• Macmurdo Noted That During the 1819 Runn of Cutch, India Earthquake Buildings on Soil Were More Affected Than Those on Rock

OBJETIVOS DE ESTA PARTE DEL CURSO

1. Hacer un breve de observaciones de efectos de sitio, con énfasis en suelos muy blandos como los de la zona blanda de la ciudad de México o de la ciudad de Guayaquil;

2. Presentar la enorme diferencia en formas espectrales de movimientos Curso de Diseño de Estructuras Sismorresistentes registrados en suelo blando;

Quinta Parte 3. Presentar factores de reducción para estructuras cimentadas en suelo blando; Análisis y Diseño Simplificado de Edificios Basado en Desplazamientos

4. Presentar cocientes de desplazamientos inelásticos a elásticos para estructuras cimentadas en suelo blando.

PROF. EDUARDO MIRANDA

DEPARTMENT OF CIVIL & ENVIRONMENTAL ENGINEERING STANFORD UNIVERSITY

Ejemplos de Documentación de Efectos de las Ejemplo de Estudio Análitico de Efectos de Suelo Blando Condiciones del Suelo en el Nivel de Daño

• MacMurdo noted that during the 1819 Runn of Cutch, India earthquake buildings on soil were more affected than those on rock • The effect of soil conditions on the intensity of ground motions was also presented by Wood in 1908 in his study of the distribution of damage and apparent intensity of shaking during the 1906 San Francisco earthquake. • Multiple authors noted important effects of soil conditions on building and pipeline performance in the 1923 Kanto earthquake in Japan.

1 27 de Julio de 1957 27 de Julio de 1957

(Source: Steinbrugge collection, NISEE, UC Berkeley) http://simerida.com/img/Angel%20de%20la%20independencia%2001.gif 2014 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA

Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda by soil deposits and comparison of the computed time histories of surface accelerations and associated response spectra with those of the recorded surface accelerations form the subject of this paper. In addition, the influence of the amplitude of the base rock motions on the response of the soil deposit underlying one of the recording stations is discussed and illustrated. GROUND MOTIONS DURING 1957 SAN FnANC~SCO EARTHQUAKE 2021 ANALYTICAL PROCEDURE comparison of the response spectra for the computed motions with those previously GROUND MOTIONS DURING 1957 SAN FRANCISCO EARTHQUAKE 2023: In cases where the surface motions are due primarily to the upward propagation of obtained by Hudson and Housner (1958, 1959) for the recorded motion. Both accelera- Primerashear evidencia waves from an instrumentalunderlying rock surface, de the losseismic efectos response of dea soil sitiodeposit tionPrimera and velocity response evidencia spectra, for a damping ratio instrumental of 2.5 per cent, are shown inand dethe computed los time efectos history of ground surface de acceleration sitio is shown in Figure 9 Figure 8 for the two motions. The velocity spectra show reasonably good agreementwhere it is compared with the recorded acceleration. Again the computed accelerations may be evaluated using an equivalent linear elastic lumped-mass method of analysis. for periods less than about 0.4 seconds, for periods between about 0.55 and 1.1 seconds,compare well with the recorded values. This comparison is better illustrated in Figure and for periods greater than 1.6 seconds. The acceleration spectra show a high degree11, where the acceleration and velocity response spectra for the computed ground of agreement for almost the entire range of periods. motions are compared with those for the recorded motions previously obtained by

O.IO F Recorded Surface ' 1 Acceleration Recorded Surface o.o5r- O.lO ~- Acceleration ~J o o-o5i- ~ 0.05 v 80.5 Oakland City Hall .... 0 I 2 5 4 5 6 7 0.10 0 I 2 :5 4 5 6 7 8 0.10 [ Computed Surface 005" Accelerotion 0.10 r Computed Surface Southern Pacific BId9,,,~~ L 005 / ~ Acceleration ~J AlexanderBldgg~, ~" Surface Motion -----" Surface Motion = State B,dg.~//// o.o t Sandy fill ~ ~I" //z ~ -- ./" < o.,% i ~ ~ ~ ~ ~ <010 1 , , , , , , , , • 0 I 2 3 4 5 6 7 8 GatePork,,, 20 Time-Seconds Soft cloy 40 Time -Seconds ,' .of:" Medium clay 40 80 \% Z' ,?,' .-" Clayey and Silty Sand \ iI 'x./// / Stiff clay "$6o "~ 120 \ Sand i ..& \ Stiff clay C~ 160 \ # .-" a80

20C Estimated center ioc of rupture zone \ instrumental 0 Epicenter Sand and gravel 240 N 120 Sand ? 0]0 [ x ,~ O.lO [ 280 k ...... Base Motion 0 I 2 3 4 5 ! 14c 285, ~4e~z,~'/4eff,@f/Xez/#Y/~/4~'Y4 Base Motion --- oo5°/ ..... Ilv t" ...... ~ I.... ~IV~F'I'" °" ,r...~ ...... k o.os[ i 1 ' j k SCALE IN MILES 0.,% i i ~ ~ ~ ~ -~ ~' Time-Seconds l k Time-Seconds FIG. 9. Recorded and computed surface accelerations--Alexander Building site. FIG. 12. Recorded and computed surface accelerations--Southern Pacific Building FIG. 1. Location of U.S.C.G.S. strong motion accelerometer stations relative to Ss,n Andreas site. Fault--San Francisco earthquake of March 22, 1957. (2) Alexander Building Site. The soil conditions underlying the Alexander Building site, as shown in Figure 9, consist of about 100 ft of clayey and silty sand and 40 ft ofHudson and Housner (1958, 1959). The high degree of agreement is readily appar- sand overlying bedrock. Since the clayey and silty sand dominates the response of theent. The soil deposit, which may consist of several sublayers of varying properties, is (3) The soil conditions at the Southern Pacific Build- (After Idriss and Seed, 1968) deposit to the base motion, the analysis was performed using the approximation that Southern Pacific(After Building Site:Idriss and Seed, 1968) idealized as shown in Figure 2 by a series of discrete (lumped) masses interconnected this material extended to the full 140 ft depth. Furthermore, since the presence ofing site are shown in Figure 12. A sandy fill extends to a depth of about 20 ft and is clayey and silty components in the soil are likely to make it more deformable than underlaina by a 35 ft layer of soft clay followed by a 50 ft layer of medium clay. Below this is a 120 ft layer of stiff clay interbedded with a 10 ft layer of dense sand at a depth by springs that resist lateral deformations. These springs represent the stiffness prop- clean sand, the shear moduli of the deposit were considered to be about sixty percent Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 of 125 ft. The stiff clay is underlain by a layer of© very2017 Prof. dense sandEduardo Miranda and gravel which ex- of those indicated by the data for clean sands shown in Figure 3. From the data shown erties of the soil between any two discrete masses. tends to bedrock at a depth of about 285 ft. The shear moduli for these various layers The equations of motion of the dynamic system shown in Figure 2 can readily be in Figure 5, it was estimated that the base rock motions at this site would be similarwere estimated from the data in Figures 3 and 4 as follows: to those recorded at Golden Gate Park but with ordinates multiplied by a factor of (a) Moduli for the sandy fill and the sand lens at a depth of 125 ft were determined written. The solution of these equations for the duration of the applied base motion 0.65. using the average data for sands shown in Figure 3. provides response vlaues for each level at each instant of time. These response values The response of the deposit was calculated using the lumped-mass solution in- include acceleration, velocity, displacement, shear strain and shear stress. Details of this analytical procedure and a computer program for performing the necessary computations are available elsewhere (Idriss and Seed, 1967, 1968). The actual soil properties (dynamic shear modulus and damping ratio) utilized for 2 each sublayer of the deposit are chosen on the basis of the variations of these properties with the shear strains developed. These variations are ascertained from the results of vibratory or cyclic loading tests performed on representative soil samples. Typical test data for fine to medium sand and for saturated clay (Seed and Idriss, 1968) are RESPONSE SPECTRA ON VERY SOFT CLAY 865

The period corresponding to the maximum ordinates and the general shape of the spectra of moderate to strong earthquakes on soft cohesive ground can be predicted using the linear viscoelastic theory of multiple wave reflections. For mild or very strong earthquakes further evidence is required.

ACKNOWLEDGMENT Thanks are given to Emilio Rosenblueth and Luis Esteva respectively Director and Profes- 214 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA 214 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA sor of the Institute of Engineering, UNAM, for their suggestions and revisions to improve the first hard sand stratum, Tarango Sand 1, at 33 meters depth and the total depth of original manuscript. The author is also grateful to the Director and personnel of the Western firstthe rigid hard foundationsand stratum, structure Tarango is 13 Sand meters. 1, at The 33 metersweight depthof the andfoundation, the total basement depth of Data Processing Center, UCLA, where most of the computations were made, to Mr. Abelardo thefloor rigid and foundationground floor structure approximately is 13 meters. compensate The weight the excavated of the foundation, effective basementpressure. Rodriguez-Sullivdn and Mr. Josd Osorno, of Agropecuaria Industrial, where the program floorThe totaland groundweight offloor the approximatelybuilding is 23,130 compensate metric tons, the whereasexcavated the effectivetotal uplift pressure. water testing was performed~ to the Director and personnel of the Computing Center of the Uni- Thepressure total is weight approximately of the building 11,150 is tons. 23,130 Hence, metric the tons, net whereas weight onthe thetotal piles uplift is aboutwater Sismos de Mayo de 1962 Sismos de Mayopressure11,980 tons. is approximately de 1962 11,150 tons. Hence, the net weight on the piles is about versity, where part of the computations were done and to the personnel of the Numerical 11,980The buildingtons. has the following computed periods of vibration (ref. 5) : T1 = 3.66, Analysis Section of the Institute of Engineering for their enthusiastic cooperation. T2The = 1.54,building T3 = has 0.98, the T4following = 0.71. computed These were periods computed of vibration under the (ref. assumption 5) : T1 = 3.66,that T2 = 1.54, T3 = 0.98, T4 = 0.71. These were computed under the assumption that SECONDS APPENDIX 1--INSTRUMENTAL CHARACTERISTICS OF THE FOUR EARTHQuAKEs 5 I0 15 20 25 30 2 35 40 45 50 55 60 Amax = 50 cm/s SECONDS STUDIED 5 I0 15 20 25 30 35 40 45 50 55 60 A AALA a a N I0° 46'W E A AALA a a N I0° 46'W Coordinates of the o ~oE / i i I ! , I ~ I Date (Mexico City) Epicenter Reference o ~oE / i i I ! A.P. EARTHQUAKE , MAY I II 1962 ~ I

/ A.P. EARTHQUAKE MAY II 1962

N.79 ° 14' E December 10, 1961 20,1°N 99.1°W 80 km 15 kin 4-5 5 Figueroa (1963) N.79 ° 14' E

May 11, 1962 16.6°N 99.4°W Figueroa (1963) NOTE~ MOTION 8EleMO@RAPHSTARTED TO RUN. NOTE~ DATA OBTAINED FROM AN ENLARGEDCOPY OF. THE 1~4NAL RE6~ID. 17.0°N 99.7°W -- 25 km -- 7-7~ USCGS (1962) MOTION 8EleMO@RAPHSTARTED TO RUN. DATA OBTAINED FROM AN ENLARGEDCOPY OF. THE 1~4NAL,SECONDS RE6~ID.

,SECONDS May 19, 1962 17.3°N 99.4°W 240km [ 30 kin 6 6.5 Figueroa (1963) 17.2°N 99.5°W -- 20 km -- 7-7~ I USCGS (1962) N 10046 ' W N 10046 ' W November 30, 1962 17.3°N 99.4°W 250 km 25 km 4-5 5½ Figueroa (1963) 17.4°N 99.6°W -- i 51 km -- 5~-5~ USCGS (1963) I r F I i I , I r F I i I , N 79014' E N 79014' E !~EFERENCES Berg, G. V. FIG. 5. Plots of the accelerations recorded at the Alameda Park site. (After Bustamante, 1964) FIG. 5. Plots of the accelerations recorded at the Alameda Park(After site.Zeevaert , 1964) 1963. "A Study of Error in Response Spectrum Analyses", Preprints of the First Chilean the structural frame alone contributed to the stiffness of the building; the curtain Sessions on Seismology and Earthquake Engineering, Chile, B1.3, 1-11. theand structuralpartition wallsframe were alone considered contributed as tofloating the stiffness elements of andthe building; added only the mass curtain to Bustamante, J. I. and Prince, J. andthe structure.partition wallsIn the were calculation considered of the as periodsfloating it elementswas assumed and that added the only base mass of the to 1963. "Correcci6n de los acelerogramas de cuatro macrosismos obtenidos en la Ciudad de thebuilding structure. was rigid. In the calculation of the periods it was assumed that the base of the Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistentebuilding, Ecuador Julio 3 was rigid. -6, 2017 © 2017 Prof. Eduardo Miranda Mdxico", Bol. Soc. Mex. Ing. Sis., Mdxico, 1 : 61-82. ACCELEROGRAMS Figueroa, J. The accelerograms were digitizedACCELEROGRAMS every 0.1 sec and the ordinates were scaled to 1963. Personal communication to the author. theThe nearest accelerograms 0.1 mm. Thewere sensitivities digitized every of the 0.1 instruments sec and the were ordinates taken wereto be scaled exactly to Jennings, P. C. the12.5 nearest and 25.0 0.1 gals/mm mm. The for sensitivitiesthe LAT and of AP the instruments, instruments respectively. were taken Theto be accelero- exactly grams,12.5 and to 25.0 an enlargedgals/mm scale,for the are LAT shown and in AP figures instruments, 5 and 6. respectively. A count was The made accelero- of the 1962. "Velocity Spectra of the Mexican Earthquakes of 11 May and 19 May 1962", Earth- grams, to an enlarged scale, are shown in figures 5 and 6. A count was made of the quake Engineering Research Laboratory, California Institute of Technology.

218 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA Marsa], R. J., and216 Mazari,BULLETIN M. OF THE SEISMOLOGICAL SOCIETY OF AMERICA The absolute acceleration spectrum is given by 1962. El Subsuelonumber ofde waves la versus Ciudad period for well dedefined M~xico, waves having an Secondamplitude greater Edition, Instituto de Ingenierla, Uni- Sismos de Mayothan 5 gMs. A plot de of this data 1962is shown in figure 7 where it may be seen that the Sismos de MayoEARTHQUAKES de 1962 OF MAY llTH AND 19TH, 1962 IN MEXICO CITY 223 versidadmost National frequent period ofAut6noma the motion are in the regionsde M~xico.from 2.0 to 2.5 see and near S(, = COd [# a(r)e -x'°~(t-~) sin cod(t -- r) dr 0.9 see. It was found that the amplitudes of the larger period waves were approxi- (7) Newmark, N. M. mately 50 % greater than those of the shorter periods. The maximum acceleration recorded at the LAT site during the two earthquakes was 25 gals and at the AP + V'~2X fo a(r)e-X'°"(t-') cos coe(t -- r) dr ..... COMPARISON RELATIVE VELOCITIES 1952. "Computationsite the maximum of wasDynamic 48.75 gals. Structural Response in the Range Approaching Failure",1 CALTECH AND BULL 220 BULLETIN OF THE SEISMOLOGICAL SOCIETY OF AMERICA Proc. SymposiumEARTHQUAKES on OF MAYEarthquake llTH AND 19TH, 1962 and IN MEXICO Blast CITY Effects219 on Structures, 114-129. MAY II 1962 N-S Rosenblueth, E. 50 ALAMEDA PARK 606o / LATINO AMERICANAALANEDAPARK, TOWER 4o -- ,~ -0.10 30 LATINO AMERICANAlAY II, 1962.TOWER 1953. "Teoria del disefio sismico sobre mantos blandos", Ediciones ICA, B, 14, MAYM~xico, II, 1962. N I0 ° 46' W MAY 19, 1962. o / \\ 50 MAY II, 1962. N 9OE 30 pages~o (August 1953). \ N 81° W 50i4o'!}° /~ N 79 ° 14' E ~oE ?~\// \ CALTECH Mayo 11, 1962 40 \ 40 40 ~ \~.-0 ,2 M6.7 30 t,' ":L,, 30 330 km J X:oos 2O 120 ~j /// 20 )~-o lo X-o.lo / Z/),.=0.20 SECONDS // / \ I 2 3 4 SECONDS I 2 3 4 0 FIG. 8.~ C.I.T. relative displacementSECONDS spectrum curves for Alameda Park, 11 May 1962, I 2 3 4 I 2 component NllW.3, 4 Fro. 10. C.I.T. relative displacement spectrum curves for Alameda Park, Ii May 1962, FIG. 13. C.I.T. relative displacementcomponent spectrum N79E. curves for Latino Americana Tower, FIG. 11. C.I.T. relative displacement spectrum curves for the Latino Americana Tower, 19 May 1962, component N9E. I0 -- ~-'= --- 11 May 1962, component N81 W. SPECTRUM CALCULATIONS The relative displacements spectra, Sd, are shown in Figs. 8-15. The calculations From the accelerograms the spectra are obtained as the response of a one degree were done by Mr. Paul C. Jennings at the California Institute of Technology, of freedom system to the earthquake. Let the accelerogram be represented as a (C.I.T.), on an IBM 7090 digital computer using the digitized data furnished by function of time by a(r), then it is well known that the maximum relative displace- 60the author, (reference 6). The spectra were obtained by integrating the equation , , I , I LIT t , i 30 II ALAMEDA PARK ALAMEDA PARK ment (reference 7), of a one degree of freedom system with small damping is of motion MAY 19, 1962. MAY 19, 1962. ;I , I so N I0 ° 46' W 50 2 + 2Xco2 +r~ co2x = --a(r) N 79 ° 14' E (s) I 2 3 4 5 0 7 Se = 1 f0 t a(r)e -x~"(t-') sin o~e(t -- r) dr m~x, (3) O~d Mayo 19, 1962 from which PERIOD , seoonds where ~0 40 zd = max Is I FIG. 17. BULL relative velocity spectrum curves compared with C.I.T. relative velocity M6.5 r: is an integration variable and, spectrum curves, h = 0.1, components N-S. t:& is the time at which the integral is evaluated 240 km X: is the fraction of critical damping 30 S. = max I2 I (9) “The spectrum amplitudes of the AP accelerograms were MAY II 1962 20 2o found to be consistently larger thanN I0 ° 46' W those of the LAT site % 160 A. P. only 600 meters distant.” x=O, I0 (After Zeevaert, 1964) I I 2 3 4 I 2 3 4 -- CALTECH i o° Fro. 12. C.I.T. relative displacement spectrum curves for Alameda Park, 19 May 1962, FIG. 14. C.I.T. relative displacement spectrum curves for Alameda Park, 19 May 1962 component N11W. (After component N79E.Zeevaert , 1964) 5

0 2o ~> ~ so I- < iO ~: 3

"~v2

o L ~ [ ..... [ • I 2 ,5 4 5 6 PERIOD, ~eoonci8 FIG. 18. Relative velocity spectrum, S~, pseudo-velocity spectrum, R,, relative displacement spectrum, Sd, and absolute acceleration spectrum, Sa, for Alameda Park, 11 May 1962, k = 0.1, components N-S. Sismos de Mayo de 1962 Sismos de Mayo de 1962

Sa#[g]# 0.20#

0.18# #N#79º#14'#E# 0.16# #N#10º#46'#W# 0.14#

0.12#

0.10#

0.08#

0.06#

0.04#

0.02#

0.00# 0# 1# 2# 3# 4# Period#[s]# (After Zeevaert, 1964)

Espectros del sismo del 11 de Mayo de 1962 en la Estado del Arte sobre efectos de sitio a finales de los Alameda vs espectros de diseño de 1966 70’s y principios de los 80’s

Sa#[g]# 0.20#

0.18# #N#79º#14'#E# 0.16# #N#10º#46'#W# Sa/PGA=3 0.14# #1966#Code,#Shear#Walls# Sa/PGA=2.3 #1966#Code#,#Frames# 0.12#

0.10#

0.08#

0.06#

0.04#

0.02#

0.00# 0# 1# 2# 3# 4# Period#[s]# (After Zeevaert, 1964) (After Seed et al. 1976) Dichos estudios no reflejaban las características de los espectros de respuesta calculados a partir de registros de suelo blando!!!

4 1985 Mexico City: 1985 Mexico City:

ACC. 1.20 [cm/s2] 200 Firm Site CU EW COMP. 1.00 100 Soft Soil 0 0.80 -100 0.60 -200 200 0.40 Acceleration [g's] Acceleration 100 SCT EW COMP. 0.20 0 -100 0.00 -200 0 1 2 3 4 5 0 50 100 150 200 PERIOD [s] TIME [s] Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

Espectro de Respuesta SCT-EW-85 vs Espectro de Diseño Espectro de Respuesta SCT-EW-85 vs Espectro de Diseño

A [cm/s2] 1.0 A [cm/s2] 1976 CODE 1.0 2004 CODE 0.8 1976 CODE SCT E-W 1985

0.8 SCT E-W 1985 0.6

0.6 0.4

0.4 0.2

0.2 0.0 0 1 2 3 4 5 0.0 Period [s] 0 1 2 3 4 5 Los espectros de diseño, aun los propuestos después del sismo, tampoco Period [s] reflejaban las características de los espectros de respuesta calculados a partir de registros de suelo blando!!! Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

5 EFFECT OF SITE CONDITIONS EFFECT OF SITE CONDITIONS October 17, 1989 Loma Prieta earthquake 0.08 g (Rock) October 17, 1989 Loma Prieta earthquake

0.26 g (Bay mud)

0.25 g (Alluvium)

0.06g (Rock)

(Modified from Boore et al. 1990) (Modified from Boore et al. 1990)

EFFECT OF SITE CONDITIONS EFFECT OF SITE CONDITIONS

October 17, 1989 Loma Prieta earthquake Treasure Island

PGA 0.16g

Yerba Buena Island

PGA 0.06g

(After Seed et al, 1990) (After Boore et al. 1990) (After Kramer, 1996) Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

6 EFFECT OF SITE CONDITIONS EFFECT OF SITE CONDITIONS

Treasure Island Yerba Buena Island

PGA 0.16g PGA 0.06g

Treasure Island

Yerba Buena Island PGA 0.16g PGA 0.06g

1989 Bay Area: 1989 Bay Area:

ACC. [cm/s2] Sv [cm/s] 300 RINCON HILL NS COMP. 150 120 0 100 Rock -150 80 -300 Soft 300 60 Soil 150 FOSTER CITY NS COMP. 0 40 -150 20 -300 0 0 20 40 60 0.0 1.0 2.0 3.0 TIME [s] PERIOD [s] Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

7 Latitude 82 MD TX Latitude 86 N N

16 Tepeyac 19.50 100 s IM 17 ES 19.50 64 14 CIRES Insurgentes CENAPRED 27 Reforma I de I (UNAM) 15 cm 39 Earthquake: 14/09/95 Periférico 23 55 08 09 C. Interior M = 7.3 19.45 28 19.45 52 76 44 01 62 Aeropuerto 48 12 Component: NS 07 25 50 58 72 59 02 05 45 ZA CH 56 06 41 Cd. Neza RM VG UK 03 31 RO 04 49 43 20 19.40 TY 57 42 19.40 30 SC ZA Firm soil 10 Zaragoza 11 32 53 29 Transition 46 68 47 P. d e l 78 CD CF VI 51 Marqués Soft soil 21 24 19.35 19 19.35 CY Ixtapalapa 40 18 22 Firm soil C. de la 37 CU 84 Estrella Transition CN 54 38 33 Soft soil 19.30 74 19.30 13 Tlalpan 80 CM TD 15 Periférico TB 35 36 Xochimilco Tláhuac 19.25 19.25 5 km Sierra de las Cruces 34 26 Sn. P. Actopan -99.20 -99.15 -99.10 -99.05 -99.00 -98.95 -99.20 -99.15 -99.10 -99.05 -99.00 -98.95 (After E. Reinoso) Longitude Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 Longitude © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

MEXICO CITY Latitude EFFECT OF SITE CONDITIONS N SAN FRANCISCO BAY AREA MEXICO CITY 19.50 Soft soil Soft soil

1985 EARTHQUAKE SCT PGA 0.16g

19.45 x 2.7 x 4.6 Rock Firm soil

19.40

S [g] Sa [g] a 1.0 1.0 19.35 Soft soil 0.8 0.8

0.6 0.6 19.30 UNAM 0.4 x 3.9 0.4 Soft soil

PGA 0.03g 0.2 0.2 x 11.6 19.25 x 2.7 Firm soil x 4.6 Firm soil 0.0 0.0 0 1 2 3 4 5 0 1 2 3 4 5 PERIOD [s] -99.20 -99.15 -99.10 -99.05 -99.00 -98.95 PERIOD [s] Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 Longitude © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

8 Variaciones en espectros de respuesta en zona de lago Comparación con las recomendaciones de Seed et al. para el diseño de estructuras en suelos blandos

Elastic response spectra at three soft sites in Mexico City

(After Seed et al. 1976)

Propuesta de binormalización espectral INELASTIC STRENGTH SPECTRA

Inelastic spectra, by definition, are smaller than their elastic counterpart.

Cy El Centro NS, 1940 1.0 µ = 1.0 0.9 µ = 3.0 0.8 0.7 0.6 Strength required to maintain the system elastic 0.5 0.4 Strength required to avoid µ > 3 Sa/PGA ≈ 5 0.3 T / Tg 0.2 0.5 1.0 1.5 2.0 0.1 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD [s]

9 Estimación de fuerzas laterales con fines de diseño Espectros no lineales calculados de registros de suelos blandos

Con fines de diseño, podemos estimar ordenadas espectrales no lineales a partir de factores de reducción Q’ (R en muchos otros países)

S F = a W y R

S F = a W y Q'

Factor de Reducción de Resistencia Factor de Reducción de Resistencia

El factor de reducción por comportamiento no lineal (Q’ o R) se define como el cociente de la resistencia lateral necesaria para permanecer en SUELO FIRME SUELO BLANDO intervalo elástico entre la resistencia lateral necesaria para controlar la demanda de ductilidad a un cierto nivel.

F (µ =1) Q’ = y Rµ = Fy (µ = µt )

O bien

C (µ =1) Q’ = y Rµ = C y (µ = µt )

10 Factor de Reducción de Resistencia SUELO BLANDO Una observación importante:

Obtener una buena estimación del espectro de respuesta elástico NO ES SUFICIENTE CON FINES DE DISEÑO SISMORRESISTENTE para estructuras cimentadas ROCA en suelos muy blandos

Mi primera propuesta de investigación J Factor de Reducción de Resistencia Calculados a Partir de Registros Obtenidos en Suelos Blandos

Podíamos aprender y usar resultados de la Cd de México para la Región de la Bahía de SF y viceversa !!!

11 Factor de Reducción de Resistencia Calculados a Partir Mi tercer artículo en una revista arbitrada J de Registros Obtenidos en Suelos Blandos

Periodo Predominante del Movimiento de Terreno, Tg Periodo Predominante del Movimiento de Terreno, Tg

12 Factor de Reducción de Resistencia Calculados a Partir de Factor de Reducción de Resistencia Calculados a Partir de Registros Obtenidos en Suelos Blandos en Función de T/Tg Registros Obtenidos en Roca o Suelo Firme

STATISTICAL RESULTS ON STRENGTH REDUCTION FACTORS

Rµ 7.0 µ = 6.0 6.0 µ = 5.0 5.0 µ = 4.0 4.0 µ = 3.0 3.0 µ = 2.0 2.0 µ = 1.5 1.0 µ = 1.0 264 ground motions 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD, T [s]

Factor de Reducción de Resistencia Calculados a Partir de Factor de Reducción de Resistencia Calculados a Partir de Registros Obtenidos en Roca o Suelo Firme Registros Obtenidos en Roca o Suelo Firme

Miranda 1991 STATISTICAL RESULTS ON STRENGTH REDUCTION FACTORS

Rµ 7.0 µ = 6.0 6.0 µ = 5.0 µ = 4.0 5.0 µ = 3.0 µ = 2.0 µ = 1.5 4.0 µ = 1.0 µ = 6.0 3.0 µ = 5.0 µ = 4.0 2.0 µ = 3.0 µ = 2.0 1.0 µ = 1.5 264 ground motions µ = 1.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD, T [s]

13 Factor de Reducción de Resistencia Calculados a Partir de Factor de Reducción de Resistencia Calculados a Partir de Registros Obtenidos en Suelos Blandos en Función de T/Tg Registros Obtenidos en Suelos Blandos en Función de T/Tg

R m Mayores 20.0 Reducciones 18.0 MEDIA DE 32 MOVIMIENTOS Mayores EXPRESIÓN ANALÍTICA 16.0 Reducciones m = 5 14.0 Menores m = 4 Reducciones 12.0 Menores 10.0 m = 3 Reducciones 8.0 m = 2 6.0 m = 1.5 4.0 2.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Ts

Factor de Reducción de Resistencia Calculados a Partir de Recomendaciones de Factor de Reducción de Resistencia Registros Obtenidos en Suelos Blandos en Función de T/Tg para Estructuras Cimentadas en Suelos Blandos

Comparación con los factores de reducción de los reglamentos de 1976, 1987, 2004 (actual)

Q', Rm 14.0 m= 4 T = 2.0 s R 12.0 s m Q' 10.0 m= 3 8.0

6.0 m= 2 4.0 Q = 4 Q = 3 Q = 2 2.0

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Periodo [s]

14 Demandas de Desplazamiento en Con Rµ(µ,T/Tg) mejoramos nuestra estimación de fuerzas lateral de estucturas en suelo blando Sistemas Elásticos e Inelásticos

D [cm] Elastic 18 T = 1.15s, x 0 =5%, Elastic 12 6 0 -6 Dmax = 10.47 cm -12 Dapprox = C Dmax = 1.13 (10.47) = 11.8 cm -18 0 5 10 15 20 25 30 35 40 TIME [s]

D [cm] Inelastic 18 T = 1.15s, =5%, =4 12 x0 µ Di = 11.4 cm 6 0 -6 -12 Pero que produce este daño ? -18 0 5 10 15 20 25 30 35 40 TIME [s] Fuerzas o Deformaciones ? DDiiµ CCµµ== DDe e Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

Demandas de Desplazamiento en Cocientes de Desplazamientos Calculados a Partir de Sistemas Elásticos e Inelásticos Registros Obtenidos en Roca o Suelo Firme

Mean constant relative strength inelastic displacement ratios: Si conocemos estos cocientes entonces es posible obtener una estimación de la demanda de desplazamiento en el sistema con CR comportamiento no lineal a partir de la demanda en el sistema 4.0 R = 6.0 elástico utilizando la siguiente expresión R = 5.0 3.0 R = 4.0 R = 3.0 R = 2.0 Δi = CR ⋅ Δe = CR ⋅ Sd 2.0 R = 1.5 R = 1.0

1.0 MEAN 264 ground motions 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD [s]

15 Cocientes de Desplazamientos Calculados a Partir de Cocientes de Desplazamientos Calculados a Partir de Registros Obtenidos en Roca o Suelo Firme Registros Obtenidos en Roca o Suelo Firme

C1 4 SOIL PROFILE: C R=8.0 R=6.0 3 R=4.0 Approximate CR (Ruiz-Garcia and Miranda 2003) R=3.0 R=2.0 R=1.5 Approximate CR 2 (Ruiz-Garcia and Miranda 2003) é 1 1 ù 1 CR = 1+ ê b - ú(R -1) ë a(T/Ts ) c û 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD [s] C1 4 C1 SOIL PROFILE: B 4 R=8.0 SOIL PROFILE: D R=8.0 R=6.0 R=6.0 3 R=4.0 3 R=4.0 R=3.0 R=3.0 R=2.0 R=2.0 2 R=1.5 R=1.5 2

1 1

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 PERIOD [s] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD [s] Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

Cocientes de Desplazamientos Calculados a Partir de Cocientes de Desplazamientos Calculados a Partir de Registros Obtenidos en Roca o Suelo Firme Registros Obtenidos en Roca o Suelo Firme

ASCE/SEI 41-06 Standard for Seismic Mean inelastic displacement ratios: Rehabilitation of Existing Buildings C1 Dinelastic Cµ = 4.0 C Delastic R = 8 µ SITE CLASS C (a=90) 4.0 3.5 R = 6 µ = 6.0 R = 4 3.5 µ = 5.0 3.0 R = 3 µ = 4.0 3.0 µ = 3.0 2.5 R = 2 µ = 2.0 R = 1.5 2.5 µ = 1.5 2.0 µ = 1.0 2.0 1.5 1.5

1.0 1.0 0.5 0.5 (mean of 264 ground motions) 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD PERIOD [s]

16 Table 3. Parameter Summary for Eq. (3)

Ensemble Hysteretic behavior ␪1 ␪2 ␪3 SE͑␪1͒ SE͑␪2͒ SE͑␪3͒ SFBA EPP 0.00 10.5 −0.50 0.0131 0.3735 0.0450 SD −0.06 11.0 −0.45 0.0103 0.295 0.0355 MEXC EPP 0.04 10.5 −0.68 0.0069 0.1976 0.0238 SD −0.04 12.0 −0.60 0.0127 0.3642 0.0439 displacement ratios for each normalized period of vibration, each ratio, the level of inelastic deformation, ␮, and a set of param- displacement ductility ratio, and each ground motion, ratios of eters, ␪គ, as follows maximum inelastic displacement of stiffness degrading systems to T −4.2 maximum inelastic displacements were computed. A total of C˜ =1+͑␮ − 1͒ ␪ + ␪ ␮ ͫ 1 2ͩ T ͪ ͬ 38,400 of such ratios were computed in this study. This ratio is a g measure of how larger or smaller are the lateral inelastic displace- T 32 T 2 ments demands on systems with stiffness degradation compared + ␪ ͑␮ − 1͒0.5 g exp 2.3 − ln − 0.1 3 ͩ T ͪ ͫͩ ␮ ͪͩ T ͪ ͬ ͭ g ͮ to those on nondegrading systems. Cocientes de Desplazamientos Calculados a Partir de Cocientes de Desplazamientos Calculados a Partir de Fig. 8(a) shows mean stiffness degrading to elastoplastic in- T T 2 Registros Obtenidos− 0.08 eng expSuelo− 70 Blandoln + 0.67 ͑2͒ Registros Obtenidos en Suelo Blando elastic displacement ratios computed for ground motions recorded ͩ T ͪ ͫ ͩ ͩ T ͪͪ ͬ g in soft soil sites in the San Francisco Bay Area. A similar plot where Tϭperiod of vibration; T ϭpredominant period of the corresponding to the Mexico City ground motion set is shown in g Cocientes de desplazamientos para diferentes niveles de probabilidad de excendencia Fig. 8(b). Again, ratios for the Mexico City ground motion set are ground motion; and ␪1 ,␪2 ,␪3,ϭconstants which depend the type CµEP ‘BAY MUD’ SAN FRANCISCO BAY AREA CµEP ‘LAKE BED’ MEXICO CITY smoother as a result of the significantly larger sample size. In 6 of hysteretic behavior and the ground6 motion ensemble. Param- (mean(Mean of of 16 16 ground records) motions) µ = 6.0 (mean(Mean of of 100 100 ground records) motions) eters ␪1 ,␪2 ,␪3 were computed through a nonlinear regressionµ = 6.0 general, it can be seen that there are spectral regions in which 5 µ = 5.0 5 analysis using the Levenberg–Marquardt µ = 4.0 method (Batesµand = 5.0 inelastic displacements of stiffness degrading systems are on av- µ = 4.0 4 µ = 3.0 4 erage larger than those of elastoplastic systems (typically for Watts 1988) as implemented µ = 2.0 in MATLAB (Mathworks 2001)µ =for 3.0 µ = 1.5 µ = 2.0 3 3 small values of T/Tg), while in other spectral regions the opposite each set of ground motions and each hysteretic behavior.µ The = 1.5 is true (primarily for T/Tg Ͼ1). It can be seen that limiting T/Tg 2 resulting values of these parameter2 estimates and the standard ( ) period ratios that separate spectral regions where inelastic dis- 1 error of these parameters are given1 in Table 3. Eq. 2 corresponds placements are larger for stiffness-degrading system from spectral to a surface in the C␮ −␮−T/Tg space that provides estimates of 0 0 regions where inelastic displacements are larger for elastoplastic 0.0 mean0.5 inelastic1.0 1.5 displacements2.0 2.5 3.0 ratios0.0 as0.5 a function1.0 1.5 of ␮ 2.0and T2.5/Tg 3.0 systems are not constant and, in general, decrease as the level of (Fig. 9). TheT / firstTg and second terms in this equationT / Tg capture the variation of C␮ as a function of ␮ and T/Tg with a similar trend as inelastic behavior increases. For the Mexico City ground motion Vs = 120 m/s Vs = 70 m/s Fig. 4. Inelastic displacement ratios corresponding to various percen- set these range from about T/T =1.0 for a displacement ductility that observed in firm sites (Miranda 2000). Meanwhile,w = 400% the third tiles computed with 100 ground motions recorded in Mexico City (a) g w = 80% ␮=3 and (b) ␮=5 ratio of 1.5 to about T/Tg =0.6 for a displacement ductility ratio and fourth term account for the local reductions(After Ruiz-García in C␮and Miranda, 2004) that take (After Ruiz-García and Miranda, 2004) of 6. As shown in this Fig. 8, for periods of vibration larger than place at periods close to the fundamental period of vibration of the predominant period of the ground motion, the inelastic dis- the soil deposit ͑T/Tg Ϸ1͒ and at periods close to the second notice that a stronger asymmetry of the probability distribution is Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda mode of vibration of the soil deposit ͑T/Tg Ϸ1/3͒. observed in this spectral region as the level of inelastic behavior placements of elastoplastic systems are on average larger than increases. those of stiffness-degrading systems, and the difference increases There are two kinds of errors associated with the use of Eq. (2) as an estimator of the population mean of C . The first is related Effect of Earthquake Magnitude These error bounds have been computed using the central value as the level of inelastic behavior increases. In some cases, the ␮ theorem and by approximating the coefficient of variation by the maximum displacements of elastoplastic systems are on average to the error associated using a sample mean (the mean computed It is well known that the amplitude of elastic spectral ordinates sample coefficient of variation. This approximation is good and frequency content of the ground motion is modified as the enough for most practical purposes (Bates and Watts 1988). from a finite number of records) as the basis of the nonlinear magnitude of the earthquake increases. Therefore, it is important 50% larger than those of stiffness degrading systems. These dif- Maximum estimation errors occur where the COV is maximum Recomendaciones de Diseño para estimar Desplazamientos Recomendaciones de Diseño para toestimar evaluate if modification Desplazamientos of linear elastic spectral ordinates is ferences in displacements are, in general, larger than those ob- Fig. 4. Inelastic displacement ratios corresponding to various percen- enough to estimate inelastic displacement demands or if also in- (for periods close to the predominant period of the ground mo- served by previous investigators who have studied the effectsLaterales of de Estructuras en Suelo Blando Lateralestiles computed with de 100 groundEstructuras motions recorded en in Mexico Suelo City (a) Blandoelastic displacement ratios are modified by earthquake magnitude. tion). The average estimation errors (average over all T/Tg) are ␮=3 and (b) ␮=5 Miranda (2000) has shown that earthquake magnitude has a neg- Fig. 5. Effectsmaller of level than of inelastic 4.2% for behavior a ductility on inelastic of displacement 1.5 and smaller than about stiffness degradation using rock or firm soil records, suggesting ligible effect of inelastic displacement ratios computed from ratios 11.8% for a ductility ratio of 6 (both also with a 90% confidence ground motions recorded on rock or firm soil sites. However, it is level). While these estimation errors can be decreased by increas- that the effects of stiffness degradation are more important for not known if the same holds true for ground motions recorded on motionsing corresponding the sample to four size, seismic unfortunately events with there different are mag- not too many records structures on soft soil than for structures on rock or firm soil sites. Comparaciónnotice that a stronger asymmetryde ofEstudios the probability distributionEstadísticos is very softcon soils. la Expresión Propuesta nitudes.from The range very of soft surface-wave soil sites magnitudes in California. in these For events the sample of records varies from 6.3 to 7.1. A comparison of mean inelastic displace- observed in this spectral region as the level of inelastic behavior To study the effect of earthquake magnitude, mean inelastic from Mexico City, in which the sample size is 100, then the increases. displacement ratios were computed from four groups of ground ment ratios for elastoplastic systems undergoing displacement ductilityestimation ratios equal errorsto 4 when are subjected significantly to ground smaller. motions Namely, re- there is a 90% These error bounds have been computed using the centralcorded value inconfidence these four events level are that shown the in maximum Fig. 6. It can estimation be noticed, errors are smaller Effect of Earthquake Magnitude theorem and by approximating the coefficient of variation bythan the 3.2% for a ductility ratio of 1.5 and smaller than 6.1% for a Nonlinear Regression Analysis 2056 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2004 It is well known that the amplitude of elastic spectral ordinates sample coefficient of variation. This approximation is goodductility of 6. Similarly, there is a 90% confidence level that enough for most practical purposes (Bates and Watts 1988). and frequency content of the ground motion is modified as the average estimation errors (over all T/Tg) are smaller than 1.4% Table 3. Parameter Summary for Eq. (3) magnitude of the earthquake increases. Therefore, it is important Maximum estimation errors occur where the COV is maximumfor a ductility of 1.5 and smaller than 4.2% for a ductility ratio of For displacement-based design and, in general, in earthquake re- to evaluate if modification of linear elastic spectral ordinates is (for periods close to the predominant period of the ground mo- Ensemble Hysteretic behavior ␪1 ␪2 ␪3 SE͑␪1͒ SE͑␪2͒ SE͑␪3͒ 6. enough to estimate inelastic displacement demands or if also in- tion). The average estimation errors (average over all T/Tg) are sistant design, it is desirableSFBA to have EPP a simplified 0.00 equation 10.5 to −0.50 0.0131 0.3735 0.0450 Fig. 10 shows the fitted expectation function of inelastic dis- elastic displacement ratios are modified by earthquake magnitude. smaller than 4.2% for a ductility of 1.5 and smaller than aboutplacement ratios for ␮=3 and ␮=6 compared to mean C values estimate mean constant-ductility inelasticSD displacement−0.06 ratios 11.0 that −0.45 0.0103 0.295 0.0355 Miranda (2000) has shown that earthquake magnitude has a neg- Fig. 5. Effect of level of inelastic behavior on inelastic displacement ␮ 11.8% for a ductility ratio of 6 (both also with a 90% confidencecomputed for elastoplastic systems and the Mexico city ground MEXC EPP 0.04 10.5 −0.68 0.0069 0.1976 0.0238 ligible effect of inelastic displacement ratios computed from ratios can facilitate the estimation of maximumSD displacements−0.04 in12.0 sys- −0.60 0.0127 0.3642 0.0439 level). While these estimation errors can be decreased by increas-motion set. Inference bands computed from a 95% confidence ground motions recorded on rock or firm soil sites. However, it is ing the sample size, unfortunately there are not too many records not known if the same holds true for ground motions recorded on motions corresponding to four seismic events with different mag- interval using a Student t distribution are shown in the same fig- tems with inelastic behavior from maximum displacements of from very soft soil sites in California. For the sample of records very soft soils. nitudes. The range of surface-wave magnitudes in these events ure. It can be seen that, in general, the analytical estimation pro- displacement ratios for each normalized period of vibration, each ratio, the level of inelastic deformation, ␮, and a set of param- systems with linear elastic behavior. To study the effect of earthquake magnitude, mean inelastic variesfrom 6.3 Mexico to 7.1. A City, comparison in which of mean the inelastic sample displace- size is 100, thenvided the by Eq. (2) provides very good results and is able to capture displacement ductility ratio, and each ground motion, ratios of eters, ␪គ, as follows ment ratiosestimation for elastoplastic errors are systems significantly undergoing smaller. displacement Namely, there is a 90% In a recent study, Mirandamaximum inelastic(2001 displacement) concluded of stiffness degrading that systems expressions to displacement ratios were computed from four groups of ground the effects of both ␮ and T/Tg on inelastic displacements ratios. T −4.2 Fig.ductility 10. Comparisonconfidence ratios equalof level to mean 4 when that inelastic the subjected maximum displacement to ground estimation motions ratios re-( errorsMexico are smaller maximum inelastic displacements were computed. A total of C˜ =1+͑␮ − 1͒ ␪ + ␪ corded in these four events are shown in Fig. 6. It can be noticed, In particular, this equation is able to capture very well the local ␮ ͫ 1 2ͩ T ͪ ͬ than 3.2% for a ductility ratio of 1.5 and smaller than 6.1% for a derived directly from38,400 statistical of such ratios analyses were computed of in this mean study. This inelastic ratio is a dis- g City, EPP) to those computed with Eq. (2)(a) ␮=3 and (b) ␮=6 reductions in inelastic displacement ratios produced at periods measure of how larger or smaller are the lateral inelastic displace- T 32 T 2 2056 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2004 ductility of 6. Similarly, there is a 90% confidence levelclose that to first and second modes of vibration of the soil deposit. placement ratios producements demandsbetter on results systems with than stiffness using degradation expressions compared de- + ␪ ͑␮ − 1͒0.5 g exp 2.3 − ln − 0.1 (After Ruiz-García and Miranda, 2004) (After Ruiz-García and Miranda, 2004) 3 ͩ T ͪ ͫͩ ␮ ͪͩ T ͪ ͬ average estimation errors (over all T/Tg) are smaller than 1.4%In this study, two commonly used measures of “goodness-of- ͭ g ͮ to those on nondegrading systems. rived from mean strength reduction factors (i.e., R␮-␮-T relation- regressionfor analyses, a ductility and of the 1.5 second and smaller is related than 4.2% to the for fact a ductility that ratio of Fig. 8(a) shows mean stiffness degrading to elastoplastic in- T T 2 fit” of the results of nonlinear regression analyses were computed: − 0.08 g exp − 70 ln + 0.67 ͑2͒ regression6. analyses do not yield a perfect match of the sample ships). A nonlinearelastic equation displacement ratios to computed estimate for ground motions mean recorded inelastic ͩ TFig.ͪ ͫ 9. ͩMeanͩ T inelasticͪͪ ͬ displacement ratios (Mexico City, EPP) com- The standard error of the residuals and the correlation coefficient. g in soft soil sites in the San Francisco Bay Area. A similar plot Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 mean (goodness-of-fitFig. 10 shows error) the. The fitted first© 2017 Prof. expectation kind ofEduardo Miranda error function is called of inelastices- The dis- standard error of the residuals, SE, is defined as follows ˜corresponding to the Mexico City ground motion set is shown in where Tϭperiodputed of vibration; withTgϭ Eq.predominant(2) period of the placement ratios for ␮=3 and ␮=6 compared to mean C values displacement ratios, C␮, is proposed as a function of the T/Tg timation error and it can be shown (see for example Benjamin ␮ Fig. 8(b). Again, ratios for the Mexico City ground motion set are ground motion; and ␪1 ,␪2 ,␪3,ϭconstants which depend the type and Cornellcomputed 1970) that for the elastoplastic COV of the systems sample and mean theMexico is directly city ground n smoother as a result of the significantly larger sample size. In of hysteretic behavior and the ground motion ensemble. Param- ˜ 2 proportionalmotion to the set. COV Inference of the bands random computed variable from and inverselya 95% confidence ͑C␮,i − C␮,i͒ general, it can be seen that there are spectral regions in which eters ␪1 ,␪2 ,␪3 were computed through a nonlinear regression ͚ 2058 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2004 interval using a Student t distribution are shown in the same fig- i=1 inelastic displacements of stiffness degrading systems are on av- analysis using the Levenberg–Marquardt method (Bates and proportional to the square root of the number of records used to SE͑␮͒ϵͱ ͑3͒ erage larger than those of elastoplastic systems (typically for Watts 1988) as implemented in MATLAB (Mathworks 2001) for estimate theure. sample It can be mean. seen Hence, that, in the general, estimation the analytical error will, estimation in pro- n − np small values of T/Tg), while in other spectral regions the opposite each set of ground motions and each hysteretic behavior. The general, increasevided by as Eq. the( ductility2) provides ratio very increases good results and will and decrease is able to capture resulting values of these parameter estimates and the standard ϭ is true (primarily for T/Tg Ͼ1). It can be seen that limiting T/Tg as the numberthe effects of records of both considered␮ and T/Tg inon computing inelastic displacements the mean ratios.where C␮,i mean inelastic displacement ratio computed from error of these parameters are given in Table 3. Eq. (2) corresponds period ratios that separate spectral regions where inelastic dis- Fig. 10. Comparison of mean inelastic displacement ratios (Mexicosample increases.In particular, this equation is able to capture very well the localnonlinear response history analysis for the ith period of vibration; placements are larger for stiffness-degrading system from spectral to a surface in the C␮ −␮−T/Tg space that provides estimates of 17 City, EPP) to those computed with Eq. (2)(a) ␮=3 and (b) ␮=6 For thereductions case of inelastic in inelastic displacement displacement ratios ratios computed produced from at periodsC˜ ϭinelastic displacement ratio calculated from the nonlinear regions where inelastic displacements are larger for elastoplastic mean inelastic displacements ratios as a function of ␮ and T/Tg ␮,i systems are not constant and, in general, decrease as the level of (Fig. 9). The first and second terms in this equation capture the San Franciscoclose Bay to first Area and records, second in modes which of the vibration number of therecords soil deposit.regression model [i.e., with Eq. (2)] for the ith period of vibra- variation of C as a function of ␮ and T/T with a similar trend as inelastic behavior increases. For the Mexico City ground motion ␮ g is relatively small,In this there study, is two a 90% commonly confidence used level measures that the of maxi- “goodness-of-tion; nϭnumber of data points (e.g., fifty T/Tg ratios); and that observed in firm sites (Miranda 2000). Meanwhile, the third set these range from about T/Tg =1.0 for a displacement ductility regression analyses, and the second is related to the factmum that estimationfit” of the errors results are ofsmaller nonlinear than regression 7.1% for a analyses ductility were ratio computed:n ϭnumber of parameters in the model which in the proposed and fourth term account for the local reductions in C that take p ratio of 1.5 to about T/Tg =0.6 for a displacement ductility ratio ␮ regression analyses do not yield a perfect match of the sampleof 1.5 andThe increase standard to a error 90% of confidence the residuals level and that the the correlation maximum coefficient.model is equal to three. The standard error is a measure of the of 6. As shown in this Fig. 8, for periods of vibration larger than place at periods close to the fundamental period of vibration of mean (goodness-of-fit error). The first kind of error is calledestimationes- The errors standard are smaller error of than the residuals, 17.9% forSE a, is ductility defined of as 6. follows average error in the model. Standard error of the residuals com- the predominant period of the ground motion, the inelastic dis- the soil deposit ͑T/Tg Ϸ1͒ and at periods close to the second mode of vibration of the soil deposit ͑T/T Ϸ1/3͒. timation error and it can be shown (see for example Benjamin placements of elastoplastic systems are on average larger than g n those of stiffness-degrading systems, and the difference increases There are two kinds of errors associated with the use of Eq. (2) and Cornell 1970) that the COV of the sample mean is directly C − C˜ 2 as the level of inelastic behavior increases. In some cases, the as an estimator of the population mean of C␮. The first is related proportional to the COV of the random variable and inverselyTable 4. Computed Measures of “Goodness-of-Fit”͚ ͑ Used␮,i in␮, thisi͒ Study maximum displacements of elastoplastic systems are on average to the error associated using a sample mean (the mean computed proportional to the square root of the number of records used to i=1 from a finite number of records) as the basis of the nonlinear Ensemble Hysteretic behaviorSE͑␮͒ϵͱ Parameter ␮=1.5 ͑3␮͒ =2.0 ␮=3.0 ␮=4.0 ␮=5.0 ␮=6.0 50% larger than those of stiffness degrading systems. These dif- estimate the sample mean. Hence, the estimation error will, in n − np ferences in displacements are, in general, larger than those ob- general, increase as the ductility ratio increases and will decreaseMEXC EPP r 0.966 0.982 0.991 0.995 0.996 0.996 served by previous investigators who have studied the effects of where C ϭmean inelastic displacement ratio computed from stiffness degradation using rock or firm soil records, suggesting as the number of records considered in computing the mean ␮,i SE 0.078 0.086 0.082 0.079 0.091 0.125 sample increases. nonlinear response history analysis for the ith period of vibration; that the effects of stiffness degradation are more important for SD r 0.963 0.966 0.977 0.983 0.988 0.989 structures on soft soil than for structures on rock or firm soil sites. For the case of inelastic displacement ratios computed from C˜ ϭinelastic displacement ratio calculated from the nonlinear ␮,i SE 0.102 0.147 0.170 0.181 0.188 0.230 San Francisco Bay Area records, in which the number of records regression model [i.e., with Eq. (2)] for the ith period of vibra- is relatively small, there is a 90% confidence level that the maxi-SFBAtion; nϭnumber EPP of data points (e.g.,r fifty T/T0.882ratios); and 0.908 0.951 0.962 0.974 0.980 Nonlinear Regression Analysis g mum estimation errors are smaller than 7.1% for a ductility ratio npϭnumber of parameters in the modelSE which in0.102 the proposed 0.137 0.164 0.192 0.209 0.236 For displacement-based design and, in general, in earthquake re- of 1.5 and increase to a 90% confidence level that the maximum model is equalSD to three. The standardr error is a measure0.892 of the 0.950 0.982 0.980 0.989 0.989 sistant design, it is desirable to have a simplified equation to estimation errors are smaller than 17.9% for a ductility of 6. average error in the model. StandardSE error of the0.227 residuals com- 0.202 0.087 0.138 0.161 0.184 estimate mean constant-ductility inelastic displacement ratios that can facilitate the estimation of maximum displacements in systems with inelastic behavior from maximum displacements of JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2004 / 2059 systems with linear elastic behavior. Table 4. Computed Measures of “Goodness-of-Fit” Used in this Study In a recent study, Miranda (2001) concluded that expressions Ensemble Hysteretic behavior Parameter ␮=1.5 ␮=2.0 ␮=3.0 ␮=4.0 ␮=5.0 ␮=6.0 derived directly from statistical analyses of mean inelastic displacement ratios produce better results than using expressions de- MEXC EPP r 0.966 0.982 0.991 0.995 0.996 0.996 rived from mean strength reduction factors (i.e., R␮-␮-T relation- SE 0.078 0.086 0.082 0.079 0.091 0.125 ships). A nonlinear equation to estimate mean inelastic Fig. 9. Mean inelastic displacement ratios (Mexico City, EPP) com- ˜ puted with Eq. (2) SD r 0.963 0.966 0.977 0.983 0.988 0.989 displacement ratios, C␮, is proposed as a function of the T/Tg SE 0.102 0.147 0.170 0.181 0.188 0.230 2058 / JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2004 SFBA EPP r 0.882 0.908 0.951 0.962 0.974 0.980 SE 0.102 0.137 0.164 0.192 0.209 0.236 SD r 0.892 0.950 0.982 0.980 0.989 0.989 SE 0.227 0.202 0.087 0.138 0.161 0.184

JOURNAL OF STRUCTURAL ENGINEERING © ASCE / DECEMBER 2004 / 2059 Dos Formas de Calcular los Cocientes de Desplazamientos Cocientes de Desplazamientos Calculados a Partir de Registros Obtenidos en Roca o Suelo Firme

INELASTIC DISPLACEMENT RATIOS FOR SOFT SOIL SITES

Di D(µ = µt ) C = = normalized period of vibration, lateral strength ratio, earthquake magnitude and distance to µ CR De D(µ =1) the source on inelastic displacement ratios were ÿrst investigated using SDOF systems with elastoplastic hysteretic4.0 behaviour. Afterwards, the in"uence of hysteretic behaviour on inelastic R = 6.0 displacement demands was investigated by comparing displacementR = 5.0 demands of elastoplastic Su cálculo requiere de iteración systems to those of3.0 the MC, SD, MSSD and SSSD systems.R = 4.0 In particular, the e#ects of R = 3.0 post-yield sti#ness, sti#ness and strength degradation are examinedR = 2.0 in Section 5.5. 2.0 R = 1.5 R = 1.0 5.1. Central tendency of CR 1.0 Di D(R = Rt ) Figures 3(a) and (b) show mean constant-relative strength inelastic displacement ratios as a MEAN 264 ground motions CR = = function of normalized periods, T=Tg, computed for elastoplastic systems subjected to accel- 0.0 De D(R =1) eration time histories recorded in the San Francisco Bay Area and in the former bed-lake of Mexico City. In spite0.0 of the0.5 di#erences1.0 in1.5 frequency2.0 content2.5 between3.0 these two ground Su cálculo NO requiere de iteración pero motion sets and of the di#erence in geologicPERIOD characteristics [s] (e.g. Mexico City soft soil deposits además tienen muchas otras ventajas have much smaller shear wave velocities and much higher water contents than soft soils in the San Francisco Bay Area), it can be seen that inelastic displacement ratios follow a similar trend. From this ÿgure, three spectral regions with distinctively di#erent characteristics can be identiÿed. In the ÿrst region, corresponding to T=Tg ratios smaller than 0.75 for the San Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda FranciscoCurso de Diseño BaySismorresistente Area set, Ecuador Julio 3 and smaller-6, 2017 than 0.85 for the Mexico City set, maximum© 2017 Prof. inelasticEduardo Miranda displacements are on average larger than maximum elastic displacements. In this spectral region inelastic displacement ratios increase as the lateral strength ratio increases and as normalized periods T=Tg decrease. The second spectral region corresponds to systems with periods of vibration relatively close to the predominant period of the ground motion where peak lateral EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS deformation of inelastic systems are, on average, smaller than peak lateral deformations of Earthquake Engng Struct. Dyn. (in press) Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/eqe.552 elastic systems. This second spectral region extends from the end of the ÿrst region to T=Tg ratiosCocientes of approximately de Desplazamientos 1.55 for both ground motion Calculados sets. A third a region, Partir corresponding de to pe- riodsRegistros of vibration Obtenidos approximately en 1.55 Suelo times Blando larger than the predominant period of the ground Inelastic displacement ratios for evaluation of structures motion, is characterized by peak deformations in inelastic systems being on average approx- built on soft soil sites imately equal to peak elastic deformations, regardlessD D(R of= R the) level of lateral strength ratio, C = i = t for the San Francisco Bay Area set. However,R for the Mexico City ground motion set 1; ; 2 De D(R =1) Jorge Ruiz-GarcÃ"a ∗ † and Eduardo Miranda

1Graduate School of Civil Engineering; Universidad Michoacana de San NicolasÃ de Hidalgo; Ediÿcio C; Planta Baja; Cd. Universitaria; 59610 Morelia; MexicoÃ 4.0 4.0 R = 6.00 2Department of Civil and Environmental Engineering; Stanford University; Stanford; CA 94305-4020; U.S.A. R = 6.0 3.5 R = 5.0 3.5 R = 5.00 R = 4.0 R = 4.00 3.0 3.0 R = 3.0 R = 3.00 2.5 R = 2.0 2.5 R = 2.00 SUMMARY R = 1.5 R = 1.55 R R 2.0 2.0 C This paper summarizes the results of a comprehensive statistical study aimed at evaluating peak lateral C inelastic displacement demands of structures with known lateral strength and sti#ness built on soft 1.5 1.5 soil site conditions. For that purpose, empirical information on inelastic displacement ratios which are deÿned as the ratio of peak lateral inelastic displacement demands to peak elastic displacement 1.0 1.0 demands are investigated. Inelastic displacement ratios were computed from the response of single- degree-of-freedom systems having 6 levels of relative lateral strength when subjected to 118 earthquake 0.5 0.5 ground motions recorded on bay-mud sites of the San Francisco Bay Area and on soft soil sites located in the former lake-bed zone of Mexico City. Mean inelastic displacement ratios and their 0.0 0.0 corresponding scatter are presented for both ground motion ensembles. The in%uence of period of 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 vibration normalized by the predominant period of the ground motion, the level of lateral strength, earthquake magnitude, and distance to the source are evaluated and discussed. In addition, the e#ects (a) T / Tg (b) T / Tg of post-yield sti#ness and of sti#ness and strength degradation on inelastic displacement ratios are also investigated. It is concluded that magnitude and distance to the source have negligible e#ects on constant-strength inelastic displacement ratios. Results also indicate that weak and sti#ness-degrading Figure 3. Mean inelastic displacement ratios computed for: (a) the San Francisco structures in the short spectral region could experience inelastic displacement demands larger than those corresponding to non-degrading structures. Finally, a simpliÿed equation obtained using regression Bay Area ground motion set (mean of 18 ground motions); and (b) the Mexico City analyses aimed at estimating mean inelastic displacement ratios is proposed for assisting structural ground motion set (mean of 100 ground motions). engineers in performance-based assessment of structures built on soft soil sites. Copyright ? 2006 John Wiley & Sons, Ltd. (After Ruiz-García and Miranda, 2006) Copyright ? 2006 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. (in press) KEY WORDS: soft soil sites; inelastic deformation demand; inelastic displacement ratio; sti#ness Curso de Diseño Sismorresistente, Ecuador Julio 3degradation -6, 2017 © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda

∗Correspondence to: Jorge Ruiz-GarcÃ"a, Posgrado de IngenierÃ"a Civil, Universidad Michoacana de San NicolasÃ de Hidalgo, Ediÿcio ‘C’, Planta Baja, Cd. Universitaria, 59610 Morelia, Mexico.Ã †E-mail: [email protected] Contract=grant sponsor: Consejo Nacional de Ciencia y Tecnologia Received 10 May 2005 18 Revised 30 September 2005 Copyright ? 2006 John Wiley & Sons, Ltd. Accepted 11 October 2005 J. RUIZ-GARCIAÃ AND E. MIRANDA

inelastic displacement demands are on average slightly greater than peak elastic displacement demands. The characteristics of inelastic displacement ratios in soft soil sites previously described are signiÿcantly di#erent from those reported by Ruiz-GarcÃ$a and Miranda [13] or by Chopra and Chintanapakdee [14] for rock and ÿrm soil sites. In particular, it should be noted that the well- known equal displacement approximation (i.e. peak inelastic displacements are, on average, equal to peak elastic displacements) originally suggested by Veletsos and Newmark [4, 5] is only applicable for periods that are 55% longer than the predominant period of the ground motions, but its use for periods shorter than this can lead to considerable overestimations or underestimations of inelastic deformation demands.

5.2. Dispersion of CR

Variability in the estimation of inelastic displacement ratios (i.e. dispersion of CR) was quan- tiÿed by computing coe%cients of variation (COV). This statistical parameter was computed for each normalized period of vibration and for each level of lateral strength ratio. Figure 4(a) shows COVs of inelastic displacement ratios computed from ground motions recorded in the San Francisco Bay Area. A similar plot corresponding to the Mexico City ground motion set is shown in Figure 4(b). It can be observed that COV for the Mexico City ground motion set are smoother as a result of the signiÿcantly larger sample size; then, the following observations are made on the basis of Figure 4(b). In general, dispersion of inelastic displacement ratios of structures on soft soil deposits are characterized by: (1) increment increase in dispersion as the lateral strength ratio increases, with exception of systems with T=Tg ratios smaller than 0.25; (2) larger dispersion for systems with periods of vibration shorter than the predominant period of the ground motion; and (3) increment in dispersion for periods of vibrationVariabilidad/Dispersión equal or near to those corresponding de los Cocientes to the ÿrst and de second modes of vibration of Efecto de la resistencia lateral en las theDesplazamientos soil deposit. With exception Calculados of periods a of Partir vibration de smaller Registros to the predominant period of demandas de desplazamiento theObtenidos ground motion, en levels Suelo of dispersionBlando computed from ground motions recorded on very soft soil sites are smaller than those reported by Ruiz-GarcÃ$a and Miranda [13] computed from Ductilidad Constante Resistencia Relativa Constante ground motions recorded on rock or ÿrm soil sites.

Di D(R = Rt ) Dinelastic 110 Cµ = CR = = Delastic De D(R =1) 100

1.2 1.2 90 R = 6.0 R = 6.0 80 R = 5.0 R = 5.0 120 R = 4.0 R = 4.0 70 Fig. 1. Mean inelastic displacement ratios for elastoplastic systems 0.9 100 computed for the San Francisco Bay Area ground motion set 0.9 R = 3.0 R = 3.0 60 R = 2.0 80

R = 2.0 50 R R R = 1.5 60 various percentiles computed from systems undergoing displace- R = 1.5 1 40 ment ductility ratios of 3 and 5 when subjected to ground motions 0.6 0.6 40 recorded in Mexico City. Here, it can be seen that the same gen- 30 20 COV C eral trend is observed at different probability levels. It can also be COV C 0.5 20 clearly seen that for systems with periods close to the predomi- 0 nant period of the ground motion, the equal displacement ap- 10 Inelastic displacement ofInelastic displacement EPP SDOF 5 0.3 0.3 4 proximation would result in a significant overestimation of the 3 2 1 0 Cy inelastic displacement even for a conditional probability level of 0 Period [s] 90%. Furthermore, the effect of the second mode of vibration of the soil deposit is much more pronounced for low levels of conditional probability of occurrence. 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Effect of Level of Inelastic Behavior (a)T/ Tg (b) T/ Tg In order to further study the effects of the level of inelastic be- Curso de Diseño Fig. 3. CoefficientSismorresistente of variation of, Ecuador Julio 3 inelastic displacement-6, 2017 ratios com- © 2017 Prof. Eduardo Miranda Curso de Diseño Sismorresistente, Ecuador Julio 3-6, 2017 © 2017 Prof. Eduardo Miranda havior on inelastic displacement ratios, all of the individual re- puted for (a) San Francisco Bay Area soft soil sites and (b) Mexico City soft soil sites sults were plotted for selected T/Tg ratios as a function of displacement ductility ratio. Results computed with the Mexico City Figure 4. Coe%cient of variation of inelastic displacement ratios computed for: (a) the San Francisco ground motion set for elastoplastic systems with T/Tg =0.5, 1.0, Bay Area ground motion set (dispersion of 18 ground motions); and (b) the Mexico City ground motion and 2.5 are shown in Fig. 5. It can be seen that for T/Tg =0.5 inelastic displacement ratios increase almost linearly with in- the ground motion, the effect of the levelset of inelastic (dispersion behavior of is 100 ground motions). creases in ductility demand. This linear trend is evident not only significantly different. As shown in Fig. 5, inelastic displacement for median values ͑p=50%͒ but also for high and low percen- ratios decrease as the level of inelastic behavior increases, how- tiles. For periods of vibration equal to the predominant period of everEfecto the observed de reduction la resistencia is not linearly proportional lateral to the en las Copyrightincrease in? ductility.2006 JohnIn particular, Wiley the & rate Sons, of reduction Ltd. is larger Earthquake Engng Struct. Dyn. (in press) INFLUENCIA DEL COMPORTAMIENTO HISTERÉTICO fordemandas small levels of ductility de than desplazamiento for larger levels of ductility demand. This means that for structures on very soft soil deposits whose periods of vibration are equal or close to the predominant period of the ground motion even a small level of inelastic behavior is very effective in reducing lateral displacement demands. These suggest that, as previously noted by Miranda (1991), the use of energy dissipation devices for structures in this situation could be particularlyDISPL%[cm]% beneficial by not only significantly reducing Tn=2s,%α=0% CON DEGRADACIÓN DE RIGIDEZ lateral strength demands but also by significantly reducing lateral SIN DEGRADACIÓN DE RIGIDEZ CµSD 120% CµEP 6 displacement demands. For systems with T/Tg =2.5, median in- %ELASTIC% 6 elastic displacement ratios increase only a little bit with increases (mean(Media of de 100 100 ground movimientos) motions) (mean(Media of de 100 100 ground movimientos) motions) 80% µ = 6.0 in inelastic behavior. However, for p=90%, the increase in inelas- %EPP,%Cy=0.3% 5 µ = 6.0 5 µ = 5.0 tic displacement40% with respect to elastic displacement is more sub- µ = 5.0 µ = 4.0 µ = 4.0 stantial. This shows that even though the equal displacement ap- 4 4 µ = 3.0 proximation is, on0% average, approximately valid in this spectral µ = 3.0 region, for systems undergoing large levels of inelastic behavior µ = 2.0 µ = 2.0 Fig. 2. Mean inelastic displacement ratios for elastoplastic systems 3 3 µ = 1.5 for some ground!40% motions, the inelastic displacement could be as µ = 1.5 computed for the Mexico City ground motion set large as two times the elastic displacement. It is also important to 2 2 JOURNAL OF STRUCTURAL!80% ENGINEERING © ASCE / DECEMBER 2004 / 2055 1 1 !120% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 0 0 Time%[s]% 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

T / Tg T / Tg

19 Efecto de la resistencia lateral en las INFLUENCIA DEL COMPORTAMIENTO HISTERÉTICO demandas de desplazamiento

D Inelastic displacement of Bilinear SDOF, Tn=2.0s,a =0 (EPP) CON DEGRADACIÓN DE RIGIDEZ 100 D SIN DEGRADACIÓN DE RIGIDEZ 90 Din,SD/Din,EP 1.6 CIUDAD DE MÉXICO µ = 6.0 80 µ = 5.0 1.4 µ = 4.0 µ = 3.0 70 1.2 µ = 2.0 . [cm] µ = 1.5 d

1.0 Despl 60 Subestimar la resistencia lateral 0.8 50 nos lleva a subestimar la demanda de desplazamiento 0.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 40

T / Tg 30 0 0.2 0.4 0.6 0.8 1 Cy

Efecto de la resistencia lateral y la rigidez post-elástica Importancia de las deformaciones residuales en las demandas de desplazamiento Hybrid testing with JMA record Despl. [cm] 200 180 EPP 160 α=-0.05" 140 α=-0.10" 120 α=-0.15" 100 80 60 40 20 0 0 0.2 0.4 0.6 0.8 1

(After Nagata, Kawashima, Watabe (2005)

20 Estimación de las deformaciones residuales Estimación de las deformaciones residuales

Firm soil records Fault-normal near-fault records C Cr r 2.0 2.0 RESIDUAL DISPLACEMENTS PEAK DISPLACEMENTS Site Classes AB,C,D R = 6.0 Fault-normal near-fault set R = 6.0 (mean of 240 ground motions) R = 5.0 (mean of 40 ground motions) R = 5.0 R = 4.0 R = 4.0 1.5 1.5 R = 3.0 R = 3.0 C C R = 2.0 R = 2.0 r R R = 1.5 R = 1.5 1.0 1.0

Cr,EP CR,EP 2.0 4.0 0.5 0.5 R = 6.0 R = 6.0 R = 5.0 R = 5.0 R = 4.0 R = 4.0 0.0 1.5 0.0 R = 3.0 3.0 R = 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD [s] R = 2.0 R = 2.0 T / Tp R = 1.5 R = 1.5 1.0 2.0 Cr Soft-soil SF records Cr Soft-soil Mexico City records 2.0 2.0 R = 6.0 R = 6.0 San Francisco bay Area Set Mexico City Soft Soil Set 0.5 R = 5.0 R = 5.0 1.0 (mean of 18 ground motions) (mean of 100 ground motions) R = 4.0 R = 4.0 1.5 1.5 R = 3.0 R = 3.0 R = 2.0 R = 2.0 0.0 0.0 R = 1.5 R = 1.5 1.0 1.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 PERIOD [s] PERIOD [s] 0.5 0.5 Residual displacement are more sensitive to strength than

maximum (peak) displacements 0.0 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T / Tg T / Tg

Demandas de desplazamiento en sistemas inelásticos Demandas de desplazamiento en sistemas inelásticos con diferentes comportamientos histeréticos con diferentes comportamientos histeréticos

DISPL%[cm]% DISPL%[cm]% Tn=2s,%α=0% Tn=2s,%α=0% 120% 120% %EPP,%Cy=0.3% %ELASTIC% 80% 80% %BE,%%%Cy=0.3% %BE,%%%Cy=0.3% 40% 40%

0% 0%

!40% !40%

!80% !80%

!120% !120% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% 0% 20% 40% 60% 80% 100% 120% 140% 160% 180% Time%[s]% Time%[s]%

21 Efecto del amortiguamiento en las demandas de Efecto del amortiguamiento en las demandas de desplazamiento para estructuras en suelo blando desplazamiento para estructuras en suelo blando

D /D T/Tg=1.0 Mean damping modification factor x% 5% DX%/D5% z=1% 2.5 2.5 z=2%

z=5%

2.0 z=10% Narrow band 2 z=15%

1.5 z=20% Ordinary GM, T=1.82s 1.5 z=30%

1.0

1 0.5

0.5 0.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30 z 0 0 0.5 1 1.5 2 2.5 3 T/Tg

Resumen y Conclusiones

Les he resumido brevemente los 30 años de aprendizaje de su servidor sobre la respuesta sísmica de estructuras cimentadas en suelos blandos

• Los movimientos de terreno sobre suelos muy blandos es muy diferente a los que ocurren en roca o depósitos de suelo blando

• No sólo los espectros de respuesta son muy diferentes sino muchos otros aspectos del diseño sismorresistente requieren ser modificados

PROF. EDUARDO MIRANDA