SOILS NONLINEARITY EFFECTS ON DOMINANT SITE PERIOD EVALUATION

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1 HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, Series A, OF HE ROMANIAN ACADEMY Volue 1, Nuber 3/9, pp.- SOILS NONLINEARIY EFFECS ON DOMINAN SIE PERIOD EVALUAION Diu BRAOSIN*, Flori-Ştefa BĂLAN**, Care-Ortaza CIOFLAN** * Istitute of Solid Mechaics of the Roaia Acadey, C-ti Mille Street, No.15, 1141 Bucharest 1, ROMANIA ** Natioal Istitute for Earth Physics, Călugărei Street, 1-Măgurele, ROMANIA Correspodig author: Diu BRAOSIN, e-ail: bratosi@acad.ro By uerical siulatio usig a oliear Kelvi-Voigt odel this paper propose a odelig ethod for resoat peaks dispersio provoked by oliear properties of the site aterials. his ethod leads to a better approxiatio of the oliear depedece of doiat site periods, depedece with a iportat ipact about site-structure resoace avoidace. Key-Words: Noliear soil dyaics, Noliear atural periods, Site-structural resoace, Nuerical siulatio. 1. INRODUCION he dyaic respose of a certai structure is strogly depedet of the ratio betwee the atural period of the structure ad the doiat period of the eplaceet site. It is very well kow that the ajor daages arise at resoace, whe the atural period of structure is equal or very close to the doiat period of the site. For this reaso, a correct evaluatio of site doiat period has a special iportace. he site aterials, soils or rocks, are oliear aterials with a dyaic behavior strogly depedet of loadig level ad this behavior affects the whole dyaic respose icludig the syste atural period values [], [4]. he eglect of this oliear behavior by usig the coputatioal ethods of the liear dyaics whe evaluatig the site period, ay be a source of errors for predictig the structural-site respose. he preset paper iteds to discuss the dyaic effects give by oliear aterial properties o syste atural period evaluatio. I this itet we will use the agificatio fuctios of the oliear Kelvi-Voigt odel [, 3]. By uerical siulatio with differet loadig level, we ca able to odel the effects of the soils oliearity o the shape ad resoat agitude of the agificatio fuctios. hus, we shall obtai a proper tool for odelig the resoat peak dispersio, which is a very iportat coditio for a correct atural periods evaluatio ad therefore for site-structure resoace avoidace.. NAURAL PERIOD RIGIDIY RELAIONSHIP k F = F si ωt c x I liear dyaics, a usual descriptio of a solid sigle-degree-offreedo behavior is give by the Kelvi-Voigt odel cosistig of a ass supported by a sprig (with a stiffess k) ad a dashpot (with a viscosity c) coected i parallel (fig. 1.1). he goverig equatio of this syste uder haroic excitatio (with the aplitude F ad pulsatio ω) is [15]: where: x + ζ x +ω x= qsiωt (.1) Fig.1.1 Liear Kelvi-Voigt odel c F ζ= ; q = ; ω = k (.)

2 Diu BRAOSIN, Flori-Stefa BĂLAN, Care-Ortaza CIOFLAN ζ beig a usual dapig easure (dapig ratio ζ = c/ ccr ), ad ω is the atural pulsatio, coected by atural frequecy f through the relatioship: ω = π f. he atural period of this syste is: 1 π = = = π f ω hus, fro eq. (.3) it follows that, for a costat ass, the atural period ad rigidity are i iverse ratio the rigidity icreasig leads to the atural period shorteig ad a diiishig rigidity leads to the oscillatio with larger atural periods. k (.3) 3. RIGIDIY OF HE SIE MAERIALS he site aterials are oliear aterials with a dyaic behavior strogly depedet of loadig level ad this behavior affects the whole dyaic respose icludig the syste atural period values [], [4]. he oliear behavior eas the loadig depedece of the aterial properties ad the dyaic aterial characteristics G ad ζ becoes fuctios i ters of strai level γ. herefore, aother dyaic characteristic the rigidity k - becoe strai depedet fuctio k = k( γ) too. Because the rigidity is i direct proportioal ratio with odulus G we ca write: k A G where A are the proportioal factor. Fro eqs. (.3) ad (3.1) results that the atural period aterials becoes strai depedet, too: γ = γ (3.1) of the systes which iclude oliear π π 1 B γ = = = k ( γ) A G( γ) G( γ) (3.) where B = π / A is aother aterial costat. G G he odulus-fuctio = ( γ) ca be writte i the oralized for G( γ ) = G G ( γ) where G ( γ ) is the odulus-fuctio oralized i ters of his iitial value G = G γ=. hus, the strai depedece of the atural period (3.) ca be writte i the siilar for: ( γ) B 1 γ = = iit ( γ) (3.3) G G where iit represet the iitial value of the atural period: ad the ter ( γ) iit = ( γ ) γ= = π = π (3.4) AG k represets the strai depedece of the atural period: 1 1 γ = = G ( γ) k ( γ) where k ( γ) is the rigidity fuctio k ( γ ) oralized i ters of his iitial value k = k γ γ=. (3.5)

3 3 Soils oliearity effects o doiat site periods evaluatio 4. EVALUAION OF HE SOILS NAURAL PERIOD CHANGES BY RESONAN COLUMN ESS I priciple, fro resoat colu test a certai odulus-fuctio value G for a certai strai level γ is obtaied by ediatio of shear vawe velocity v s usig the well-kow relatioship: G = ρ v s (4.1) where ρ is the ass desity of specie [4], [14]. he shear wave velocity provided by resoat colu test is obtaied i the for: ω h vs = = Ψ ω h 1 4 R R + R where ω is the specie atural pulsatio, h is the specie height ad Ψ is the root of torsioal frequecy equatio with aalytical for i ters of the ratio betwee torsioal iertia of the specie ad the torsioal iertia of the top cap syste: R = J J [4], [15]. / top Fro eqs. (.3) ad (4.) the expressio for the specie atural results: (4.) π πh ρ 1 = = ω Ψ G ad thus the proportioality costat A fro eq. (3.4) ca be ow full deteries: (4.3) πh ρ A = Ψ After several tests with differet strai level γ we ca obtai the odulus fuctio G G G G ad the atural period fuctios (3.3) γ = γ where the iitial value = ( γ ) = ( γ ) iit ( γ ) is give i eq. (3.5). has (3.4) expressio ad oralized for he atural periods obtaied by resoat colu test is the atural periods of the s.d.o.f. oscillatig syste coposed by a sigle ass (the vibratio device) supported by a sprig ad a daper represeted by the specie []. he atural period value depeds o echaical properties of the specie aterial but it is a syste characteristic ad ot a aterial characteristics. Fortuately, as ca see fro eqs. (3.4) ad (3.5), the physical ad geoetrical properties are icluded oly i the iitial value of the oliear atural period fuctio ad the oralized for (3.5) depeds iit oly o oralized aterial characteristics. hus, the resoat colu test ca offer accurate data for obtaiig the oliear depedece of the atural period. Certaily, the iitial value of the atural period fuctio ust be obtaied fro aother way by ay test perfored o ivestigated oliear syste. As ca see i the ext, for the site oliear systes which cotais soils oliear aterials ca be use the data recordig durig seisic evets. Relatioship (3.5) allows estiatig the oliear variatio of the atural periods if the odulusfuctio G = G( γ ) is previously deteried. For exaple, i fig. 4.1 there are give soe oralized odulus-fuctios obtaied fro resoat colu data ad i fig. 4. the correspodig oralized atural period fuctios obtaied fro eq. (3.5) are give. As we ca see fro this experietal data processig, the reductio of the odulus-fuctio values due to dyaic degradatio leads to the icreasig period values, the sae iverse variatio betwee rigidity ad atural period as i the liear aterial case. (4.4) iit

4 Diu BRAOSIN, Flori-Stefa BĂLAN, Care-Ortaza CIOFLAN 4 Fig. 4.1 Soe oralized odulus-fuctios Fig. 4. Noliear variatio of the atural periods 5. NONLINEAR NAURAL PERIOD FUNCIONS IN ERMS OF LOADING For practical applicatios it is ecessary to deterie the oralized atural period i ters of loadig aplitude ad because i the seisic calculus the usual loadig aplitude is PGA (peak groud acceleratio) we ust odify the fuctios i = PGA for. = ( γ ) For this coversio, we ca use the oliear Kelvi-Voigt odel [3], [4] subjected to support otio x g t = xgsiω t with differet acceleratio aplitude x g. I this loadig case, the otio equatio reads as: x + ω ζ x x +ω k x x= x (5.1) g Usig the chage of variable τ = ω t ad itroducig the ew tie fuctio ϕ() τ = xt () = x ( τ/ ω ) we ca obtai for eq. (5.1) aother for: K si ϕ+ C ϕ ϕ+ ϕ ϕ=μ υτ (5.) where the superscript accet deotes the tie derivative with respect to τ, ad: g ω ; ω ω k c x k x k x C( ϕ) C( x) = = ζ( x) ; K( ϕ) K( x) = = = k ω x μ= υ= ω he steady-state solutio of the equatio (5.) ca be writte i the for: where Φ( υ ; ζ ) is the oliear agificatio fuctio: ( x) (5.3) ϕτυζ (,, ) =μφ( υζ ; ) si( υτ ψ ) (5.4) ax,, τ ϕτυζ x Φ( υ; ζ ) = = μ x dyaic static (5.5)

5 5 Soils oliearity effects o doiat site periods evaluatio a ratio of axiu dyaic aplitude ϕax xdyaic to static displaceet μ = xstatic. For give aplitude μ the oliear equatio (5.) ca be uerically solved usig a coputer progra based o Newark algorith [4], ad the oliear agificatio fuctio Φ( υ μ ζ ) =Φ( υ ζ). ;, ; ct ζ= μ= ct. ca be obtaied. o illustrate this ethod i fig.5.1 such a uerical siulatio result is preseted. For this exaple the aterial dyaic fuctio for clay was used [6, 9]. By this siulatio oe ca poit out a softeig type behavior with resoat picks at pulsatio υ located before the liear resoat pulsatio υ=1. Fig. 5.1 Noliear agificatio fuctio Fig. 5. Natural period variatio i ters of μ Because υ=ω/ ω = 1/ fro the siulatio results give i fig.5.1 we ca obtai a relatioship = ( μ ) (fig.5.). he oralized aplitude μ is coected by abutet acceleratio aplitude x g through the eq. (5.3 3 ) μ= x g / ω (fig. 5.3) ad together with the relatio = ( μ) eables us to obtai a relatioship (fig. 5.4): = μ x.9 1.9exp( 3.51 g g = + PGA ) (4.1) μ=μ ( PGA) = = PGA ω p ωp where g is gravity acceleratio ( g = 9.81 /s ). Fig. 5.3 Relatioship μ - PGA Fig. 5.4 Natural period variatio i ters of PGA

6 Diu BRAOSIN, Flori-Stefa BĂLAN, Care-Ortaza CIOFLAN 6 6. EVALUAION OF HE SIE NAURAL PERIOD Kowledge of the oralized atural period for each site aterials as it was previously deteried is ot eough for the evaluatio of the etire site oralized atural periods. Each layer has a cotributio depedig o its geoetrical properties. For that, we accept the hypothesis of the oralized atural period depedece i ters of the layer thickess. Accordig to the thickess depedece hypothesis, the average atural period variatio for the etire av i site layers will be coposed as the average of the oralized atural period weighted with the thickess of each layer: h i av = i h h i i (6.1) o exeplify this ethod we choose the site eplaceet of the seisic statio INCERC with kow stratificatio [1] (table 6.1). Material hickess vs [] [/s] Silty clay Sad ad gravel Clays Fie sad ad clay Clays Fie sad Fig. 6.1 Noralized atural period for INCERC eplaceet usig RC data First, for each costituet layer the aterial odulus-fuctio was estiated i cocordace with i i previous resoat colu data [8] ad thus by uerical siulatio a fuctio = ( PGA) was obtaied. he, for soe PGA values (.5,.1,.15,. ad.5 g) usig eq. (6.1) the site atural period averages was obtaied. he results are give i fig Evet data able 6.1 M w Observed acceleratios [c/s ] able his evaluatio ethod of the atural period variatio usig resoat colu data for a certai site ca be validated by copariso with seisic data recordig at the sae site. he spectral aalysis of seisic data recordig durig Vracea earthquakes with differet agitudes shows a doubtless depedece of axiu acceleratios ad doiat periods o earthquake agitude ( M ) [11, 14]. Such data recordig PGA [g] w [s]

7 7 Soils oliearity effects o doiat site periods evaluatio at INCERC seisic statio are give i table 6. (after [11, 1, 13, 14]). Fro these data oe ca obtai a uerical correlatio of the type = ( PGA) (fig. 6.). As we ca see i table 6. ad fig. 6. the atural period is icreasig fuctio o seisic loadig as it was observed i the case of processig resoat colu data, too. Because i the both processig ethods the iforatio provided fro the sae site was used, a copariso betwee the is perissive. Such a copariso is give i fig. 6.3, which plots together the sae fuctio = PG provided by both colu resoat data (fig.6.1) ad seisic data (fig.6.). ( A) Fig. 6. Depedece PGA fro seisic records Fig. 6.3 Depedece PGA provided both resoat colu data ad seisic record As oe ca see fro fig 6.3, the differeces betwee these differet acquisitio ethods are ot too large. I the previous, for the validatio of the RC ethod the geological ad seisic data of the INCERC site was used because for this site there are ultiple seisic recordig with differet agitudes begiig with low evets util the strog March 4, 1977 earthquake. However, for a large ajority of the usual sites oly seisic recordig of the low ad oderate evets are available. I these cases, the evaluatio of the doiat period for strog earthquakes usig oly seisic low ad oderate data presue a extrapolatio procedure with iheret large errors. o exeplify, i fig. 6.4 are show few uerical extedig fuctios for INCERC site usig oly low ad oderate data. As ca see, the obtaiig periods correspodig to strog March evet are Fig. 6.4 Extrapolatio fro partial data ad RC iterpolatio very differets. I the resoat colu device, the saple ca be loaded with a large excitatio doai equivalet to low util to strog earthquakes. he evaluatio ethod of the oliear depedece of doiat periods usig resoat colu data presues a iterpolatig process with a better approxiatio, which replace the ucertai extedig process. Fro this reaso, we thik that the resoat colu deteriatio of the oliear variatios i oralized for: ( PGA) = together with the deteriatio of the oralizatio value iit fro seisic recordig ca leads to a better approxiatio of the doiat periods for large PGA values whe oly low ad oderate seisic data are available.

8 Diu BRAOSIN, Flori-Stefa BĂLAN, Care-Ortaza CIOFLAN 8 CONCLUDING REMARKS he atural period ad rigidity are iversely proportioal both for liear ad oliear aterials. he site geological aterials have a degradable rigidity fuctio of the strai level, ad the icrease excitatio level leads to the rigidity decrease ad the icrease of the atural period values. = PGA - ca be obtaied he depedece of the doiat site period o the excitatio level - fro recorded seisic data if these data cover the etire PGA value rage. he depedece of the atural period o the excitatio level i oralized for - ( PGA) = - ca be odeled by iterpolatio of the resoat colu data. his ethod described i this paper has bee validated usig recordig earthquake data. Whe oly low ad oderate seisic data are available, the resoat colu deteriatio of the = PGA together with the deteriatio of the oliear variatios i oralized for: iit oralizatio value fro seisic recordig ca leads to a better approxiatio of the doiat periods for large PGA values. ACKNOWLEDGEMENS he author wish to express his gratitude to the Miistry of Educatio ad Research for supportig this work through PN II project 316/9. REFERENCES 1. BĂLAN Stefa, CRISESCU Valeriu, CORNEA Io (Editors), Roaia earthquakes fro March 4, 1977(i Roaia), Publishig House of the Roaia Acadey, BRAOSIN Diu, A dyaic costitutive law for soils, Proceedigs of the Roaia Acadey Series A: Matheatics, Physics, echical Scieces, Iforatio Sciece, 1-, pp.37-44,. 3. BRAOSIN Diu, SIREEANU udor, Hysteretic dapig odellig by oliear Kelvi-Voigt odel, Proceedigs of the Roaia Acadey Series A: Matheatics, Physics, echical Scieces, Iforatio Sciece, 3, pp.99-14,. 4. BRAOSIN Diu, Soil dyaics eleets (i Roaia), Publishig House of the Roaia Acadey,. 5. BRAOSIN Diu, Passive cotrol i structural dyaics, cap.1 i opics i applied echaics, vol.3, (Editors: CHIROIU Veturia, SIREEANU udor), Publishig House of the Roaia Acadey, BRAOSIN Diu, Seisic base-isolatio - o-liear iplicatio i period-shift choice, Proceedigs of the Roaia Acadey Series A: Matheatics, Physics, echical Scieces, Iforatio Sciece, 6, 3, pp.49-58, 5 7. BRAOSIN Diu, No-liear atteuatio i soils ad rocks, Proceedigs of the Roaia Acadey Series A: Matheatics, Physics, echical Scieces, Iforatio Sciece, 7, 3, pp.9-14, BRAOSIN Diu, BĂLAN Stefa-Flori, CIOFLAN Care-Ortaza, Soils oliearity evaluatio for site seisic icrozoatio, Proceedigs of the Roaia Acadey Series A: Matheatics, Physics, echical Scieces, Iforatio Sciece, 8, 3, pp.35-4, BRAOSIN Diu, Noliear restraits i seisic isolatio of buildigs, Proceedigs of the Roaia Acadey, 9, 3, pp.17-, CHEN Wai-Fah (Editor), Structural Egieerig Hadbook, CRC Press LLC, CIOFLAN C.O., Ivestigatios regardig soil local effects o assesset ad reductio of the seisic risk (i Ruaia), Ph.D. hesis, Uiversity of Bucharest, MĂRMUREANU Gheorghe (Project director), Deteriatio ethods of the atural site periods usig oise records (i Ruaia), Scietific project PN II 316, MĂRMUREANU Gheorghe (Project director), Deteriatio ethods for doiat periods of the seisic ipute (i Ruaia), Scietific project PN II 316, MÂNDRESCU Nicolae, RADULIAN Mircea, MĂRMUREANU Gheorghe, Geological, geophysical ad seisological criteria for local respose evaluatio i Bucharest area, Soil Dyaics ad Earthquake Egieerig 7, pp , DeSILVA, C., Vibratios: Fudaetals ad Practice, CRC Press,. 16. * * * * * Drevich Log-or Resoat Colu Apparatus, Operatig Maual, Soil Dyaics Istruets Ic., (1979). Received February 3, 9