MICHAEL FARDIS Seismic Isolation Principles and Practice

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PROTA 28th Anniversary Symposium “Seismic Isolation Methods and Practices” Ankara, Feb. 28 & March 1, 2013

Seismic Isolation Principles and Practice in the Context of European Standards Michael N. Fardis University of Patras, Greece

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European Standards (ENs) Design standards : The Eurocodes Material standards (steel, ETAs: European Technical concrete, etc.) Approvals Product standards (Special isolation or (Structural bearings, dissipation devices, FRPs, Antiseismic devices, etc.) prestressing systems, etc.) Execution standards (e.g., standards for the execution of concrete or steel structures) Test standards

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THE EN-EUROCODES EN 1990 Eurocode: Basis of structural design

EN 1991 Eurocode 1: Actions on structures EN 1992 Eurocode 2: Design of concrete structures EN 1993 Eurocode 3: Design of steel structures

EN 1994 Eurocode 4: Design of composite (steel-concrete) structures EN 1995 Eurocode 5: Design of timber structures

EN 1996 Eurocode 6: Design of masonry structures EN 1997 Eurocode 7: Geotechnical design EN 1998 Eurocode 8: Design of structures for earthquake resistance

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EN 1999 Eurocode 9: Design of aluminium structures

INTERRELATION OF EUROCODES EN1990

Structural safety, serviceability and durability

EN1991

Actions on structures

EN1992

EN1993

EN1994

EN1995

EN1996

EN1999

EN1997

EN1998

Design and detailing

Geotechnical and seismic design

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• •

• •

• •

FLEXIBILITY IN THE EUROCODE SYSTEM Eurocodes (ECs) or National Annexes cannot allow design with rules other than those in the ECs. National choice can be exercised through the National Annex, only where the Eurocode itself explicitly allows: 1. Choosing a value for a parameter, for which a symbol or range of values is given in the Eurocode; 2. Choice among alternative classes or models detailed in the Eurocode. 3. Adopting an Informative Annex or referring to alternative national document. Items of national choice in 1-2: Nationally Determined Parameters NDPs National choice through NDPs: – Wherever agreement on single choice cannot be reached; – On issues controlling safety, durability & economy (national competence) & where geographic or climatic differences exist (eg. Seismic Hazard) For cases 1 & 2, the Eurocode itself recommends (in a Note) a choice. The European Commission will urge countries to adopt recommendation(s), to minimize diversity within the EU. If a National Annex does not exercise national choice for a NDP, designer will make the choice, depending on conditions of the project.

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EC8 Parts EC8 Part

Title

CEN date

1: EN1998-1 General rules, seismic actions, rules for buildings

Dec. 04

2: EN1998-2 Bridges

Nov. 05

3: EN1998-3 Assessment and retrofitting of buildings

June 05

4: EN1998-4 Silos, tanks, pipelines

July 06

Foundations, retaining structures, geotechnical 5: EN1998-5 aspects

Nov. 04

6: EN1998-6 Towers, masts, chimneys

June 05

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EN 1998-1:2004 General rules, seismic actions, rules for buildings No. of NDPs 1. General _ 2. Performance Requirements and Compliance Criteria 2 3. Ground Conditions and Seismic Action 8 4. Design of Buildings 7 5. Specific Rules for Concrete Buildings 11 6. Specific Rules for Steel Buildings 6 7. Specific Rules for Steel-Concrete Composite Buildings 4 8. Specific Rules for Timber Buildings 1 9. Specific Rules for Masonry Buildings 15 10. Base Isolation 1 Annex A (Informative): Elastic Displacement Response Spectrum 1 Annex B (Informative): Determination of the Target Displacement for Nonlinear 1 Static (Pushover) Analysis Annex C (Normative): Design of the Slab of Steel-Concrete Composite Beams at Beam-Column Joints in Moment Resisting Frames

Total:

_ 57

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EN 1998-2:2005: Bridges no of NDPs 1. Introduction 2. Performance Requirements and Compliance Criteria 8 3. Seismic Action 4 4. Analysis 2 5. Strength Verification 3 6. Detailing 6 7. Bridges with Seismic Isolation 4 Annex A (Informative): Probabilities related to the reference seismic action 1

Guidance for the selection of the design seismic action during construction Annex B (Informative): Relationship between displacement ductility and curvature 1 ductility factors of plastic hinges in concrete piers 1 Annex C (Informative): Estimation of the effective stiffness of reinforced concrete ductile members Annex D (Informative): Spatial variability of earthquake ground motion: Model and 1 methods of analysis Annex E (Informative): Probable material properties and plastic hinge deformation 1 capacities for non-linear analyses Annex F (Informative): Added mass of entrained water for immersed piers 1 Annex G (Normative): Calculation of capacity design effects Annex H (Informative): Static nonlinear analysis (Pushover) 1 Annex J (Normative): Variation of design properties of seismic isolator units 2 Annex JJ (Informative): -factors for common isolator types 1 Annex K (Informative): Tests for validation of design properties of seismic isolator units1 Total: 38

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Application of EN1998 rules for seismic isolation  The rules in Section 10 of EN 1998-1:  apply to buildings (and building-like structures).  The rules in Section 7 and Annexes J, JJ & K of EN 1998-2:  apply to bridges only;  are more detailed & up-to-date than those in EN1998-1.  The parts of Eurocode 8 on tanks, silos, pipelines, towers, etc:  do not include specific rules for seismic isolation;  for such structures the rules in EN 1998-1, EN 1998-2 on seismic isolation may be applied by analogy.

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From EN1990 (Eurocode – Basis of structural design): «Seismic design situation»:

G j 1

k, j

"" P"" AEd ""  2,i Qk ,i i 1

 Gk , j : Permanent actions (characteristic or nominal values) j 1 : Prestressing P

 2,iQk ,i : Quasi-permanent values of variable actions (live loads) AEd    AEk : Design Seismic action AEk: Characteristic Seismic action,   : Importance factor of structure

From EN1990:

AEk :«Reference Seismic action» («Reference» exceedance probability PR in design life TL of structure, or «Reference» return period TR).

ψ 2,i: Residential & office buildings: 2 =0.3;

Shopping or congregation areas in buildings: 2 =0.6; Storage areas in buildings:2 =0.8; Roofs: 2 =0.0 (but 2 =0.2 for snow, at altitudes >1000m or in Scandinavia); Bridges of motorways/roads of national importance:2=0.2 on uniform load; Bridges for intercity rail links or high speed trains: 2 =0.3; Other bridges and footbridges: 2 =0.

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Design seismic action in Eurocode 8 • The Reference Return Period of the Reference Seismic action is a NDP, with recommended value of 475yrs (Reference Probability of Exceedance in a design life of 50yrs: 10%). • The Reference Seismic action is described (in the national zonation maps) in terms of a single parameter: the Reference Peak Ground Acceleration (PGA) on Rock, agR. • The design ground acceleration on rock, ag, is the reference PGA times the importance factor: ag = γIagR

• In addition to the Reference Peak Ground Acceleration on Rock, the Reference Seismic action is defined in terms of the Elastic Response Spectrum for 5% damping.

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Elastic Spectra in Eurocode 8  Spectral shape: Given in National Annex as NDP as function of: Ground type (surface layers, a few tens of m) Earthquake Magnitude (possibly) deep geology below surface deposits.  Spectral shape: Has regions of:  Constant response spectral pseudo-acceleration  Constant response spectral pseudo-velocity  Constant response spectral displacement • Recommended: Two types of horiz. spectra from S.European data:  Type 1: High & moderate seismicity (distant EQs, Ms> 5.5);  Type 2: Low seismicity; local EQs (Ms< 5.5). (High amplification at low T; falls-off sooner with T).  Detailed ground classification (5 standard ground types defined on the basis of shear-wave velocity in top 30m, plus 2 special ones).  Site specific spectra required for Importance Class IV isolated buildings (“essential”) near potentially active faults giving Ms>6.5

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Standard Ground types

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Standard elastic response spectral shape • Ranges of constant spectral pseudo-acceleration, pseudo-velocity, displacement, start at corner periods TB, TC, TD. • Uniform amplification of spectrum by soil factor S (including PGA at soil surface, to Sag). • Damping correction factor: • Constant spectral acceleration = 2.5 times PGA at soil surface for horizontal, 3 times for vertical. • TB, TC, TD, S: NDPs

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Recommended horizontal elastic spectrum, Type 1, ξ=5% 4

E

D C

Se/ag

3

B A

2 1

0 TB TC Ground motion max acceleration: agS

1

max velocity: vg = agSTC/(2π)

TD

3 (s)

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max displacement: dg = 0.025 agSTCTD

Recommended vertical elastic spectra, ξ=5%

• Corner periods TB, TC, TD: NDPs • Recommended: – Independent of ground type (no

– – – –

data) TB = 0.05s TC = 0.15s TD = 1.0s Peak vertical ground acceleration • avg = 0.9ag, if Type 1 spectrum used; • avg = 0.45ag, if Type 2 spectrum.

• Vertical component mandatory: – for all isolated bridges; – if avg > 0.25g in buildings. • .

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Ground motion records for time-history analysis

• Historic or simulated records preferred over artificial ones. – Simulated records: from mathematical model of the source

dominating the seismic hazard (rupture event, wave propagation via the bedrock to the site and via the subsoil to the ground surface). – Historic records: from seismic events with magnitude, fault distance & mechanism of rupture consistent with those dominating the hazard for the design seismic action. Travel path & subsoil conditions of recording station should resemble those of the site. – Artificial (“synthetic”) records: mathematically derived from the target elastic spectrum (unrealistic if rich in all frequencies in the same way as the target spectrum; perfect matching of spectrum to be avoided). • Component records scaled so that the elastic spectra values are ≥ 90% of the code spectra (in the range of 1.5x to 20% of the fundamental period along the component). For pairs of horiz. components this is applied to SRSS of spectral values, taking 0.9√2 ~1.3. • ≥ 7 independent seismic events (component or pair time-histories) needed if analysis results for peak response quantities are averaged; • 3-6 if most adverse peak response from all the analyses is used.

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Strong discontinuity in heightwise distribution of lateral stiffness uncouples deformations in the superstructure from the ground motion

Superstructure

Isolation device/unit

Isolation system @ isolation interface

Isolation device/unit

Substructure

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Isolation strategies to reduce lateral forces on the superstructure damping↑

acceleration

Period T

resistance↓

Period T

displacement

Period ↑

displacement

acceleration

Period ↑

damping ↑

Flexible isolators lengthen period & reduce forces. Damping reduces displacements @ Period T isolation interface damping ↑

Limiting the force resistance of isolators reduces the force input (cf. capacity design). Damping reduces displacements @ Period T isolation interface

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Sd (m)

Sa (m/sec2)

Acceleration, Displacement & ADRS (Acceleration-Displacement) Type 1 elastic response spectra recommended in EC8 (for damping ξ = 5% & PGA = 1m/sec2 on Soil A – rock) 4

Soil A Soil B Soil C Soil D Soil E

3.5 3 2.5

Sd Sa T      

2

2 

2π

    

0.16 0.14 0.12 0.1 0.08

1.5

0.06

1

0.04

0.5

0.02

0

0

0.5

1

1.5

2

2.5

3

3.5

4

Period (sec)

• Spectrum reduction factor for damping in EC8:

  10 (5   %) • η ≥ 0.55 (ξ ≤ 28%) in buildings. • η ≥ 0.40 (ξ ≤ 57%) in bridges.

Sa (m/sec2)

0

4

Soil A Soil B Soil C Soil D Soil E

0 0.5 T=0.25 T=0.50

1

1.5 T=1.0

3.5

2

2.5 3 3.5 4 Soil A Period (sec) Soil B

Soil C Soil D

3

Soil E

2.5

T=1.5

2 1.5

T=2.0

1

PROTA T=3.0 T=4.0

0.5 0 0

0.02 0.04 0.06 0.08

0.1

0.12 0.14

Flexibility strategy • Common and relatively inexpensive. • Isolators ~elastic and re-centering: – Elastomeric (rubber) bearings: • with low damping (~5%, LDRB), or high damping a) (10-20%, HDRB). – If damping is low, supplemental damping (eg, fluid viscous dampers) may be used to reduce the displacements.

• Less effective if: – motion is rich in low-frequencies (eg, on soft soils); or – superstructure is flexible (high-rises); or – substructure is flexible (tall/flexible piers, flexible piles).

Force-limitation strategy • • • • •

More effective in the cases where flexibility strategy is less effective. Convenient for retrofitting superstructures with low force resistance. Isolators with force limitation (eg, flat sliding isolators). No re-centering. Supplemental damping may be used to reduce the displacements if energy dissipation (eg, by friction) is low.

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Combination: Flexibility after force limit • Most common in practice. • Possibly more expensive. • Isolators: – Lead-Rubber Bearings (LRB); – Units with spherical sliding surface(s); – Sliding surface with yielding (elasto-plastic) steel device; etc.

• Hysteretic energy dissipation after force limit exceeded. • Re-centering depends on details of the hysteretic loop. • Don’t need supplemental damping.

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Isolator hysteretic behavior idealized as bilinear Basic parameters F0: Force at zero displacement Ke: elastic stiffness Kp: post-elastic stiffness Derived parameters Fy: yield force = F0/(1-Kp/Ke) dy: yield displacement = (F0/Ke)/(1-Kp/Ke) Response (& design) values 1  ED  ξ eff    dd (or dbd): design displacement 2π  Fmaxd d  Fmax: maximum force = F0+Kpdd ED: dissipated energy/cycle at displacement dd (area inside hysteresis loop) = 4F0(dd-dy) ξeff: damping =(2/π)(Fy/Fmax-dy/dd)

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Analysis methods in EN 1998 Reference method (always applicable): – Non-linear time-history analysis in 3D.

Simplifications (subject to certain conditions): – Equivalent-linear analysis: • Multi-modal equivalent-linear spectral analysis; • Simplified equivalent linear ("fundamental mode" spectral)

 In bridges, the displacements & forces from any analysis are scaled up to reach at least 80% of the displacement at center of isolation system & of the total base shear from a fundamental mode analysis (per hor. direction, if the piers are tall or the longitudinal eccentricity of deck mass to the stiffness center of isolation system is > 0.1L).

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Non-linear time-history analysis • Only the isolators are modelled as nonlinear. • Two concurrent horizontal components, considering interaction of response in the two horizontal directions and the effects of overturning moments; masses moved by ±accidental eccentricities. • The effects of the vertical component may be computed separately and linearly - with the response spectrum approach and combined to those of the horizontal via the 1:0.3:0.3 rule. • Raleigh damping (C = αΚ + βΜ) should not interfere with the hysteretic damping of the isolated modes (with longest T); it should dampen-out very short periods: – β = 0; – α > ξT/π = 0.1x0.05/π = 0.0016 give ξ>5% for T10% due to loading rate or vertical load variations in the range of the design values; – when displacement of isolation system increases from 0.5dd to dd, its force increases > 2.5% of total superstructure gravity (for certain recentering).

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Iterative equivalent linear analysis

Assume dcd,a  dbd,i From device i monotonic F-d relation: Fmax,i  Keff,i=Fmax,i/dbd,i,

Fmax Fy

Kp

F0 Ke

ED: dissipated energy/cycle at dbd,i

Effective period & damping:

dy

 Iterations till dcd,r ≈ dcd,a. 

2 1 ( 1  p)(μ  1) ~const. π μ 1  p(μ  1)

dbd

ED

ΣED,i M 1 Teff  2π ,..ξ eff  K eff,i 2π ΣFmax,id bd,i dcd,r from displacement spectrum: 10 ≤ 0.55 in buildings ηeff  5  ξ eff (%) ≤ 0.40 in bridges

Nb: ξ eff 

Keff

for ↑μ

p=Kp/Ke:

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Multi-modal equivalent linear spectral analysis • Full modal analysis of complete structural system, separately in the two horizontal directions (also separately for the vertical). – to capture possibly significant contributions of higher modes. • Isolators i with their effective stiffness, Keff,i, from the fundamental mode method in the direction considered. • The substructure & superstructure w/ their normal stiffness (uncracked in bridges, 50% of uncracked in buildings). • Modal damping ξ = 5% in all modes with T < 0.8Teff – Teff from fundamental mode method; – ξeff of that mode for all modes w/ T >0.8Teff. • Torsional response due to accidental eccentricities computed statically and superimposed to results of modal analysis.

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Simplified equivalent linear analysis ("fundamental mode" spectral in EN1998-2)

• Superstructure rigid (forces on it proportional to spectral acceleration & mass) • Buildings & structures ≠ bridges: – 2 DOFs: uncoupled horiz. translations of isolation system stiffness center, with static torsional effects (about the vertical) on isolation system due to natural & accidental eccentricities, but neglecting overturning effects . – If horiz. eccentricities ex, ey between mass center & isolation stiffness center (incl. accidental)  7.5% of plan dimensions: • Torsional effects of eccentricities ex, ey (natural & accidental) by 2 2 multiplying isolator displacements by: δxi  1  e y yi / ry ,..δ yi  1  ex xi / rx (rx, ry: torsional radius of isolation system in x, y) – Consider vertical component & DOF (separately) only if avg > 0.25g. • Bridges: – 3 DOFs: 2 uncoupled horiz. translations & one vertical; – static torsional effects (about the vertical) on isolation system due to longitudinal eccentricity ex (natural & accidental) of transverse earthquake ex (y) by multiplying transverse isolator displacements etc by: δ yi  1  xi (r: radius of gyration of superstructure mass about vertical). rrx • Combination of components w/ SRSS or 1:0.3(:0.3) rule.

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Applicability of simplified equivalent linear analysis ("fundamental mode" spectral, in EN1998-2)  Conditions for bridges (EN1998-2) – Distance >10km from potentially active faults.

 Conditions for structures ≠ bridges (EN1998-1) – – – – –

Distance >15km from potentially active faults producing Ms  6.5 Max plan dimension  50 m Rigid substructure All isolators: above substructure elements supporting vertical load Effective period Teff 3s & ≥3-times1st mode period on fixed base.

 Additionally, for buildings (EN1998-1): – – – –

Superstructure stiffness regular & symmetric in plan Negligible rocking at the base of the substructure Vertical-to-horizontal stiffness ratio of isolation system Kv/Keff 150 1st vertical vibration period: Tv  0.1 s

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Low Damping Elastomeric Bearings (LDEB) in EN1998-2 • “Normal” laminated elastomeric Bearings per EN 1337-3 (rubber layers & bonded steel plates) • Narrow hysteresis loops, ~elastic. • Damping ξ not much larger than 5%. 50

200

45

Rubber Bearing Force(kN)

150 100 50 0 -50 -100 -150

PGA:25% (#5 Couple)

40 35 30

7.5% KM 15% KM 20% KM 20% 10KM 20% 0.1KM 20% 0.5KM 20% 2KM 20% 2KM (repeated) 25% KM 25% KM (repeated) 25% 10KM 25% 0.5KM 25% 2KM

25 20 15 10

0.5

PGA:25% (#1 Couple) PGA:20% (#4 Couple)

5

PGA:20% (#1 Couple)

-200

3.5 7.5% KM 15% KM 3 20% KM 20% 10KM 20% 0.1KM 2.5 20% 0.5KM 20% 2KM 2 20% 2KM (repeated) 25% KM 25% KM (repeated) 1.5 25% 10KM 25% 0.5KM 25% 2KM 1

G (MPa)

Equivalent Viscous Damping (%)

250

PGA:15% (#4 Couple) PGA:15% (#1 Couple) PGA:7.5% (#1 Couple)

0

PGA:7.5% (#1 Couple)

-250 -150

-120

-90

-60

-30

0

30

Displacement (mm)

• • • •

60

90

120

150

0

25

50

75

100

125

Strain (%)

150

175

200

0

0 225

25

50

75

100

125

Strain (%)

150

175

d

200

b Fb Horizontal stiffness Kb=GbAb/tb Ab: plan area, tb: elastomer thickness Gb: shear modulus (≈ 0.9 to 1.3 MPa, “scragging” if Gb>0

α=0.15, C=3000kN/(m/sec)α

0

0,5

1

1,5

2

Velocity (m/s)

}

λ(α)  2 α

2 α

Γ 2 (1  0.5α) , Γ (2  α)

0.01 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00

λ(α) 3.988 3.882 3.774 3.675 3.582 3.496 3.416 3.341 3.270 3.204

π

2.876 2.667

( ): gamma function

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If α>μd the period is T ≈2π√(Rb/g) ~ independent of the mass of the superstucture.

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Layout of isolators per EN1998 • To minimize twisting: effective stiffness center of isolation system as close as possible to the horizontal projection of the center of mass of the superstructure (met automatically for sliding isolators). • To minimize differential behavior of isolators: they should share ~uniformly the gravity loads of the superstructure. • Rigid diaphragm above isolation interface - in buildings, below it as well. • Sufficient space around isolating devices for inspection, maintenance, replacement. • Use dampers, shock-absorbers, etc., if shocks is an issue.

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Design properties of isolators used in analysis

 Nominal (“mean”) design properties are determined via tests of prototype devices, to confirm the range specified in the design.  The variation of design properties due to aging, temperature, contamination, cumulative travel/wear, scragging, etc, considered; design is carried out using both: • Upper Bound Design Properties (UBDP), for maximum forces in the superstructure & substructure; and • Lower Bound Design Properties (LBDP), for maximum displacements of the isolators & the superstructure.  The Bounds of Design Properties from tests, or modification (λ) factors (Annex J of EN15129, J & JJ of EN1998-2, from AASHTO Guide Specs).  Properties obtained for the quasi-permanent variable actions, but for temperature the frequent value is taken into account.  In bridges, multimode spectral or nonlinear time-history analysis may be based just on nominal design properties, if the displacements from Fundamental mode analyses with UBDPs & LBDPs differ from those for the nominal ones by < 15%.

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Increased reliability required of the isolation system Why? • The superstructure and the substructure have safety margins, because their ULS resistance is calculated from characteristic (5%-fractile) values of material strengths divided by the material partial safety factors (1.5 for concrete, 1.15 for steel). • By contrast, the ULS of isolators is defined by their nominal displacement capacity, without margins. • Thanks to redundancies in the superstructure and the substructure, attainment locally of their ULS resistance does not have catastrophic consequences. • By contrast, failure of isolators may be catastrophic for the superstructure.

How? • Multiplicative factor applied on the seismic displacement of the isolators from the analysis, dE, with recommended values: • γx = 1.2 in buildings; • γIS= 1.5 in bridges. dE,a = (γx or γIS) dE

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Verifications – “Full isolation” Superstructure: Verified at the ULS for forces from the analysis reduced by (behavior factor) q=1.5 for overstrength - even for nonlinear time-history analysis.

Foundations & substructure: Verified at the ULS for forces from the analysis (reduction or behavior factor q=1), except for bridges, where the piers are designed in flexure for q=1.5 & detailed as "limited ductile“(but in shear, q=1).

No detailing for ductility (except in bridge piers) or capacity design Horiz. clearance between superstructure & surrounding elements dEd= dE+dG+0.5dT  dG: due to permanent & quasi-permanent actions (shrinkage, creep)  dT: due to design thermal actions.

Isolating system: Accommodate total displacement:

dEd= dE,a+dG+0.5dT [dE,a=(γx or γIS)dE ] Interstory drifts in buildings for “damage limitation” earthquake:

PROTA 0.8  OK.  Total shear in X: Vdx/Vfx =6929/6292=1.10 >0.8  OK.  Total shear in Y: Vdy/Vfy =6652/6292=1.06 >0.8  OK.

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Displacement demand on isolators: dEd=dE,a+dG+0.5dT o o

• Design uniform ΔΤ: –45 C/+55 C. • Fixed point of thermal expansion/contraction at one pier: • Expansion/contraction LT: 140 m for abutments, 80 m for pier bearings. • Thermal movement 0.5dT at pier (+ towards abutment, - towards bridge center): • 0.5LTαΔΤ=0.5×80000×1.0×10–5×(-45) or ×(+55)= -18 mm or 22 mm • 0.5dT at the abutments: • 0.5LTαΔΤ=0.5×140000×1.0×10–5×(-45) or ×(+55)= -31.5 mm or 38.5 mm • Displacements dG due to (quasi-)permanent actions (shrinkage, creep): • At the piers: -3 mm • At the abutments: -8 mm • Offset displacements dG+0.5dT at the piers: • Towards bridge center: -3-18=-21 mm • Towards abutments: +22 mm • dG+0.5dT at abutments: • Towards bridge center: -8-31.5=-39.5 mm • Towards abutment: +38.5 mm  Total resultant displacement for combined components - LBDP analysis:  Longitudinal, pier bearings: dm =√[(1.5x193+22)2+(1.5x63)2] = 325 mm  Longitudinal, abutment units: dm =√[(1.5x193+39.5)2+(1.5x63)2] =342mm  Transverse, pier bearings: dm =√[(1.5x57+22)2+(1.5x207)2] = 329 mm  Transverse, abutment units: dm =√[(1.5x57+39.5)2+(1.5x207)2] = 335 mm

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Restoring capability of isolation system  Ratio dcd/dr ≥ 0.5 ? 

Maximum static residual displacement dr=F0/Kp



Post-elastic stiffness Kp =W/R



dr=F0/Kp=Wx(F0/W)/(W/R)=(F0/W)xR



Longitudinal direction - LBDP: dcd/dr=0.193/(0.051x1.83)=2.07>0.5.



Transverse direction - LBDP: dcd/dr=0.207/(0.051x1.83)=2.22>0.5.



Longitudinal direction - UBDP: dcd/dr=0.149/(0.09x1.83)=0.90>0.5.



Transverse direction - UBDP: dcd/dr=0.138/(0.09x1.83)=0.84>0.5.

 UBDP more unfavorable, as dr larger and dcd smaller.  Sufficient restoring capability without increasing the displacement capability of devices

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Thank you !

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