1. Introduction. Eduard Khachiyan. address:

Transcription

1 Earth Scece 8; 7(4): do:.648/j.earth ISSN: (Prt); ISSN: (Ole) Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface Eduard Khachya Buldg Mechac Departmet, Natoal Uverty of Archtecture ad Cotructo of Armea, Yereva, Republc of Armea Emal addre: o cte th artcle: Eduard Khachya. Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface. Earth Scece. Vol. 7, No. 4, 8, pp do:.648/j.earth Receved: May 4, 8; Accepted: July, 8; Publhed: Augut 3, 8 Abtract: Problem of the predcto of dplacemet ad accelerato value for trog ol dplacemet are codered for the cae where a earthquae regarded a a tataeou mechacal rupture of the Earth urface. We have attempted to develop, baed o recet cocept of earthquae geerato proce, mplfed theoretcal method for the quattatve predcto of ol dplacemet parameter durg trog earthquae. A a llutratve example, we coder a earthquae orgatg a a coequece of relatve dplacemet of uddely ruptured bloc a horzotal drecto wth a gve tal velocty. A emprcal relatohp betwee ol partcle moto velocty ear the rupture ad at a certa dtace from t, o oe had, ad the earthquae magtude, o the other had, wa etablhed. It aumed that the mpact of ertal moto of a deep ol tratum o the ertal moto of upper uburface ol tratum at tataeou brea of a medum ca be eglected. By olvg a wave problem for a multlayer ear urface tratum, aalytcal relato were developed for a ol emogram ad accelerogram o the urface depedg o the phycal mechacal ad dyamc charactertc of the ol at all layer of the tratum; atteuato coeffcet of mechacal ol vbrato; the dtace to the rupture; ad the magtude of the predcted earthquae. he reult obtaed eable u to determe the maxmum dplacemet ad accelerato value of the ol, tag to accout local ol codto ad ther varato over tme, a well a the value of the predomat vbrato perod the ol. he method wa appled for old ad looe ol baemet. Keyword: Earthquae, Itataeou Rupture, Ital Velocty, Multlayer Stratum, Wave Problem, Predomat Perod, Semogram, Accelerogram. Itroducto Oe of the ma objectve of earthquae egeerg (egeerg emology) to predct patter of ol vbrato (a cotructo te, for example) ad ther ampltude frequecy charactertc depedg o local geologcal codto durg trog earthquae. Relable data o the character of earthquae duced ol vbrato the ol ca be obtaed oly by record made durg a real earthquae. Owg to the large umber of record of pat earthquae for the area of tudy, charactertc type of earthquae ca be dtguhed. But, the umber of uch record tll uffcet for full tattcal proceg. h cocer both the etre terrtory of the former USSR ad the Republc of Armea partcular. he term of vbrato record of the ol uually refer to the record of ol moto (emogram) ad accelerato record of the ol moto (accelerogram). Sol dplacemet duced by trog earthquae the epcetral area ca be o eetal that accurate trumetal reproducto become techcally feable. Sol dplacemet ca be recorded wth a hgh degree of relablty at epceter of maly wea earthquae ad at a large dtace from the epceter durg trog earthquae. Nowaday, all emcally actve rego have bee characterzed by a large umber of uch record. I both cae, thee record (emogram), term of ther applcato to the evaluato of the emc retace of buldg, are of lttle real teret, ce the tre level tructural ut of a buldg cotructo duced by wea earthquae, relatve to maxmum permble level,

2 Earth Scece 8; 7(4): gfcatly low. It hould be emphazed that there are o problem fdg accelerato record of ol moto. At preet, etwor of tato equpped wth hgh-performace three-compoet accelerograph for recordg trog ol dplacemet ad accelerato of the ol moto both a trog earthquae epcetral area ad at a coderably great dtace from t epceter, are may emcally actve rego. Durg a earthquae, alog wth permaet (ow weght) ad temporary (the total weght of a buldg cotructo, ow, wd preure) load, o-called emc load affect buldg ad cotructo. Actually, there are o load (force) the uual ee of the word. Whe the ol move beeath the tructure, the tructure telf fall behd the moto of the ol, a a reult of erta, ad bed. h le the effect of horzotal force o a buldg perpedcular to t ax. he value of thee force (alteratg ertal load) are prmarly determed by the alteratg ol accelerato magtude, a well a weght ad tffe of a tructure. All load, except for emc load, create a drect phycal mpact o a object ad have cotat drecto. te. Whe degg repoble cotructo, egeer ofte ytheze artfcal accelerogram ug real accelerogram recorded a area wth mlar geotechcal codto. he accelerato ordate of a real accelerogram uually reduced or creaed. I addto, there are computer program that allow oe to create a artfcal accelerogram of a trog earthquae for the obervato te baed o record of real emogram ad accelerogram of wea earthquae. Whe olvg problem of the emc tablty of complex ad exteded tructure, deger eed ot oly accelerogram of a trog earthquae, but alo emogram. All of the above how the relevace of tude that allow oe to develop approache to creatg artfcal emogram ad accelerogram of trog ol dplacemet, adequately reflectg the properte of real emc evet recorded. h artcle dcue a partcular approach to predctg the dplacemet ad accelerato value of trog ol moto coderg a earthquae a a tataeou brea of the Earth urface. A a example, we coder a earthquae duced by dplacemet of ruptured bloc relatve to each other a horzotal drecto (tre lp fault). I th cae, the tal velocty of ol partcle ear the rupture after a earthquae the ma parameter characterzg the earthquae magtude. Baed o theoretcal tude o determg the maxmum velocty ad accelerato value of the ol ear a pot ource J. Brue, tudy reult gve wor by G. Reed, Lomca, S. K. Sgh, L. Eteva, K. Kaahara, D. Well, ad K. Coppermth, a well a our ow data, we etablhed a emprcal relatohp betwee the maxmum velocty of ol partcle ear the rupture ad at ay dtace from t ad the predcted earthquae magtude.. Ital Parameter of the a.. Ital Velocty of Dplacemet the Bloc Fgure. Schematc repreetato of the low (log-term) deformato of a medum durg a perod of earthquae preparato: (a) tra tate of the medum before faultg; (b) the medum dplacemet dtrbuto the drecto perpedcular to the rupture before a earthquae, h a depth of a future rupture; U / tatc deformato of bloc at rupture; R the extet of the deformato feld the perpedcular drecto to the rupture; W area tae a odeformed by a future earthquae due to the relatvely mall tra value. he arrow demotrate the drecto of low movemet of bloc; dotted le future rupture le. Dyamc emc load oly vald durg a earthquae. Hece, parameter characterzg the level of emc hazard of the gve area are determed prmarly by horzotal (vertcal, rotatoal) ol-dplacemet accelerato ad t varato over tme. However, a oted above, the umber of uch parameter for mot emcally actve area are uffcet. here are umerou emprcal formula allowg oe to determe oly maxmum ol accelerato value, depedg o the magtude, focal depth, ad dtace to a obervato Accordg to moder cocept, a earthquae reult from a mechacal brea or rupture of the medum due to the low moto of two cotacted geologcal ut wth ueve boudare oppote drecto. Before a earthquae, cotuouly creag mechacal tree, caued by the low deformato of roc, urroudg a future focu, appear a area of the medum (Fgure a). he troget tatc deformato U / wll be recorded the vcty of a future rupture. At ome dtace R perpedcular to drecto of the rupture (Fgure b), the magtude of thee deformato ca be regarded eglgble compared wth U / ear a propoed rupture [, 6,, 5]. Accordg to Reed ad Bru [, 5]: he oly way for the abovemetoed tree to be dcharged relatve ol dplacemet o the oppote ed of a fault zoe ad at a dtace from t [5]; he tatc dplacemet feld, whch determe the double dpole, wthout momet decreae wth a dtace proportoally to [] (where a dtace from a pot ource (focu) to a obervato pot). Method for determg dtace R ad ther quattatve value for 44 trog earthquae are gve [, 3]. I

3 85 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface partcular, R value ca be determed by a mea (over the legth of a rupture) dplacemet value U [], reultg from a earthquae (Fgure a) by formula (R ad U m): 3 ( 5 5) R = U + () he proce of breag the medum uually occur wth a certa hort perod of tme. I th paper we coder the cae mot ufavorable from the pot of vew of emc acto, where the brea codered to be tataeou. Fgure. Schematc repreetato of the medum after earthquae rupture: (a) cheme of the rupture developmet ad the mechacal model of the upper trata; (b) acceptable cheme of horzotal deformato of the vertcal ecto alog the O O' le; (c) the cheme of upper trata the form of a homogeeou urface trata; (d) horzotal deformato of a homogeeou urface layer. H a total thce of the uburface trata; dtace from the ecto le to a obervato pot; v max velocty of dplacemet of the bloc ear the rupture zoe; v ( ) velocty of dplacemet of bloc at a dtace from the rupture zoe; U (z, t) a fucto of dplacemet the drecto parallel to the rupture; ρ, G, H the dety, modulu of hear, ad a thce of layer, repectvely; (H + H ) the depth of the rupture. - Drecto of movemet of the bloc after the rupture; -the drecto of the rupture; 3-drecto of the frt ertal dplacemet of layer after the rupture (compreo ad exteo the medum). After a tataeou brea (ee Fgure a), the ol partcle of each bloc wll have ome tal velocty, the value of whch wll be determed by the legth ad depth of the rupture, the relatve dplacemet U, ad mechacal properte of the medum, that, the earthquae emc momet M ad, accordgly, the earthquae magtude M. It evdet that ol partcle wll have the hghet velocty ear the rupture zoe. By aalogy wth the abovemetoed cae of tatc deformato, oe ca aume that the tal moto velocty of ol partcle at a certa dtace from the rupture wll be much lower tha that ear the rupture (Fgure a). Proceedg from the extg theoretcal tude ad record of real trog earthquae, we propoe to tae the followg relato betwee the tal value of the velocty at a dtace from the rupture zoe ad the maxmum velocty ear t: v( ) = v R max, () where value R, depedg o the relatve average dplacemet U are tae from the formula (). Accordg to Bru reearch, accelerato exceedg g ad velocty v max of more tha cm/ are poble old ol ear a earthquae focu. He argue that the upper lmt of the real tal velocty of ol partcle the majorty of trog earthquae 5 cm/ec. If we aume that th value vmax = cm/ec correpod to the dtace from the rupture zoe of a trog earthquae wth a magtude of 8.5 ad tae to accout that, accordg to [4],

4 Earth Scece 8; 7(4): able. he value of the velocte of ol partcle vbrato cm/ec, depedg o the magtude M ad the dtace from the rupture m. Earthquae magtude M Mea lp u, m Value of R from (), m Value velocte for rupture v, max cm/ Value velocte v( ) cm/, depedg o the magtude M ad dtace from the rupture (3) m ( <R) Earthquae magtude M able. Cotue. Value velocte v( ) cm/, depedg o the magtude M ad dtace from the rupture (3) m ( <R) M the maxmum velocty of ol partcle proportoal to e, where M a earthquae magtude, the value v max at other magtude ca be repreeted a follow: vmax 8.5 = e M cm/ec. h equato eable u to evaluate the value v max o the urface ear the rupture for trog earthquae wth a magtude of M > 6., the cae where the rupture expoed o the urface. Accordg to () ad tag to accout the above formula tal velocty of ol partcle at a dtace Δ from the rupture, we obta: ( ) М 8,5 v = e R. (3) Here aother emprcal formula [6], led to a earthquae magtude of M ad relatve moto U : lgu =.55M 3.7 (4) O the ba of formula () (4) value U (m), R (m) ad the correpodg value v max (m/ec) for dfferet magtude M were calculated (able ). he magtude ot v max lted able, amely the value of U, R, ad, were alo calculated by the formula () (4). I able are lted the value of the groud vbrato velocte (cm/), depedg o the earthquae tety o the MSK-64 cale. Comparo of able ad able how that the value of the groud partcle velocte calculated by the formula (5) are comparable wth ther MSK-64 value for VII-X-pot tete the epcetral zoe, ad area outde the zoe wth V-X-pot tety ad 6. M 8. magtude earthquae.

5 87 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface able. he value of the velocte (cm/) of ol partcle ocllato depedg o the earthquae tety o the cale MSK-64. Itety V VI VII VIII IX X Velocty Fgure 3 how the v ( ) depedece for ome magtude M ad earthquae tety o the MSK-64 cale... Ifluece of No-tataeou Rupture I cae of o-tataeou detructo fracture, the tal value of ol partcle velocty, accordg to [] ca be repreeted a: = е t t, v v t where: v t the above obtaed value of the tal velocty at tataeou detructo of the medum (3), t the tme of complete brea, cludg the tme of rupture, t ca be wrtte a: L t =, where: L the legth of the future rupture, v the propagato velocty of the travere wave. he durato of detructo ca lat up to te ecod. he rppg beg the lat tage of complete detructo, whe a mall part (wth te to hudred meter) of the rupture le rema ubroe. Utl th pot, the dyamc effect of the earthquae - the geerato of ertal force the groud ha ot yet mafeted telf. he lat tage of detructo (the begg of the earthquae) -wrecg, accordg to expert [, 5], occur wth uperoc peed ad lat a fracto of a ecod, wth the rage t =. to.3 ecod (depedg o the tregth of roc). v

6 Earth Scece 8; 7(4): Fgure 3. Depedece of the rate of velocty of ol partcle from the epcetral dtace ad tety o the cale MSK-64, for dfferet magtude. a.-for magtude 6. M 9., b.- for magtude of 6. M 8.5. herefore, by formula for v, for example, we have: at M = 7., L = 3 m, v = 4 m/, 5 t = = 3.75, 4 at M = 8., L = 8 m, v = 4 m/, 4 t = =. 4 Aumg the rppg of the rupture, repectvely, t =. ec ad t =.3, accordg to the formula for v, we obta: v = v v = v t = e.975 v cae M=7.,.3 t = t e.97 v cae M=8.. t A we ee, t ot the tataeou dcotuty that lead to a gfcat decreae the tal velocty of ocllato of ol partcle ad, coequetly, to the ame mtgato of the earthquae effect o the earth' urface. 3. Mechacal a Model he ear urface roc layer dffer gfcatly from deep oe ther ematc codto ad phyc mechacal properte. hey are le dee, ad ther elatcty coeffcet at exteo E ad hear modulu G are early twofold le tha thoe of the uderlyg roc. I addto, the urface layer are ubject to eetally lower vertcal compreve tre, ther volume gfcat compared to a total volume of ruptured bloc. Owg to thee feature, they are more ubject to hear deformato the horzotal ad vertcal drecto compared to deep layer. herefore, at tataeou brea of the crut, the layer of upper ol wll be ubject to much more teve alteratg dyamc dplacemet duced by the erta of roc tha deep ol layer. Durg a real earthquae, emograph ad accelerograph record thee alteratg movemet ad accelerato of ol partcle o the urface. (Statc ol deformato occurrg durg the log-term preparato of a earthquae are ot recorded by emograph ad accelerograph, a the bae plate of ther cae of a teady ma, held up by a prg, alo move lowly together wth the ol wthout a vbrato of the ertal ma.) If we choe a ear urface prmatc area wth a uform wdth ad a thce of H<<H, where H + H the rupture depth (Fgure a) a bloc at a dtace from the rupture, the t preet a multlayer lthologcal colum (tratum) ay part alog the rupture legth L wth dfferet phyc mechacal ad geometrcal charactertc: dety ρ, hear modulu G, ad thce H (Fgure b). he depth H mea a dtace from the urface (grade elevato of a cotructo)

7 89 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface to dee roc (a deep a 3 m) at a propagato velocty of travere emc wave v S of more tha m/ec. here are umerou cae of multaeou record of real earthquae o the urface ad at deep level up to 5 m, dcatg a gfcat decreae of emc effect at depth [5]. h alo cofrmed by gfcat damage from trog earthquae to udergroud tructure [7]. It alo well ow that the tructure of the uburface multlayer trata gfcatly affect the parameter of emc vbrato. Gve the above, we aume that at the tataeou brea of the medum the fluece of ertal movemet of deep roc layer o that of urface layer ca be eglected (Fgure b). hu, we coder a multlayer lthologcal colum wth a heght H (Fgure b d), all layer of whch at the momet t = have tal velocty v ( ), accordg to formula (a) ad the data gve able, a the patter for the calculato of varato dplacemet (emogram) ad accelerato (accelerogram) of ol partcle movemet o the urface over tme. 4. Mathematcal a Model ad It Soluto he mathematcal tatemet of the ta a follow. Frtly, let u fd a expreo of the pecewe cotat fucto of U z,t (Fgure c) the dplacemet of layer ( ) ( ) U z,t ( ) ( ) U z,t при z h U z,t при h z h... =, (5) U ( z,t ) при h z h... U ( z,t ) при h z h whch atfe the followg wave equato of travere hear vbrato of layer tag to accout the vcoty of the medum [, 8]: G U 3 ρ U U η + = z t z t, (6) =,... ( ) at z = U, t =, z at z = H U H, t =, ( ) ad ( ) codto for cotuty dplacemet ad tagetal tree at the terface betwee layer: G U ( h, t) = U+ ( h, t) (, ) U ( h, t) U h t + = G + z =,,3... z Equato (6) mut alo atfy the followg tal codto: where ( ) ( ) (,) v at t = U z, = U z at t = = t =,... ( ) (7) (8) (9) v the tal rate of moto of the bae of a ecto. he equato (6) wll be olved ug the method of eparato of varable followg the ext formula: (, ) = U ( z) q ( t) U zt () = =,,3... where U (z) a fucto oly of the coordate z, ad q (t) a fucto oly of tme t. Subttutg () to (6) ad tag to accout that () mut be atfed for ay, we obta GU q ρu q + ηu q = () η Dvdg () by U q + q we obta the equato G G U q = = p ρ U η q, () q + G where G the hear modulu of roc of layer; η, the dety;, the coeffcet of vcoty; H, the thce of layer ;, the umber of layer; h, h =, h = H, h = = H H. = Equato (6) mut atfy the followg two boudary codto o the urface (z = ) ad at a depth H: ρ where p a potve umber. From () two equato follow: U p ρ + λu = = λ G =,... ηp + + = = G q q p q =,,3... he oluto of equato (4) wll be a follow: (3) (4)

8 Earth Scece 8; 7(4): 83-9 * * ( ) = t ( + co ) q t e c pt c pt (5) * where c ad c are uow cotat ad crcular frequece of -type free vbrato a etre multlayer tratum wth coderato of vcoty of roc, whch are determed by the formula: p * p =, (6) where a crtcal dampg coeffcet of type free vbrato for the etre heterogeeou tratum. For the majorty of roc << p, therefore, the fluece of compared to p o the agular frequece formula (6) ad trgoometrc fucto of formula (5) ca be eglected. * ag p p, formula (6) ca be a follow: ( ) = t ( + co ) q t e c pt c pt. (7) he oluto of equato (3) ca be wrtte a ( ) λ co λ,... U z = A z+ B z = (8) I order to determe of uow coeffcet A ad B ad crcular frequece p, let u ue boudary codto (7) ad cotuty codto (8). A a reult, we obta the followg ytem of homogeeou lear equato relatve to of uow coeffcet of A ad B : A =, A λ H + B co λ H =, A λ h + B co λ h = = A λ h + B co λ h, A G λ co λ h B G λ λ h = = A G λ co λ h B G λ λ h =,..... (9) Sce the ytem of equato (9), relatve to uow coeffcet A ad B homogeeou, there a otrval oluto oly for the determat of the degree beg zero formed by uow varable coeffcet. I the cae of a two-layer tratum t wll be the determat of the 4 degree, for three ad four-layer trata-6 ad 8 degree, repectvely. After expao of the determat, we obta a complex tracedetal equato cotag oly oe uow varable-the free vbrato frequecy p. For gve G, ρ, ad H t erve a a ba for calculatg the crcular frequecy p ( =,, 3, ) of all form of free (predomat) vbrato of the multlayer thce. It almot mpoble to get uch a explct equato for a large umber of layer. Frequecy value p are ofte calculated ug dfferet computer program drectly from correpodg equato a the determat of the ytem (9). For two-, three- ad four-layer trata tracedetal equato are gve explctly [9, ]: p at ρg ρ at = ρg, () tgp α tgp α = ρg tgpαtgpα tgpαtgpα3 G ρ3g3 ρg ρ ρg + tgp α tgp α3 = ρ G 3 3 at = 3 ρ G + +, () tgpαtgpα tgpαtgp α3 G ρ3g3 3 3 α α4 tgp α3 tgp α4 4G4 ρ4g4 ρg ρ3g3 tgp α tgp α tgp α3 tgp α4 = ρ G ρ G 4 4 at = 4 ρg + + ρg ρg + tgpαtgp α4 + tgpαtgp α3 + ρ4g4 ρ3g3, () ρ G ρg + tgp tgp + ρ ρ α = H, =,, 3... G Subttutg the calculated frequecy value p =π/ (=,, 3 ) to equato ()-(), we obta ytem of homogeeou lear equato relatve to A ad B wth a major zero determat. For otrval oluto of the ta we reject oe of homogeeou equato ad, the, gve oe of the uow varable, for example, B, ( geeral, B ca be tae a equal to ) from the ret of the ytem ( ) of ohomogeeou equato ad determe all ( ) uow relato of A /B ad B/B. I order to determe the uow coeffcet c ad c let u ue the tal codto (9). From the frt codto (9), tag to accout (), (7), ad (8), we have: U ( z) ( c + c ) = (3) = =,.... Sce the codto (3) mut be vald for ay pot z at the heght of the lthologcal colum, ad coeffcet of A ad B, a a part of U (z), ca ot be equal to zero, t follow from (3) that c =, =,.... From the ecod codto (9), we have

9 9 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface = U z ( ) ( c p p c ) = v( ) =,... (4) Multplyg both de of equato (4) by ρ, Uj ( z ) ad tegratg from zero to H, replaced advace wth the um of tegral from h to h, accordace wth (5), we obta: h ( ) ρ ( ) c = v( ) ρu ( z) dz. (5) U z U z pdz j j = h = = h h Gve the orthogoalty of the fucto (, ) U ( z, t ), accordg to [] j h U zt ad ρ U ( zu ) j ( z) dz =, (6) = h j for all combato of ad j vbrato, we obta from (5) v ( ) ρ ( ) = h c = h. (7) p = h h ( ) U z dz ρu z dz hu, baed o (), (7), (8), ad (7), the geeral oluto of the ta, tag to accout all mode of vbrato the ol tratum, a follow: ( ) U z t (, ) v( ) δ p = U zt = e pt. (8) Gve the relatve malle of the dampg coeffcet, U z,t we obta from (8) for accelerato ( ) t (, ) v( ) ( ) δ = U zt = U z p e pt (9) where: =,.... h = h = h = h ρu ( z) δ. (3) ρu ( z) For moto of ol partcle (emogram) ad accelerato (accelerogram) o the urface we have dz dz = ( ) U t U (, t) = v( ) δe pt p ( ) ( ) ( ) U, t = v U p δe pt = t (3) For the mot mple (oe layer) tratum wth parameter of G, ρ, H (Fgure d) the ta eetally mpler: ( ) ( ) U zt, = U zt, z H For oe-layer tratum we have the ext equato atfyg the boudary ad tal codto (7) ad (9) ( ) π v δ U ( z ) = B co z, A =, c =, cb =, H p ad the geeral oluto a follow: U U where o δ ( z,t) = v( ) π p = = π H π co zdz + H 4( ) =, (3) π ( )π co zdz H π 4H = = p v ( ), (33) the perod of type free vbrato a layer; the crtcal dampg coeffcet of type free vbrato a layer; ad G = π v ρ H, =,,..., = = H H o δe π π πz πv co H H πz H πv H ( z,t) ( ) e t = v δ co t = o t the propagato velocty of travere wave a layer. A ca be ee from formula (8) ad (33), the ma dfferece betwee homogeeou ad heterogeeou trata δ (cotructo te) coeffcet ad relato of,,...,. o o o o o o For a homogeou layer thee relato (ee (8)) are odd umber, 3, 5, 7,, ( ), ad the g of coeffcet δ for eve wll be +, wherea for odd t wll be. herefore, for the ummato of ere (33), δ g ad maxmum of trgoometrc fucto v π t for ay wll be the ame. h lead to ther H algebrac ummato ad a gfcat creae the total maxmum value of the ere. he probablty of uch cocdece for a heterogeeou colum very mall, whch reult the creae of the total of the ere. Formula (33) ca be terpreted term of wave. It ow that a real emogram the reult of the ummato of travere emc wave repeatedly refracted ad reflected from lower v = G ρ t

10 Earth Scece 8; 7(4): 83-9 layer ad the urface. h cocluo alo follow from formula (8) ad (33). I fact, we ue the well-ow tatemet [3] that ay tadg wave ca be replaced by a par of travelg wave, a follow from the equalty π π π co z ct = H H H, (34) ad, o the cotrary, ay travelg wave ca be repreeted a π a par of tadg wave wth phae hfted by, ce π H ( z + ct ) πz πct πz πct = co + co, (35) H H H H where c the velocty of wave propagato. he, the oluto of (33) ca be repreeted the form of a wave: U π H U π H ( z,t) = v( ) ( z + vt) + ( vt z) ( z,t) = v( ) = = oδe 4π π H π δe t t π H ( z + vt) + ( vt z) ( z + ct) + ( ct z) o π H, (36) where v the velocty of travere wave a homogeeou layer. A follow from equato (36), the emogram ad accelerogram of the ol repreet the um of a et of cdet ad reflected from the urface wave wth varyg ampltude ad frequece. Accordg to moder cocept of the ol dyamc, there are the followg relato betwee the crtcal dampg coeffcet (%), the dampg decremet θ, the aborpto coeffcet ψ, ad free ocllato perod : θ ψ = θ, =. (37) he water aturato of the ol lead to a early twofold creae the decremet of dampg θ compared wth that the dry ol. he θ value of dry ad at mea tra reache. (Faccol ad Reédz, 976). Accordg to expermetal tude by Stavter the value for dfferet type of the ol wth a dety (.63 ρ (to/m 3 ).) ad a level of humdty from 5 to 3%, the θ value chage from.7 to.64 []. hu, equato decrbg ol dplacemet o the urface (at z = ) (emogram) ad t accelerato (accelerogram) at a dtace Δ from the rupture zoe at a magtude M of a predcted earthquae tag to coderato (3) ad (3) for a oe layer bae wll be: o U U (,t) = e M 8. 5 R R = o δe π π π t π M 8. 5 о (,t) = e δe t = o θ t о θ t o o (38) where the value R, depedg o a magtude M, tae from able. Equato (3) allow u to tae ay other relato ad codto for the gve area, tead of value v ( ) (3), ce t cluded (3) a a cotat tme depedet (t) factor. Due to th, t ca ot fluece the ummato of the mode of vbrato. Accordg to (38), the accelerato value homogeeou old ol at gve M ad hgher tha that of a homogeeou looe ol layer of the ame thce. h pheomeo ha bee repeatedly cofrmed by trumetal record durg wea ad moderate earthquae. Sce the ampltude of all harmoc (38) are greatly reduced over tme, Ut ad whe earchg for the maxmum of expreo of ( ) U ( t) oe ca oly tae to coderato the frt three member of ere (the frt three mode of free travere vbrato a layer of H thce). hu, accelerato of the ol dplacemet o the urface (ad depth of a layer) wll be a uperpoto of dumpg harmoc ocllato wth perod equal to thoe of free ocllato a layer. Aother feature of thee formula (36) that accelerato of ol dplacemet deped eetally o t emc type, whch determed by value of predomat perod of a cotructo te formula (38). I order to determe the emc type of the ol, t recommeded [] to ue, addto to a mea value v, the perod value a a tegral charactertc of heterogeeou ol, a wa dcued above, ce t determed by phyc mechacal feature ad the thce value of all layer. I addto, t ow that the relato of perod of duced (ol) ad free vbrato (cotructo) play a gfcat role for all dyamc effect. I accordace wth accepted tadard of earthquae egeerg the Republc of Armea [9], ol type etablhed accordg to value of v ad [7]) (able 3). hu, the cae of a large umber of layer, approxmate value о ad mea value v are recommeded to be determed by formula H 4H = G о =, v =,,. (39) v H = H v = H = ρ v = he ma feature of formula (38) that baed o a predcted earthquae magtude M gve ad a ow dtace from a obervato pot before rupture (actve fault) oe ca evaluate ot oly maxmum dplacemet ad accelerato value of the ol, tag to accout the ol codto ad varato of thee parameter over tme ad,

11 93 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface whch qute mportat, the vbrato perod. I th cae, accordg to (38), hort-perod vbrato wll domate old ol, ad log perod vbrato, looe ol. he ma feature of ay complex vbrato proce t frequecy pectrum. I our opo, vbrato perod (o accelerogram) play a major role the behavor of the abovegroud part of a buldg cotructo durg a earthquae. Semc mpact belog amog the dyamc effect where reoace pheomea (the cocdece or mlarty of vbrato perod the ol ad free vbrato the abovegroud part of cotructo) duce the greatet effect. here are umerou tude coderg value of predomat perod of trog earthquae duced ol vbrato [5,, 3], accordg to whch the pectra of earthquae repoe (the maxmum accelerato value o the pectrum) baed o record of real ol accelerogram obtaed area wth dfferet geologcal codto, whch are regarded a a ource of the mot geeral formato about thee value. At the curret tme, there are a large umber of uch pectra recorded. A comparatve aaly how that predomat perod for old ol durg a earthquae are maly a arrow rage, from.5 to.4 ec, wherea thoe for the looe ol are a wde rage, from.5 to. ec, ometme reachg 3. ec or more. he value of predomat perod [7, ] determed by formula () () for dfferet geologcal ecto (wth the value of phycal ad mechacal properte of layer ρ, G, ad H, obtaed durg expermetal drllg) ad ther comparo wth predomat perod of pea accelerato value o repoe pectra for trog earthquae recorded the ame ecto [] have good correlato. able 3. Sol type, accordg to ther emc properte depedece o v ad, accordg to [, 9]. ype v, m/ec, ec I >8.3 II 5 v 8.3 <.6 III 5< v < 5.6 <.8 IV <5 >.8 v - the mea velocty of traveral wave propagato wth the etre heterogeeou tratum H, from the grade elevato of a cotructo to dee roc at 8 m/ec accordg to formula (39); the predomat perod for the etre tratum H from the grade elevato to dee roc wth v v 8 m/ec. he maxmum (the larget of the maxmum value) accelerato value the repoe pectrum ca be recorded durg the frt predomat perod, o the oe had, ad durg hgher perod of ad 3, o the other had. A how [7], the real value of prevalg perod for earthquae wth M > 6 are cloe to thoe determed by a harmoc aaly of ol mcro vbrato at the obervato te. It hould be oted that there are other wor [6], howg that the type of dplacemet the dtace from the rupture zoe, magtude, ad dtace from the rupture dtace greatly affect the formato of value of predomat perod. It hould be alo oted that the value of predomat vbrato perod the ol ad free vbrato of buldg ad tructure of ma cotructo are the ame rage of.. ec. h fact eetally creae the probablty of the occurrece of reoace pheomea. I addto, th probablty for trog earthquae creae due to ther loger durato. I our opo, reoace vbrato are the reao for mot cae of collape ad erou damage to buldg ad cotructo durg trog earthquae. h cofrmed by the erou damage to dvdual cotructo recorded durg earthquae at large dtace from ther epceter, eve at low level of ol accelerato ampltude (<.g). he detructo recorded durg earthquae 985 Mexco Cty ad the Spta earthquae 988 tow of Leaa ad Krovaa, are regarded a clacal example [, ]. he above mea that the deg of ew buldg ad tructure hould be preceded by relable predcto of ot oly the maxmum ampltude of ol accelerato, but alo the value of predomat perod of vbrato durg a earthquae. h predcto wll allow the choce of the mot favorable groud te for each cotructo wth a certa frequecy pectrum of ow vbrato. Accordg to [, 9], th codto expreed by the followg equalty: >. 5 or 5. <, where the perod of the frt mode of free vbrato of the abovegroud part of a buldg cotructo, ad the domat vbrato perod a ol layer by formula ()-(). Sce buldg ad tructure ad ther multlayered baemet have g of hear deformato, the relato betwee the value of perod of hgher mode of vbrato both type of object are the ame. h ugget that havg avoded reoace cotructo for the frt mode of vbrato, oe ca avod reoace vbrato for hgher mode of vbrato. 5. Applcato of the Propoed Method of Predcto of the Surface Groud Vbrato Parameter 5.. Sythetc Semogram ad Accelerogram of Varou Groud A llutratg example, emogram ad accelerogram of the bae for ol emc properte of I-IV categore wth the predomat perod from. ec to. ec, accordg to the code of quaeproof cotructo of the Republc of Armea [9] have bee calculated (able 3.). he ummary data of the maxmum value U max ad U max obtaed by formula (38), tag to accout the three form of ocllato ad oly the frt form at M = 7., = 5 m, (R =.9m), gve able 4. he correpodg artfcal emogram ad accelerogram are preeted Fgure. 4 ad 5.

12 Earth Scece 8; 7(4): For a uform urface layer, a ee from (3), the value of ocllato perod dffer 3 ad 5 tme, therefore at the that pot of occurrece of the maxmum ampltude of the dvdual term of the ere (38) cocde tme ad the drecto (the g δ ). h lead to a gfcat creae the total value of the ere, epecally for accelerato U (. h how graphcally Fgure 6 for accelerato U ( t) calculated by the formula (38) by the frt, ecod ad thrd wave form dvdually ad ther combed acto (uperpoto) for the cae of =.5 at the ame put parameter: M = 7., = 5 m, Θ = Θ = Θ 3 =.3. A ca be ee the data lted able 4 ad how Fgure 4. ad 5 for hard ol ( <.45) ad looe ol ( >.45), tag to coderato the hgher form of ocllato of the bae lead to a creae the maxmum value of the urface accelerato by.53 tme o the average, calculated oly for the frt form of ocllato, ad jut by. tme for maxmum value of groud dplacemet. At the ame tme, the durato of teve ocllato o looe ol up to tme greater tha o hard ol. Oe of the reao for th pheomeo that the dplacemet drectly deped o the perod, ad accelerato are verely proportoal to. I addto, for hard ol, ( =.5), the perod of the ecod ad thrd form of ocllato, accordg to the formula (3) are: =.83, 3 =.5. It readly ee that they are of hgher-frequecy comparo wth ther value for looe ol. Several vetgator coder the occurrece of uch hgh-frequecy ocllato o the groud urface hghly ulely evet ad doe ot poe a dager to mot of the above-groud tructure. herefore, apparetly, ther preece formula (38) for rocy ol wth perod ad 3 le tha., ca be codered urealtc,.e. for the calculated value of the accelerato of old ol wth perod.5, ca be tae oly tag to accout the frt form of ocllato. For the examed example t wll be.36g. Let u ote oe more crcumtace favor of uch a cocluo. he value of the teral frcto coeffcet ad 3 for the ecod ad thrd waveform (due to the lac of expermetal tude relatg to the procee of teral frcto the hgher form of travere roc vbrato) were adopted both for the frt form, whch ca alo lead to great error. A far a th pheomeo cocered there are dfferet vew the lterature tatg that eve f uch hgh-frequecy ocllato o the groud urface occur, they qucly decay ad may have ome effect o the formato of the total accelerogram of the ol oly a arrow focal zoe of the earthquae. Accordg to the ature of the ocllato, the artfcal emogram ad accelerogram how Fgure 4.5 are cloe to the emogram ad accelerogram obtaed real earthquae Port Gueeme o March 8, 957 ad Par feld o Jue 7, 966, preeted wor [8, 7]. able 4. Maxmum value of dplacemet U max ad accelerato U " max of the ol at magtude M = 7. ad at the dtace = 5 m, depedg o the predomat bae perod. he predomat perod of the bae ocllato, ec Dampg decremet of the groud, Θ he bae category by emc properte accordg to [9] Gve the three form of ocllato by (38) Maxmzato Maxmum Maxmum accelerato U tme t, dplacemet U max, cm " max, fracto of g..3 I I I II II II II III III III IV IV IV IV IV able 4. Cotue. ag to accout oly the frt form of Relato betwee dplacemet ad accelerato, tag to accout he predomat perod of the ocllato, by (38) the three form of ocllato to ther value, oly for the frt form bae ocllato, ec U max, cm U max, fracto of g For dplacemet For accelerato

13 95 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface ag to accout oly the frt form of Relato betwee dplacemet ad accelerato, tag to accout he predomat perod of the ocllato, by (38) the three form of ocllato to ther value, oly for the frt form bae ocllato, ec U max, cm U max, fracto of g For dplacemet For accelerato

14 Earth Scece 8; 7(4): Fgure 4. Sythetc emogram of varou bae ( ), at M = 7, = 5 m.

15 97 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface Fgure 5. Sythetc accelerogram of varou bae () at M = 7., = 5 m.

16 Earth Scece 8; 7(4): Fgure 6. Accelerogram of the groud at =.5, tag to accout the uperpoto of the frt three form of bae vbrato accordg to formula (7) ad the frt, ecod ad thrd form of vbrato. 5.. Dplacemet ad Accelerato of Sol the Epceter Zoe able 5 how the maxmum value of dplacemet ad accelerato of old ad looe ol at the rupture = ad at a dtace = 5 m, from the dcotuty depedg o the magtude of the earthquae M. able 5 how that the dcotuty ( the epceter zoe) the value of the accelerato of old ol ca already reach oe g at magtude M 7., ad wth looe ol, oly at magtude M 8.. At the extreme magtude of a trog earthquae M = 9., the value of accelerato of old ol ca reach up to 8g, ad wth looe ol, the accelerato ca reach up to g. able 5. he maxmum value of dplacemet ad accelerato of old ad looe ol at rupture = ad at a dtace = 5 m, depedg o the earthquae magtude M Earthquae magtude, M Maxmal dplacemet ad accelerato of the groud at the rupture = for hard groud =.4 U max, fracto U max, cm of g Maxmal dplacemet ad accelerato of the groud at the rupture = for looe groud =.8 U max, fracto U max, cm of g Maxmal dplacemet ad accelerato of the groud at the dtace =5 m for hard groud =.4 U max, fracto U max, cm of g Maxmal dplacemet ad accelerato of the groud at the dtace =5 m for looe groud =.8 U max, cm U max, fracto of g From able 4 t alo become clear that the value of dplacemet ad accelerato of old ad looe ol at a dpero of = 5 m are approxmately half that of the rupture, regardle of the magtude of the earthquae. Wth a creae the magtude of the earthquae by oe ut, the magtude of dplacemet ad accelerato of ol creae.73 tme. he dplacemet ad accelerato of ol wth a earthquae wth a magtude M = 6. tme maller tha a earthquae wth a magtude of M = O the Lmtg Value of Dplacemet ad Accelerato of Sol he maxmum dplacemet ad accelerato of the ol, gve the above able 4, are obtaed by a aalytcal method, aumg the deal elatc wor of the ol. Whe coderg the elatc-platc tage of the ol wor, they wll be completely dfferet. At the ame tme, gfcat chage, toward the creae, wll udergo the value of the maxmum dplacemet U max of the ol wth the prevalece of the redual deformato. For example, accordg to the average tattcal emprcal

17 99 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface etmate at M = 7. the focal zoe, wth geeral PGD = 4cm ol dplacemet, the redual dplacemet for rocy ol 37 cm, therefor, the elatc dplacemet wll be 4-37 = 3 cm, ad for looe ol wth PGD = 45 cm, the redual dplacemet 4 cm, coequetly the elatc dplacemet wll be 45-4 = 5 cm [6]. Below a mall calculato howg that for roc the begg of detructo (the ed of elatc deformato) ca occur dplacemet wth.6-5 cm. I a elatc medum from a preadg travere wave U ξ ξ = v (4) ( t, ) f t, at a dtace ξat tme t a certa relatve hear deformato ad velocty of ol partcle, the value of whch wll be: U = ξ v ξ f t, v U = v t g ξ = f t, v (4) I cae of a plae travere wave, a ca be ee from (4), for the relatve hear deformato γ, we have U U vg γ = = =, vg = γ v (4) ξ v v t herefore, the proce of determg the value of deformato γ eetally mplfed, ce th ca be doe drectly ug the value of the ocllato velocty, that tead of a complex operato of ettg up the dfferece U = U U of the dplacemet of two pot of the medum, oe ca tart from the earthquae velogram U t at oe pot of the ol ad the magtude of the wave propagato velocty v. For a harmoc plae travere wave, we have U π ξ ξ, v ( t, ) = U t max co v U π g = = U max t (43) where U max the ampltude of ol vbrato, the perod of ocllato of the ol partcle, v the travere wave propagato velocty. O the ba of (4) ad (4), th cae we wll have U v π max γ =. (44) he roc of the earth' crut ca wthtad a certa hear deformato, after whch hear crac are formed the medum. If deote the lmtg hear deformato of the γ medum through lm, the from (44) the ol vbrato ampltude U max, U max v = γ lm. π (45) Smlarly, for the ol partcle ocllato velocty v = U ad accelerato of ol U uder elatc g vbrato, we have π U max = v γ lm, U max = v γ lm. (46) γ lm =.5. 4 he magtude ( ) codered the mot probable for mot earthquae [6,, 4, 6]. Accordg to the reult of laboratory tet of ol ample, t beleved that γ of the roc of the earth' crut ca reach the value γ lm 3 lm = [6]. For xtee varat of rocy ol, the value of maxmum dplacemet U max, the velocty U max of ocllato of ol partcle ad the maxmum accelerato U max of the ol, calculated by (45) ad (46), are lted able 6. able 6. he value of the maxmum elatc dplacemet, velocte ad accelerato of rocy ol. Opto of ol γ lm v, m/ Т, U max, cm U max, cm/ U max, cm/ By the way, accordg to the ame E. F. Savorey [8] methodology, for grate at v = 33см/ сек ad =.5 the varato rage of U wth cm to cm. max Whe tag to accout the dampg of wave, due to teral frcto roc, the value U, max U, max max preeted able 6 ca be reduced by approxmately -5%. hu, the cae of elatc ocllato of rocy ol, the maxmum value of dplacemet, velocte, ad accelerato of the rocy ol ca reach, repectvely, up to 3.57 cm, cm/, ad 8 cm/ wthout ther detructo. Occurrece of hgh ol accelerato o the earth urface apparetly are codtoed ot by the value of the perod of the ma form of ocllato of the ol ecto, but to the perod of the ecod or thrd waveform 3 (3), whch are repectvely 3 ad 5 tme maller tha the perod of the frt waveform. h evdeced by the value of maxmum dplacemet, velocte, ad accelerato preeted able 6, correpodg to the varat of ol -6, for whch the value of the perod have bee calculated accordg to the tradtoal formula = 4H v, wth the layer thce H = 3 m. U

18 Earth Scece 8; 7(4): 83- he ampltude of ol dplacemet alog urface wave trog earthquae propagatg at large dtace reach up to 3 cm wth a perod = ec, but they do ot repreet ay dager [5]. Such etmate for earth (looe) ol are uacceptable, ce the proce of dcotuty drupto uch ol of more complex ature. hey durg the trog earthquae are ubjected to ether lquefacto, or ueve precptato, whch ca reach everal meter. hee pheomea poe a erou dager to buldg ad tructure (they ca fall wthout detructo), f they are calculated eve for hgh horzotal accelerato of the groud. able 7. Frequece of harmoc ocllato ( hertz), correpodg to the gve dplacemet U (cm) ad the ol accelerato U ( fracto of g). U fracto of g U, cm Fat umber dcate the fluctuato expected cae of earthquae of moderate tregth. I cocluo, we ote that we hare the opo of Ch. Rchter [5] that wave wth maxmum dplacemet of the ol do ot cocde wth the wave of maxmum groud accelerato. Large value of groud accelerato are aocated wth mall groud dplacemet, ad large ol dplacemet are aocated wth low frequece ad low ol accelerato. A graphc llutrato of th provded by the data gve able. 7, borrowed from the wor of Ch. Rchter [5], wth ome addto developed by the author. 6. Ma Reult We developed a method of forecatg the accelerato ad dplacemet value durg trog ol d placemet duced by a earthquae coderg t a a tataeou rupture of the Earth urface. h method baed o the fact that ear urface roc layer have a much greater degree of horzotal hear deformato compared to deeper layer. herefore, for tataeou rupture of the Earth crut a fluece of ertal moto of deep roc layer o that of the urface layer ca be eglected. Accordg to th, the heterogeeou ol tratum of a thce H to the bed roc wth v S > m/ec, all layer of whch have the velocty v at the begg of a earthquae, are ued a a cheme for the calculato of the regularte of varato dplacemet (emogram) of ol partcle o the urface over tme. Havg aalyzed the data [, 4, 6,, 5, ], we propoe to determe a tal velocty value v a the fucto of a predcted earth quae magtude M ad a dtace from the rupture to the obervato pot by formula (3) ad the data gve able. Ug the oluto of the wave equato of tra vere vbrato layer (6) wth the boudary ad tal codto (7), (8), ad (9), the aalytcal expreo (8) of dplacemet ad accelerato for all level of layered geologcal ecto were obtaed. her ma feature that for a gve predcted earthquae magtude M ad a dtace from the obervato pot (a cotructo te) to a predcted rupture (actve fault), oe ca etablh ot oly the maxmum ol dplacemet ad accelerato value baed o the local ol codto, ad ther varato over tme, but alo the value of the predomat vbrato perod the ol. he reult obtaed are llutrated by example of a homogeou urface layer wth old or looe ol at M = 7 ad = 5 m (Fgure 4 ad 5). he propoed method ca be ued for the developmet of map of emc zog ad emc hazard aemet of cotructo te of epecally repoble dvdual object. 7. Cocluo. It how that the ot tataeou rupture lead to a gfcat decreae the tal rate of ocllato of the ol partcle ad, coequetly, to the ame mtgato of the earthquae effect o the groud urface. he value of the rate ad tete of earthquae, etablhed by the developed method, have bee compared wth the tete etablhed o the ba of the MSK-64 emc cale, howg a uffcet correlato for earthquae wth V-X tety ad magtude 6. М 8. (Fgure3). Depedece of v ( ) for value of magtude 6. M 9. ad earthquae tety o the MSK-64 cale were plotted, whch, by aalogy wth depedece of the decreae of the maxmum value of the ol accelerato a a fucto of the epcetral dtace, ca be called the earthquae tety atteuato curve (Fgure 3).. Fgure 4, 5 ad able 4 how ythetc emogram ad accelerogram plotted accordg to the bac formula (38) for varou ol bae of emc properte (able 3). he durato of teve vbrato o looe ol up to 3 tme greater tha o hard ol. ag to accout the hgher form of ocllato of the bae (the ecod ad thrd form) for hard ad looe ol lead to a creae the maxmum accelerato of ol, calculated oly for the frt form of ocllato,.53 tme o average, ad oly. tme for maxmum ol dplacemet. 3. he value of hard ol accelerato ca already reach g at M 7. magtude, ad wth looe ol oly at M 8. magtude. Wth a creae of the earthquae magtude by oe whole ut, the dplacemet ad accelerato of ol creae.73 tme. he dplacemet ad accelerato a earthquae wth a magtude M = 6. tme maller tha for a

19 Eduard Khachya: Predctg of the Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface earthquae of M = 9.. he value of dplacemet ad accelerato of hard ad looe ol at a dtace of = 5 m from the rupture are about half that of the rupture, regardle of the magtude of the predcted earthquae. 4. I preet developed method determg the lmtg value dplacemet, velocte ad accelerato for rocy ol, aumg the deal elatc wor of the ol (able 6). Referece [] J. N. Brue, he phyc of earthquae trog moto, Semc R ad Egeerg Deco, Lomtz, C. ad Roeblueth, Edtor, New Yor: Elever, 976, pp [] G. Butcher, D. Hop, R. Jury, W. Maey, G. McKay, ad G. McVerr, he September 985 Mexco earthquae: Fal Report of the New Zealad Recoaace eam, Bull. N. Z. Soc. Earthquae Eg., 988, vol., o.. [3] H. Jeffrey, ad S B. Wrle, Method of Mathematcal Phyc, Cambrdge: Cambrdge Uv., 95, d ed. [4] L. Eteva, Semcty, Semc R ad Egeerg Dec o, Lomtz, C. ad Roeblueth, Edtor, New Yor: Elever, 976, pp [5] E. Faccol, ad D. Reedz, Sol dyamc: Behavor cludg lquefacto, Semc R ad Egeerg Deco, Lomtz, C. ad Roeblueth, Edtor., New Yor: Elever, 976, pp [6] K. Kaahara, Earthquae Mechac, Cambrdge: Cambrdge Uverty, Pre, 98. [7] Khachya E. Y. O Bac Cocept for Developmet of Uted Iteratoal Earthquae Retat Cotructo Code. Earthquae Hazard ad Semc R Reducto. Edtor S. Balaaa, A. Ctera ad M. Melumya, Kluwer Academc Publher, Netherlad,, pp [8] N. M. Newmar ad E. Roeblueth Fudametal of Earthquae Egeerg. Pretce-Hall, Ic. EglewoodClff, N. Y. [9] E. Y. Khachya A Method of Determato of Domat Vbrato Perod Value for Nohomogeeou Multlayer Groud Ste. Horzo Reearch Publhg Corporato, USA Uveral Joural of Egeerg Scece 3, vol. (3), pp [] E. E. Khachya, Prladaya emologya (Appled Se mology), Yereva: Gtutyu, 8. ( Rua). [] E. Y. Khachya O a Smple Method for Determg the Potetal Stra Eergy Stored the Earth before a Large Earthquae. ISSN , Joural of Volcaology ad Semology,, Vol. 5, No. 4, pp Pleade Publhg, Ltd.,. [] E. E. Khachya, O Determg of the Ultmate Stra of Earth Crut Roc by the Value of Relatve Slp o the Earth Surface after a Large Earthquae. Scece Publhg Group, Earth Scece, do:. 648/j.earth ISSN: (Prt); ISSN: (Ole) Receved: October 6, 6; Accepted: November, 6; Publhed: December, 6 Vol. 5, No. 6 pp -8, [3] E. E. Khachya, Method for Determg the Potetal Stra Eergy Stored the Earth before a Large Earthquae. Scece Publhg Group, USA Earth Scece, vol.,, 3 pp [4] E. Y. Khachya O the Poblty of Predctg Semogram ad Accelerogram of Strog Moto of the Sol for a Earthquae Model Codered a a Itataeou Rupture of the Earth Surface. ISSN 747_939, Semc Itrumet, 5, Vol. 5, No., pp Allerto Pre, Ic., 5. [5] C. Lomtz, ad K. S. Sgh, Earthquae ad earthquae predcto, Semc R ad Egeerg Deco, Lomtz, C. ad Roeblueth, Edtor., New Yor: Elever, 976, pp [6] N. N. Mhalova, ad F. F. Aptaev, Some correlato relato betwee parameter of emc moto, J. Earth quae Pred. Re., 996, vol. 5, o. 5, pp [7] Oamoto S. Itroduato to Earthquae Egeerg. Uverty of oyo Pre, 973. [8] E. F. Savare, Semchee voly (Semc Wave), Mocow: Nedra, 97. ( Rua) [9] SNRA II Semotooe trotel tvo: Normy proetrovaya (Cotructo Regulato of Republc of Armea II EarthquaeRetat Buldg: Deg Stadard), Yereva, 6. [] L. R. Stavter, Semotoot oova fudametov (Semc Retace of Foudato), Mocow: Izd. Aot. trot. vuzov,. ( Rua) [] A. N. hoov, ad A. A. Samar, Uraveya matemat cheo fz (Equato of Mathematcal Phyc), Mo cow: Naua, 977. ( Rua) [] D. L. Well, ad K. I. Coppermth, New emprcal rela tohp amog magtude, rupture legth, rupture wdth, rupture area, ad urface dplacemet, Bull. Semol. Soc. Am., 994, vol. 84, o. 4, pp [3] V. Zeleovch, ad. Paalev, Yugolav code for aemc deg ad aaly of egeerg tructure emc rego, Proceedg of the 8th Europea Coferece of Earthquae Egeerg, Lbo, 986, vol., pp [4] Rtae. Earthquae Predcto. Elever Scetfc Publhg. CoAmterdam, 976, 357p. [5] Ch. F. Rchter Elemetary Semology. W. H. Freema ad Co., Sa Fracco, 958, 768p. [6] Mog K. Earthquae Predcto Academc Pre, 985.