprising PTFE sliing elements (sometimes referre to as sliers).

5 Figure. Seismic Isolation System with aequate self-centering capability t Figure 3. Seismic Isolation System with poor self-centering capability t To correctly formulate the problem, some elementary cases are examine here below that will serve to introuce an further unerstan the propose criterion as well as interpret the requirements aopte to ate by the Norms to guarantee goo self-centering capability. The Friction Penulum Let us first consier the case of the Friction Penulum outline in Figure 4 below. R α F t W F n Figure 4. Moel of Friction Penulum With the symbols of Figure 4, the restoring force is equal to the tangential component of the supporte weight W: F t = W sin α () while the resisting force is that prouce by friction, which is equal to: F f = µ S F n = µ S W cosα () where µ S is the static coefficient of friction between the articulate sliing element an the concave plate. Any position that results in F t F f that is: W sin α µ S W cosα (3) is in fact an equilibrium position. tan α µ S (4)

6 At this point, one coul spontaneously conclue that the restoring force is active up to a limit isplacement equal to: 0 = R sinα 0 (5) ( α 0 = arctan µ S ), inasmuch as from this point on, the restoring force becomes lesser than the resisting friction force. Actually, things are not that simple but quite ifferent. In fact, suppose the sliing element is isplace to a position α > α 0 an then release. With the progression of the motion, the initial potential energy E P = W R (-cosα 0 ) (referre to the inferior point of the spherical surface) becomes partly kinetic energy an is partly issipate by friction into heath. On the basis of the Energy Conservation Principle, the motion will cease when the variation in potential energy equals the energy issipate through friction, namely: α W R sinα α = µ W R cosα α (6) α that is: cos α + µ sin α = cos α + µ sin α (7) For example, for a ynamic friction coefficient µ = 0,07, the angle at which the restoring force equals the friction force is α 0 = 4. Nonetheless, for an initial angular isplacement α = 7, there results a final angular isplacement α = < α 0. Sliers In secon place, let us examine the case of Sliers, i.e. a combination of conventional sliing bearing an a polyurethane or rubber spring unit that can be represente by the moel illustrate in Figure 5 below: α α W k Figure 5. Moel of a Slier The restoring force is furnishe by the spring an is equal to F t = k x, where x is the isplacement, while the resisting force is ue to friction an is equal to F f = µ W. Even in this case, any position that results in: is in fact a position of equilibrium. F t F f that is: x µ W k If x 0 = µ W/k represents the limit value of the possible positions of equilibrium an the evice unergoes a isplacement x > x 0, there will evelop motion up to point x upon release, where the variation in elastic strain energy will equal the energy issipate through friction, namely: (8)

7 Simplifying an substituting µ W/k = x 0, yiels: k ( x x ) = W x x µ (9) x = x 0 sgn x x (0) from which it can be euce that it always results in x < x 0. Analogous consierations can be mae with other types of hysteretic isolators such as lea rubber bearings, steel hysteretic bearings, etc. The conclusions arrive at are always the same, that is: The comparison between restoring force an characteristic strength of the isolator (i.e. friction force for sliing evices, yiel force for lea or steel hysteretic evices, etc.) serves the purpose of etermining the possible static equilibrium positions. This is sometimes referre to as Static Self-centering Capability. The criterion to establish the entity of the self-centering is base upon a comparison between the energy store by the system in a reversible form (elastic, potential etc.) an that hysteretically issipate. This criterion allows to etermine the so calle Dynamic Self-centering Capability. THE ENERGY APPROACH To better illustrate this last assertion, let us consier the energy balance equation in the following form vali for structures (Bertero, [5], [6] an Uang, [7]): where: E i = E S + E H + E V () - E i represents the mechanical energy transmitte to the structure by the seismic groun motion through its founations. - E S is the reversibly store energy (elastic strain energy, potential energy an kinetic energy) - E H is the energy issipate by hysteretic eformation - E V is the energy issipate by viscous amping Self-centering capability is quantifie through a comparison between the first two terms of the secon member. In fact, the energy E V issipate by viscous amping is associate with the forces F that epen only on the velocity v through a constitutive law of the type F = C v n () For v 0 also F 0, that is, there oes not exist a characteristic strength associate with this type of force. In this regar the AASHTO Guie Specifications state the following: Forces that are not epenent on isplacements, such as viscous forces, may not be use to meet the minimum restoring force requirements In conclusion, in the propose approach, the verification of the re-centering capability of an isolator (or an isolation system) consists in the simple comparison between the two types of energy in act uring a seismic attack.

8 In other wors, one has to check that the reversibly store energy E S is greater than a given portion of the energy issipate by hysteretic eformation E H, that is to say: E S λ E H (3) It goes without saying that the larger is the value of λ, the higher is the re-centering capability of the system. The results of numerous step-by-step, non-linear analyses emonstrate that a seismic isolation system possesses sufficient self-centering capability when λ = 0,5. More precisely, for eformations from 0 to, it shall be: E S 0,5 E H (4) The above criterion has proven to be vali an applicable to all types of existing isolation evices, as well as construction typologies. Sliing isolator with steel hysteretic elements The requirement (4) can be translate in formulae or esign criteria for each type of isolator. For example, consier the bi-linear characteristic curve of a PTFE sliing isolator (Figure 6 below) equippe with steel hysteretic elements as energy issipaters. F k p = η k e k e e = /m Figure 6. Characteristic bi-linear curve of a hysteretic system If, for the sake of simplicity, the issipation prouce by the PTFE sheet is ignore (*) an the case uner consieration has a uctility factor m (i.e. = m e ), the energy store elastically is equal to: E m S = ke + ke = k m e m ( + η ( ) ) η (5) m m where k p = η k e represents the slope of the post-elastic branch expresse as a fraction of that of the elastic branch of the characteristic curve. The energy issipate hysteretically is equal to: ke x m m Eh = = ke (6) m m m It can be conclue that requirement (4) is satisfie for:

9 m 3 η (7) ( m ) It is interesting to notice that in this case Self-centering Capability is governe by the post-elastic slope of the characteristic curve an its limit value epens only on one magnitue, i.e. the uctility factor m. Lea Rubber Bearings Let us now consier the Lea Rubber Bearings (LRBs) of the type shown in Figure 7 below (Skinner et alii,(9)). A Pb A r Figure 7: Deforme Lea Rubber Bearing Inicating with A r the cross-sectional area of the rubber bearing, with h as its total rubber thickness an G as the rubber shear moulus, the elastically store energy equals: G Ar E S = (8) h In (8) the moest contribution of the energy elastically store in the lea core was conservatively ignore. (*) Note: This is not a conservative hypothesis an in practice it is opportune to take into account the contribution of friction forces. Inicating with A Pb the cross-sectional area of the lea core an τ Pb as the shear stress at which the lea yiels, the hysteretically issipate energy then equals: E h = τ Pb A Pb (9) Placing the typical values G = 0,9 MPa, τ Pb = 0,5 MPa in (9) an (0), conition (4) is satisfie if: APb 0,7 γ A (0) r where γ = is the esign shear strain. h From the above it can be inferre that for the LRBs Self-centering Capability is governe by the ratio between lea core an rubber cross sections an its limit value epens only on the esign shear eformation.

10 Friction Penulum Consier now the case of the Friction Penulum. Using the symbols alreay use previously in Figure 4, an inicating with α the esign angular isplacement, the energy accumulate uner the form of potential energy is: E s = W R ( cosα ) () while the energy issipate through friction is: E h = µ W R α 0 cos α α = µ W R sinα () Therefore, for the Friction Penulum, requirement (4) is satisfie by: The esign angular isplacement of the spherical surface by: µ cosα 4 (3) sin α α is linke to the linear isplacement an the raius of curvature R = R sin α (4) On its turn the raius of curvature is linke to the natural perio of the structure through the formula (Zayas,(8)): T R = g (5) π From the above one can conclue that in the case of the Friction Penulum the Self-centering mechanism is governe by three parameters, of which two can be chosen at will, while the limit value of the thir is etermine by the expressions (3) (5). For example, if we choose the perio T an the maximum (esign) isplacement, by using (5) the raius of curvature R will be firstly calculate. Thereafter with (4) the angle α is etermine an finally with (3) the maximum value of the coefficient of friction µ is assesse. Conversely, if we choose µ an T, the limiting parameter becomes the minimum value of the esign isplacement. For instance, the raius of curvature of a Friction Penulum that can ensure the structure a natural perio equal to T = 3,5 s is equal to R = 3,04 m an thus, assuming a coefficient of friction µ = 0,07, expression (4) yiels to α. Finally, to attain goo self-centering capability, in the case uner examination the esign isplacement shall excee ± 06 mm. The above explains why the shake table tests have experimentally shown substantial resiual isplacements with earthquakes of lesser magnitue than the esign earthquake. COMPARISON WITH AASHTO GUIDE SPECIFICATIONS At this point, it is noteworthy to attempt a comparison with the requirements set forth by the AASHTO Guie Specifications. Let's start with the case of the Friction Penulum just examine.

11 Comparison for Friction Penulum As mentione in the Introuction, the Guie Specifications require that the restoring force at esign isplacement shall be greater than the restoring force at 0,5 by not less than W/80. The stiffness for this type of isolator is constant an equal to k = W/R. Thus, the above requirement is satisfie when: k W W F = = (6) R 80 R that is: (7) 40 It is observe that the self-centering capability verification conucte in accorance with the AASHTO Guie Specifications is inepenent from the value of the ynamic coefficient of friction µ. This is paraoxical an confirms the oubts expresse by this author in the Introuction. For R = 3,04 m it results 76 mm (vs. 06 mm). From this comparison, it can be conclue that, for the Friction Penulum, the present AASHTO Guie Specifications are less restrictive than the criterion propose in this paper. On the contrary, the 99 version woul have been stricter ( = 5 mm). This therefore etermines that for the Friction Penulum the newly propose criterion places itself right between the two versions of the Guie Specifications. Comparison for Sliing isolators with steel hysteretic elements Referring to Figure 8 on next page, the comparison is conucte with the following esign assumptions - Lateral esign force of the Isolation System: F = 0,0 W (W is the weight of the isolate structure) - uctility factor m = 0 From the esign assumption it results: = = F 0,0 W k e + η k e (8) m m The verification requirement calls for: W F = 0,5 η k e (9) 80

12 F η k F = 0,0 W e m η k e W F 80 k k e e m e = /m 0,5 Figure 8: Characteristic bi-linear curve of Sliing isolators with steel hysteretic elements Obtaining W from (7) an substituting it in (8) the fulfillment of the requirement in the Guie Specifications is obtaine if: η (30) 3 + m For the case uner examination it is η 3,%, versus the value η 4,% calculate - for m=0 - with the expression (8) η (m 3) / [(m ) ] vali for the energy approach. Similar to the case of Friction Penulum, it can be conclue that the present AASHTO Guie Specifications are less restrictive than the criterion base on energy consierations. However, the 99 version was far more restrictive (over a factor of ). Comparison for Lea Rubber Bearing Referring to Figure 9 on next page an similarly to the former case, the comparison is conucte uner the esign assumption that the esign force of the Isolation System is F = 0,0 W. For the sake of simplicity the elastic eformations of the lea core have been ignore. From the esign assumption it results: F = 0, 0 W = A τ + A G γ (3) The verification requirement calls for: A r G W F = 0,5 = 0,5 γ A r G (3) h 80 Pb Pb r

13 F F = 0,0 W k = A r G h W F 80 APb τ Pb 0,5 Figure 9: Characteristic curve of Lea Rubber Bearings Obtaining W from (3) an substituting it in (3) the fulfillment of the requirement in the Guie Specifications is obtaine if: γ Ar G APb 3 (33) τ Placing the typical values G = 0,9 MPa, τ Pb = 0,5 MPa in (3), conition (3) is satisfie if: A Pb Pb 0,57 γ A (34) Also in this case the AASHTO Guie Specifications are less emaning than the criterion base on energy consierations, by which it follows that (see (0)): A Pb 0,7 γ A r r Comparison for Sliers with constant restoring force The Guie Specifications for this case request the following: the combine constant restoring force of the isolation system shall be at least equal to.05 times the characteristic strength of the isolation system uner service conitions. The elastically store energy equals: E S =,05 µ W The hysteretically issipate energy is: E H = µ W Therefore it results: E S =,05 E H Compare with the energy approach criterion (E S 0,5 E H ), the constant restoring force requirement of the Guie Specifications appears to be extremely strict. CONCLUSIONS

14 For the sake of escription simplicity an, above all, so as to avoi uselessly complicate mathematical working out, the examples given in this paper refer to cases where only single types of evices have been consiere instea of combine entire isolate systems. It goes without saying that similar consierations are also vali regaring the latter. Self-centering capability is a characteristic of the entire isolation system, not necessarily of each of its components. Present Norms o not furnish acceptable criteria of general valiity to evaluate the self-centering capability of seismic isolation systems. The comparison conucte between the propose metho base on energy concepts an those with the existing Norms show remarkable iscrepancies that epen notably from the type of isolator. Except for the constant restoring force requirement, in all of the cases taken in consieration for the comparison it was observe that the AASHTO Guie Specifications are less restrictive, especially for Lea Rubber Bearings. Even though this paper oes not preten to give a efinitive answer to the problem of quantifying the recentering capability, it nonetheless serves to suggest a general valiity criterion (i.e.: one applicable to any type of evice) that also incorporates praiseworthy simplicity (it just involves the comparison of two calculable an measurable physical magnitues). The suggeste verification requirement can be easily translate in formulae or esign criteria for each type of isolator or isolation system. The criterion suggeste is base on energy concepts an thus couples very well with the intrinsic nature of the phenomenon in question (the earthquake). In this author s professional experience, the propose criterion has shown itself very vali to preliminarily efine the isolation system s characteristics before unertaking a burensome step-by-step non-linear analysis. The latter still represents toay the most vali metho to verify an isolation system s self-centering capability inasmuch as it permits to quantify the resiual isplacement as well as most importantly reveal any eventual rift of the system mean oscillation point that causes cumulative isplacements uring a seismic event. Nonetheless, a requirement for self-centering capability in a Norm is necessary. This serves to accommoate unpreictable averse factors - such as bearings out of level - which in the ynamic analyses are normally not taken into account. REFERENCES. Meeot, R., (993) "Self-Centering Mechanism in Elastic-Plastic Brige Isolators", Post SMiRT Seminar on Isolation, Energy Dissipation an Control of Vibrations in Structures-Capri, Italy.. Braun C., Meeot R., (000) New Design Approaches to Reuce Seismic Risk - Mitigation of Seismic Risk-Support to Recently Affecte European Countries, Belgirate - Italy 3. Meeot, R., (993) "Comments on AASHTO Guie Specifications for Seismic Isolation Design", Private communication to American Concrete Institute (ACI), Technical Committee USA. 4. Meeot R., (998) Energy Dissipation as Technological Answer to Highly Demaning Design Problems US-Italy Workshop, Columbia University at N.Y. - USA

15 5. Bertero, V.V., (988) Earthquake Hazar Reuction: Technical Capability - U.S. Perspective, Proc. n Japan-US Workshop on Urban EQ Hazars Reuction, Shimizu: Tokai University - Japan 6. Bertero, V. V., (99) Seismic Upgraing of Existing Structures, Tenth Worl Conference on Earthquake Engineering, Mari - Spain 7. Uang, C. M., (988) Use of Energy as a Design Criterion in Earthquake-resistant Design, UCB/ERC-88/8, University of California at Berkeley - USA 8. Zayas V., Stanley L., (995) Sliing Isolation Systems an Their Applications ASCE Annual Seminar, Unite Engineering Center, New York USA: Skinner, Robinson & McVerry (993), An Introuction to Seismic Isolation, John WILEY & Sons Lt., West Sussex PO9 IUD, Englan : 96-08