Earthquake Engineering by the Beach II Capri 8 10 June, 2013 Elia Voyagaki, Ioannis Psycharis & George Mylonakis

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1 Earthquake Engineering by the Beach II Capri 8 10 June, 013 Elia Voyagaki, Ioannis Psycharis & George Mylonakis Chair in Geotechnics & Soil-Structure Interaction University o Bristol, U.K.

2 m g.. R α (+)θ(t) α-θ.. I o θ m u g Ο.. u g

3 I & θ + mgr sin( α θ) = + mu& R cos( α θ), θ( t ) > 0 o g I θ& mgr sin( α + θ) = + mu & R cos( α + θ), θ( t ) < 0 o g Non-linearities Geometric, due to: (1) change o pivot points () trigonometric terms in D.E. Geometric & Material : Energy loss during impact

4 sin( α θ) ± α ± θ cos( α ± θ) 1 θ& p θ = + p u& / g p g α sgn( θ) p = 3 g /4 R ( characteristic requency )

5 Rocking Initiation Ground Acceleration Amplitude Frictional Coeicient Overturning Criteria Number o Impacts beore Overturning Role o Pulse Shape Eect o Number o Pulse Cycles Peak Rocking Response (Rocking Spectra) Comparison to Actual Near-Fault Records Solving the Non-Linear Rocking Equation

6 Overturning criterion or hal-cycle sine pulse A g t d A g Io π = + 1 α g mgr t η= d +π η=α g/a g, = ptd Approximate solution assuming that θ(t d ) =α, tanh[ (1-t up )] = 1

7 Equations o motion Period o ree rocking Geometric coeicient o restitution or rigid base Overturning criterion or hal- cycle box pulse η= 1 e A g t d

8 β β + u(t) & g d 1 β e, 0 t t/t d Ag = β (1 e ) 1 e β(1 t/t d ), d t t t t Jacobsen & Ayre 1958 d

9 .. u g A g t d t d t -A g.. u g (t)=a g [1-H(t-t d )+H(t-t d )]

10 Dimensionless Variables = p x t d η = α / α g normalized pulse duration uplit strength τ = t / t d θ/α normalized time normalized rocking angle

11 Analytical Solution θτ ( ) x1+ xh( τ 1/ ) + x3h( τ 1) = 1 α η β ( β )( 1 e 4 ) 1 1 β β + τ β β τ βτ x = β( + β)[1 η(1 e )] e β( β)[1 η(1 e )] e + (1 e ) 4β 1 x e e e e e βτ β (1 τ ) β ( τ 1/) ( τ 1/) = ( ) β ( ) x e e e β(1 τ) ( τ 1) ( τ 1) 3 = ( 4 β ) + β[( β) ( + β) ]

12 Special Cases β θτ () 1 1 = 1 cosh( τ) 1 cosh ( τ 1) 1 H( τ 1) α η η β 0 θτ ( ) 1 η = 1 (τ sinh[ ( τ )]) ( 4τ + sinh[ ( τ )]) H( τ ) α η η 1 (( τ 1) sinh[ ( τ 1)]) H( τ 1) η

13 Overturning Criterion θ a θ = α? Inadequate!

14 Overturning Criterion θ a & θτ ( ) = 0 (immobility) c θτ & ( ) = 0 (equilibrium) c θ& ( τc ) > strictly increasing point o inlexion! or, equivalently, θ = a at τ c

Areas o Saety & Overturning 1.0 1.0 η = α /α /α g 0.

15 Areas o Saety & Overturning η = α /α /α g S saety wall τ c τ c =1 O τ c =1/ = p t d β = π

16 Analytical Solution (all pulse shapes) w β β 1 e (+ β) 1 1 e β +β e ( β) η =

17 Analytical Solution (special cases) β - η = 1 e w Housner (1963) β = 0 η w = ln[e 1] β + η = w 0

18 Overturning Criterion ΑΝΤΙΣΤΑΣΗ UPLIFT STRENGTH ΣΕ ΛΙΚΝΙΣΜΟ η = α / α g β -π 0 π π -π β=0 π 0.4 π 4π = p t d ΔΙΑΡΚΕΙΑ PULSE DURATION ΠΑΛΜΟΥ 1 /

19 Areas o Saety & Overturning θ cr = a S 1 1 t c η = α /α g 0.6 t m = t d 0.4 O 3 O t c = t d / S 0. O = p t d t c = t d β = π

Õ «œ œàà««uplift STRENGHT η = α /α g 1.0 0.

20 Õ «œ œàà««uplift STRENGHT η = α /α g Limits o Overturning Ûˆ ÎÂÈ saety Ì ÙÒÔapplei appleôı 1 td<ti<td ti>td Mode 1 Mode 0 Ì ÙÒÔapplei appleôı ƒ ÀÃ œ PULSE DURATION = p t d

21 Rocking Response O S 0 1 time rocking angle θ/α pulse acceleration uplit peak response θ=α impact

22 Examples o Rocking Response ΛΙΚΝΙΣΤΙΚΗ NORMALIZED ΑΠΟΚΡΙΣΗ ANGLE : θ (t)/α όριο ανατροπής saety wall Ο 1 Ο Ο 3 saety wall.. θ 0 θ α. θ 0 τ up τ cr τ m TIME : ΧΡΟΝΟΣ : τ = t / t d S S 1 τ i 0 1

23 1 τ < τ < 1 up m Peak Response θ &( t ) = 0 m 1 1 β τ β + β β m τm ( + β)[1 η(1 e )] e + ( β)[1 η(1 e )] e = = + + β(1 τm) β ( τm 1/) ( τm 1/) e e ( e e ) τ m 1 τ m + β + β 1+ ( ) ( )[1 (1 )] β β β + e e + β e + β η e 1 = ln + β 1 β β β 4 e e ( β) ( + β)( β + e )[1 η(1 e )]

24 Rocking Spectrum saety wall 1.0 overturning θ max / α β = π τ m =1 0.7 η = = p t d 0.6 η=

25 τ m < τ < 1 i ( )( ) Time o Impact θ ( t ) = 0 i 1 1+ β β β τ β β i τi η 1 e 4 β = β[( + β)[1 η(1 e )] e ( β)[1 η(1 e )] e ] + + e e e e β(1 τi) β ( τi 1/) ( τi 1/) 4β β ( ) τi 1 η y = 1 e 4 1 β β ( )( β ) τ i = 1 ln + β 1+ β ( ) ( )[1 (1 )] β = + + β + β η y e e e yy+ y y y 3 + β 1 β ( ) ( )[1 (1 )] β = + β + + β η 3 y e e e

26 Time o Impact (Special Cases) β τ i 1 ( e 1) e (1 η) η = ln 1 η e β 0 η 1 η sinh ( τi ) sinh ( τi ) + ( τi + 1) = 0 τ = η η η η ( e e + e )( e e + e ) + 1 ln e e + e i η

8 8 Ù i = t i / t d β = π 0.1 6 0.3 0.4 4 0.5 0.6 1 0.7 0 η = 0.8 0 4 6 8 10 6 0.3 0.6 4 0.8 0.9 0.95 0.97 1 η = 0.

27 8 8 Ù i = t i / t d β = π η = η = β Ù i = t i / t d η = 0.9 β 0 β = π η = = p t d = p t d

28 1.6g PCDa EMOb E04b 0.38g E05b E06b 0.705g 0.463g E07b 0.96g 0.594g NPSa 0.36g DOWa 0.1g 0.455g PTSa 0.439g STGa 0.51g 0.515g ERZb LUCa 0.571g JFAa RRSa 0.897g SCGb 0.89g 0.77g 0.455g 0.838g 0.416g SCHa NWSa 0.46g TCU068b 1 g 0.333g TCU075a TCU076b 10 s

29 η w =-/ ln[e / -1] Õ UPLIFT œ «STRENGTH œ œàà««η = α /α g η w =3/ η = α /α g, rect rectangular pulse ÔÒËÔ ÌÈÍ Ú apple ÎÏ Ú PCDa EMOb E04b E05b E06b E07b NPSa DOWa PTSa STGa ERZb LUCa JFAa RRSa SCGb SCHa NWSa TCU068b TCU075a TCU076b w =3/4 ƒ PULSE DURATION ÀÃ œ = p t d ƒ ÀÃ œ = p t d PULSE DURATION

30 t I θθ && dt+ mgr sin[ α θ] dθ = mr u& cos[ α θ] & θdt o θ to θo to change in kinetic energy change in potential energy t g work o base acceleration 1 t [ & ] t + [cos[ ]] t = & cos[ ] o to to g t I θ mgr α θ mr u α θ & θdt critical system t = t, t, θ = 0, θ = α, & θ = & θ = 0 o up o cr o cr mgr(1 cos α) = mr u& cos[ ] & g α θ θdt t up o

31 α η = (, )cos[ (1 )] 1 cos & & w ψ g β τ α χ χdτ α τ up ψ& ( β, τ ) = u& / A, (,,,, ) = ( )/ g g g η, lin = ψ & ( β, τ) & χdτ w τ up g χβτη α θτ α α cos[ α(1 % χ)] η, = w non lin ψ & g( β, τ) & χdτ 1 cosα w τ up

32 ηw, non lin α α χ η wlin, = cos[ (1 %)] (1 cos α) ηw,non-lin / ηw,lin α 0 10 o 0 o 30 o 45 o %χ %χ lim α : ÏÔlÒÂÚ degrees %χ lim %χ = 0.1

33 ηw, non lin α α η w, lin = cos[9 /10] (1 cos α) η w, non lin η w, lin = 1 α 5 ηw,non-lin/ηw,lin = β α : ÏÔlÒÂÚ degrees β = 0 β = π

34 Νovel analytical solutions or rocking o rigid blocks were derived They cover an ininite number o pulse shapes ranging rom a perect spike (β + ), to a perect rectangle (β - ). Rocking response was ound to exhibit complex patterns even or highly idealized pulses like the ones at hand. Response is controlled by two dimensionless pulse: dimensionless pulse duration and dimensionless uplit strength η

35 Analytical ormulae were derived or distinguishing between Sae (S) and Overturning (O) regions Regions O may be divided into : O 1 corresponding to instability in response during rising pulse, O corresponding to instability in response during decaying pulse, and O 3 where critical angle is attained ater pulse. Regions S may be subdivided into:s 1 or peak rocking response attained during pulse and S or peak response attained ater pulse. O 1 & O are applicable to ground motions with time histories that extend beyond the end o the pulse.

A complete overturning criterion was established by means o a simple closed-orm expression Simple solutions or peak rocking response were derived in terms o peak

36 A complete overturning criterion was established by means o a simple closed-orm expression Simple solutions or peak rocking response were derived in terms o peak angular displacement and corresponding time. The ull non-linear equations o motion were investigated; they always lead to more stable systems than the linearized equations.

37 Voyagaki E., Psycharis I.N., Mylonakis G. (013) Rocking response and overturning criteria or ree standing rigid blocks to single lobe pulses Soil Dynamics & Earthquake Engineering, 46:

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