A Comparison and Experimental Test of the Types of Analysis of the Motions of Buildings Effected by Transient Oscillations Citation Kartzke, Paul Louis (1935) A Comparison and Experimental Test of the Types of Analysis of the Motions of Buildings Effected by Transient Oscillations. Master's thesis, California Institute of Technology. doi:10.7907/fjg5-vk82. https://resolver.caltech.edu/CaltechTHESIS:10072024-173129922 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Civil Engineering) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Civil Engineering Thesis Availability: Public (worldwide access) Research Advisor(s): Unknown, Unknown Thesis Committee: Unknown, Unknown Defense Date: 1 January 1935 Record Number: CaltechTHESIS:10072024-173129922 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:10072024-173129922 DOI: 10.7907/fjg5-vk82 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16778 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 07 Oct 2024 17:46 Last Modified: 07 Oct 2024 17:47 Thesis Files PDF - Final Version See Usage Policy. 4MB Repository Staff Only: item control page

z.n: IliTRODU~"ION - ·• • - - · - - - - • - • - • • • - - • l ·Tne liiot .e-i.Uly ·i s. ~ I I.. _ .,...,. ..,._ ..., _ '!be :,eater€11B1"d Amlyst OH6PfEB xv. - •- ••- -- •- •- - -• •- •• le o'f the ,.- e ter CRAP'fli:R v:. Ocmclus 1? rd 20 '1'tle puJ:pOae ot this SGt tenh the practieal • st. to per to three t-01c1, pl1:eat1ons o:t 't\ methods of' analystnc the mo·tiollS of build1ngs eff:ected by an earthquake. o£ th i r pec.uliaritt a. And, third., by etual shaking table eac_pell'i;m&nts, ~o eo~o.re the rooul t · of a:nalys1 tG real recorded motions. The ftrs,t me'll.od i that fOl!lUtlated by Dr. at the Oalttomta Institute of in 1'11.l 'to the 19~2. u. Blot pre ted at.tonal AppU.o4 f.lechaniea ~eet1ng 1n J'tme • • Since that ti the pra1)tical application or the t eoey ban been d iVeloped at the Cal~rnin Imtttut or Teclmol.ogy,. '!be t'ttll dorlw.tion ls oomplex probl.«ll in ice and trill not be presented her , thu • ting this to an explanation ot the p . ctieal method. to be The second analy: i s ta th t f'omul.ated by ?J:oteseoJ.< a. • ,. -~ tore;sard and presented in t'llll in the Q,t 'thla t heo:ry hss not preT1ously beon ne ring Ne . de. e conelusions put forth 1n th1s per ee based on eight complete .applications of the Btot ane.lyete and t110 appUcationa o:t the :' stergaa.Td theo?"Y• calculat:lans are &ho in this paper. Bxam_plea 0£ these ~ l. Dr . Ii . Biot 1n his IBPer states that .,the motion ot a building e tt"ooted by an earthquake ha.a the e hnracter ot a transl ent oaei 11at ion." U.s1ng the s a::oo :principle as o.ppli ed by ileavis1de in the analysis o:t transient electric currents. Dr. Btot, eonsidertng shearing tor-cos only, (l)pl l '9 t he general thod of aMlysts to the oselllations i n a bUildin Br1 tly et eory 1s, that en s.n harmonic torce aets , ,on an elastic system Ule oGctllation tends to.rard a steady state and t n the ease o resonance the maximum ompll tude is datermiD by the 1 nternal. frietl on. ,'then th syst(lll 1a ef octed by a tm.naieat umpttls , h0'/i18ver, the harmonic mot i on baa not auffie1ent time to devolop. Dr. Blot inve.~tlgates the :r0sponse ot an undemped bull.di , 0 °'o ulses i n tour 6 istlnct phases. OS'Cilla- tioa, -second, 1"orced he.rmonic oaeilllltion ,, third, sudden conatant a.c ele:rations, and fourth • oscillations due lo e.rbitrar:, horizontal aceelerntiona. Uavtag treated tbe cases s,ep-ara tel:y , when f'ully eve.lop takes the tom • ult (x) • whEn"e Uk(x) equal.s the displae , ( V ) t and Kit and F( Y k} are 1\alctions inYGl.T.lng constants and the aotue:1 earthquake I.f the anulysts is mde oomi4e:r1ng th . buil41ng tunet1cn. vibrating. primarily in the fun&.mentnl period• t hia U lxl 0 • K 0 ( Y . ). 0 • ( ( O(t) Din 8 TT'V t dt ) j ~ 0 T 0 ·• :u.a"tion 2 1 'l +( G (t}oos 2 rr'Vt dt) O t:he :tundamental period of tho bull~ in seconds per ose1Ua:tlon as found by experiment. y0 • the frequency of the t'un-damsntol of the building 1n oscillations per seeoad equals _l.,_ f lo • -¼-2 • .. 0 T 0 'Where n is t he :rumibor at st ori.&s e.bove the :fi'.'l'st end a ls the ra:tio ot th • rigidi't y or the f'irst t"loor to that or- the oth.era., 'l'b1s may be ealculated if the colmims have very constant cross•aaotions b y the tomula R • Ln3 Ll 3 (Ln is t .h e het . t Gt one of the upper st.art&& and L 1 tho hei ght ot the :first 0 2 story)• bow ,var i t uns f'o'Wld r.10re satis aotory o detmmno H. mperl.mentall.y. A mestU.10.ble lateral force ms applied at the :f'irst f'loor and a def'leeti.on measured .a.a slit>· • - j dn - I L ______, L Ln 1 • tat<o LI=======! as hen applied to one of the: upp-:u cl n d R • 0 B : 1 ts found f'ron the equation • l s found fttna the re ;(30 • goa . 1 quation B 0 ◊S? • ~ dp io • 2 ex.. lfu:1,o "~ Ao g=J Where x ts t he a:1etru1e • dmm f'1'0m the top at ab1eh the d1 splaca?mlt 1s desired a.ad h 1 s the he1ght ot the building above t.he nrst story. t • G(t) • is the ind endeD"'t var ble, ti.mo. ia e tunet:ton et t el'«)ressing the g:rou.ntl motion in tems of t . This nomooclatu.ra is t.b.e ea?&e a.a used 1.n Dr .. Biot•s paper a t en'IIS a.re t e th ame; ho~ver, tram t h the emet ph.31Siecl. e1 gn1 ficance or the t • • Th1s is due. to the fact t d e &q1Dtions s not appear•. t th~ ph.1sieal val.ms aro not as sily obtain d as the ones v-en above. In 1:h1 manner all of the constants can now be ob The di tttoul ty lie in e r dieal at • ined. luati ng the two 1nte~o under th · Thi~ as done by- nnms ~ ph!oal integra tt·on-. The sin.e and msine tuuet1ons lfhm integrated become c-os1n and sine functions rospeetivaly.. .L -"'° 411 " 111 t e o-urve or th appropriate se&le and t ri :t angles t o bo"tb of th • It earth 11 be seen t p lotted o ake plotted t the 'l'&ueo o:r t he too itttegrs.ls are then express d by the aeet:DU- lative, al braic swns of the ar s ttp to the point in question. Care,. ho ever, mu.st be takm to dlotinguish bet111een positive and neeJ1tive areas. 1th this tntor 1t1on 1t is posalble to evaluate the ootire ex:p.ression dbtaining vel time dul'ing the rthq s for detleettons at every ke and a.t eTe:ey point in the bu1ldl:ng. These "lalue c e.n b el t er positive Ol" negat 1ve and thus they provtd an euvru.ope thin wh1 ih the curve for the actual relative dis.pl.BCEm nt (due to oscillations 1n the fundamental) should r«na1n. " e above to too mathorla an provioual J,,ientioned~ have been far oo ai 11a tlon building.. in L"l the fun I f 1.t is desired t o ke an en~"'ll per1o of the tk-q.t ysis 1',r mcrr nt ny of the bal'mooics, t e subscripts i.~ the above equation ah011.l d b second, or t h i rd ha changed to ls e, or 3 tor the first, on.tea. respeet!vel.y• the mceesary con- stant a:eat n ealoula:t9d mid ftluss ot def'l. By t he t r¥ of aupe osi tion the . d tl&etions dded t o tooee ot the t'um eccn d en.vel is obtained due to tu.ndem.cntal • bered. smp:e• th-e she the first i y be entsl oscillations an d thus a in 1 the displae.eoonts m ham.onie oscillations sl:ould ra1 I n applytng this u1ethod • l • Bi o • must be r ti 01\S oompat.ed., Thet 13 . the bu r-1 aaaumpt ioua tns m 't dity nd. t -he mu ot ea-ch story bov -t be constan , mid t he most lt:l)Ortant de-formation must be i n ho-rtzo-ntal shear. }eglg of ,,b.o. Diot ' ~rds WHld in the cnlw.lation..c; Bi on belott The ga o tat ned . ran a Fartel,. ~ ne t in a photo lJ(!iu. ha ng table d aig~d by Pxofessor ere enl.ar ed to e. :ps s t P~cn phe:r's enkl"ging eamera. neeese.uiry- to e eairable ize In ootng this it fully synch:t'On.1ze the various tra•ees note the enlargemmt f'actor (Z). . R. 1e aino and e e nd ine ·c urves then plotted ha'\l'ing a per.tod of (2 1T' ~ E) • and an a.,ipl.1tud ot 8111 desi ble SiYA, for exmnple, Y• 'l'he ground mtioo sas t.hen plotted at ri.{~"t an les to those cu: ·s aA shown i n Pl.a to !Jl.,. n p tor M.nt;tngu1s as tollo • The eonvm t\oa adop't.ed as it '.I'h e ar-ses are positive when the OG."'Mnate of the grou.nd motion am the slope or the si..110 or c s hln like signs., ine curves areas are thm plotted to the lef't e arees the ordinates md slopea ve unlike st re n ga 1 ve hon end are plotted to the ri ght of the s:1ne or cosine eurv • These nr and the al .·G ~are obtained by maze o£ a plenimetar btnic sums ·ote detl&c ions were desired. t poi "' lfl1em values or Ia tlle exa!'.1ple 1 ven herein,. the model used as ot the di 0nS1.ona s hown balom 1 6" 1 • 5. 75" t 5 . 75" = t 5. 75" t 'rhe ealcu1e.t1-ene for c onstants tor tb equations ot os-eillatio Yo • a ,. 1n t he f'und ,!_, • _!.,. 1.'0 . 323 l P' 0 • l . 3232 2 •¼ • Ll o<.. :: IiD • : 2 l tat are . hon belo : 3.. 10 • 9. &0 1.1eill.Q·U~na/.:• eond., tan A o • ol... • • 0(. • , 523- x 1.,ozz 6,.28. Ao l ..,. 3 • c!o /e l+ Ao . 0554 2 X . 4'141 1.on! -- u0 .- • J 2 • 584 eos ( A o 5 ) : "YJ.,4 Ai.2 • • e. l. lt 1. 5 2 1T' X 3. 1 X . 65 = cos (l . C?"l X 0) ( /\ol/2) - •· cos (l. 077 x 1/2) • .as, ( • • . 4"75 For Roof C'Ofl ( Fol"" 3rd Floor OS For 2nd F'.l.oor eos A 0 ) Ao x l) - • cos (l. O'l'l X l) l The graphical i nt . ation 1s shown in Plat d. plotted in Plate i/12. rasults e..re tabulated in '.i'nble i/1 V5!!ues of' ,. Dlii!! ,m Oeetllations .! ! t:qe 1nt (I Ar 1 2 s ,. 16 . 16 ' . 34 • .38 1.09 .:rn . 39 ., 1. in 12 13 14 15 l . 07 16 17 1. 00 . 72 1 ..40 l.. 11 lS l . 56 l . 52 1 :00 21 22 1.ai 1. 00 23 l..46 1. 5, M .oa • lfi •• . 25 lO ll 2 .02 .01 . 39 4 5 6 8 g .2 u #1 ;, .21 •.1& . 19 .os 0 4amental f u. 31'd 'O'o, 2nd .oo .l.O . 09 . 21 . 2G . m~ . iz ·•· . 51 •• • .24 . 42 . 51 .-44 .M . 38 . 53 .. 18 . 29 . t.7 .:u .!58 . 51 . 62 l.J.2 . 41 .M . 45 1 16 1.1 1..16 l . 44 .:w .01 .&2 1. 09 . 52 . SB l ; the .oo .55 1. 23 1. 20 1.07 • r.: 1. 1.,e •.ZS . 55 . 13 .ea .e, 1.41 1. 24 1. m. ·• • 52 . tw .58 .32 1.-4 l..46 . '11 . 69 1. 30 1. 15 1.ae 1.2& 1 ...12 . 02 . 65 1.00 l. 1. 1. 91 .9 1.•.55 1.56 . 86 1.58 l.M . 91 •• . 23 . 12 1. 52 1. 5? 1. 53 1.00 1.00 . 88 2. 21 4 1. 90 1.00 ,.25 1.02 1.ss . 56 1. 56 . 64 1 ..69 1. 89 2.05 . 75 l . 62 . 97 . 90 .98 .os 1.,, . 8'1 It hamon1r:i vib s believed necesoo.ry to tions. '1ho cal k:o an a nalyeis for the tiona :tor t h (') oonstaute tor this Calqul.a.'t.\Ons Is;t Oonstuts fbr Osctllat1ons !!!. the Harmon.le : -.106 : .. A . .l.08 • By Experiment. = s. :n Oscillation /second. l 1,-2 l n R • • 2 1 Rn c:).., • 2 • .108 lC 3. fl§ s. :: . 062? cos "A, i l + t,(. do.sfr, .f\1 • 34 st • Ki F( Y1) E y 2 17' '(0 P For Roof cos <A1 ~ ) For 2nd Floor cos ( /\1 t ) -• :or 5rd Floor cos ( f\1 ~ } • ,cos cos (3. ,6 43 X 0} cos (5.,645 Xi) ' 643 xl) • • • 1 . 988 .ave 'the graphtoal tnteg;mtton is also shom in PlattJ 11. 'l'he resttl ts • re tabul te4 in the. ecnbined f'U.rld 1n late • entel. arrl ble #2 a1'Jd t he en .· •op ttor nie oooillat1 one ts plot.t Aroo. l . 09 2 3 4 5 . 45 1 ·'° ..25 1. w 1. 45 6 .16 . 7'1 'I 8 6 9 10 ll 12 13 .u 1. os 1. 56 L •ea • 21 . 16 • 1. oa .a:s 1. 45 .es . 21. •• 14 2. 0 15 16 l? . 25 . 02 18 . 56 . 89 . 49 2l .• 73 .• 57 .• 08 .es 1. 36 . 91. . 85 1. 1.36 . s:t. 1. 21 •• 20 24 ·" l.52 .&5 . 17 .n l.::S . 82 .ao .~o 23 1 _.s:, 1 . 11 2..24 . 86 l 22 13 .• 55 . . Pf! . 46 • 29 .ao .ov . 29 . 112 l.00 ., M •94 . 75 •• d'I . 84 . '5 . 43 ..6? .,a:; .74 .4& .M 1.23 .n .u . GS ·" .ae .oo •• •• .25 1.-24 . '42 . t:e .ss . vi .s •• ,.t16 .• 34 .. ,.25 ·. G? •· , .os .ea .as .n 1.os ·"•• •· 2 1. 02 . 01 .'14 •.u .oo .25 . 61 1.oa •.46 .41 .as . 34 .es .a .m "'35 . 51 . 41 .46 .s . 09 .,:48 . 36 •• .as 'file !!ff1Hffl4U'd {m§,ta1s. The the., ey t'ermulatoo by - . fossor ~ te~ ts baae4 u;on the faet that a s\14dsn hori20ntal earthquake shoek m&w a tall bu1ldtng 11:111 ause s \'ffl.Ve or detblffl,Qtiom to ~,vel t,lm:)~ tbe bulldl.ng to the top. 'l'be wave will be retl.oot'ed at the t&p and Will retu:m to the too.ndation and bo retleeted again. The energy l'llll gm.dually be d1&sipated. by dmeping and tnro~l.ete %" leotit>M tm'til the Wilding \'aim placed in matbtD..fttical : t o ~ the th«>ry ts developed ae i'bllow&. x It• = "ffll"tice.1 distame... y • hormontal d&.f1ection+ lt • total stitrnetm of the eolumns. v • veloo1 ty or the shmr wavo. h he1 ht of the buildl.ng. • i' • period ot tbe tttn&miental ot the building. By 4,aaml.c CDnelder.atlo.ns ot tore a ·a cting upon a d1fterent1al of height W'e obtain the equat1GJ1 solving this d1ffol'8llt1al equation , e g t 7 = 5(t i ' ) 19ically this means Ulat the. detleoticm ordinate r ins the value but it baa he.en moved along the ti..'lle axis by the value l. wh1eb is the t1. i t t 109 a sb-ear waff t.o tl"aTel trom X the bottom to t e position x. When the wave 1"04Cheo $he top e new wave le created ffllich starts do mrd., U the sh~m at th... top are now set eq to aero~ ffhieh 1 the eorroet boundary ,e onditio , 1 t 1s found t def'lootion 11 t t e top 1s dou·ble.d and the n t the descending wa,e ls ot the f'cmn X - AA) mum thi& reflected. ave rer:u'!li-ets the foundation another if ve 'ibis c a.se hns the bouaiary cond1 tion that ,r 2 ts =- ,-5 i'ro:llthta find ... + 2b ) v lf' the the- r efi.,e cted round 1 s u,. motion n must be a.dd d t o the actual before analysis can 1b In this ftlUD.d m tion eneral rm1u1.el" the ent'b·e ?Mtion ot tbe ln:Uding h can be plotted. by a eonst e.nt at t i.r! s or reflection n be of eonttmiea.. during a parl.od o any hm t aetor t this time the valu s1ly eonstdwed. By multiplying the eff'eet of a darrt> tng 'llhus, a. c omplete f.ll.J&ie is ~ o .of tb.e 7ff1':tf£Eri :4!!!:l;yais. The :records used 1n tho example ttg able.. obtained f'rom. the a in the previous ~ 1 vcn below W8l'O name bent m used as s~ and tho mmo ground motion .pplled. ~ nccessery oaleul.atimu, ere given below • • t 1 t t ms lt t e a sit r ave to tra l trCD 'the b •Sa to the top ot the bl11!1ng• t then plott$d as sh ee or th ground mt'>tlon 1n Plate • and ho time abeissa dlVid d into 1.ntervals ot . 0032 SQC. ru mm done s, that sh oar be banal • aves of this lenet,b l t this is .n ot done th • p the of the hear uld ve beeomi a . se ot lines in n leh 1 t ls vary D!uu•ly 1mpoas1ble to 41 tinguish anyth1n 1 ahoo.r tfav.& as th traced t the· second, third, a.ml tO!ll'th floor as a dotted line• as shown in ?late 1/3. reflected tran. the top floor- as When tbe ave aga1n reach This ave dot- dash line (Plat 3) . tba ground tt e ret'lected (.d otted line) actual gmud i!>O"U·on and tatmn ~ "'1th th-o next l't1Rpath., ~heae CUT'9'e s h d been plotted, it Wh they were similar to the oms recorded upon the shold.ng table. bui that the p-9rlod ot' the f'um(P. tel e.& t aken traii the c al• as thought bGst tor practical purpos s, to calculate reason it t !Tom the ottn p o4 of the butlding. Therefore - Using this 'Value of' t am plott.t.ng 1n the sarae ,mamior, :f1late G4 ·a s pre.paroa. . ·l"'Te· cot - emi be soe-n ·~ period o:t t me cal.eula t oi es wt th t he. t of t h actual OUl"'V'Gw .A.."'\ investigation of the ~auae o.t thi de end 1t was tmeed to the orig!. . In t his el!J)ress1on 0 height . ie the 1 t or tb unUomly distrlbut erapancy was qu t1on - :.uie equation. ts go d tor a en mo:ro or le d building per unit of in ieh the 18'1t !s ln tho e:rsmple eh en, ho evca. the we1Ght 1."l c.o ncs. trated at t he t hree floor leftls and the equatiOD la not IJ;:led• Thue. in a building of tenty storios o. so, the equation ts valid but cannot be uaed in tho p t ~ • Sine th . &l.J-eady knom , the al temato met.hod could be uaei!. Ia cases where the pe,ri.oo ls not lmown a mre refined method of ealcul.attons must be en.pl~. It ls readily seen that tm Em.tire proeeas t l e ~ upon the aecm-ac7 ot , he plottllig,. i'his must be dom as eare- tully a.s possible if gocd results are te be obtained. It is s~os:ted that a st~tl .• jut admitting to the plo.t ter-'e eye tbe wi dth. or om int·~ .'be prape. ·d. Using th:ls" .an.d templat&s or the ground motion and retleeted waves ,, a quick and simple routine c an be developed• Coagl91!:o,apt,1, The conolw, ions of this report e.re dram eh ie tl.7 ~ the Gar'9'8a. tn eampl• /fl, Plate ii2 M101f8 tha1 the Biot a.al.pis lbr oao,i llat.lona tn t he t'lmdam.ct al t'&1ls o co naidor the qulek shol"t oactllaUona a t the bag:1mttug ot the crm.te. lb.on tron tJ:usmelves ou.t ., '101"$ pmodtcally, flom9Tor. as 1Jd th4J building begins to oacillate this fit-st, et.10.I )le e.n.elys.la is Talid. It a reil MQent hJ made am an aoolysis 1\)r the har11DJ1lc oecilh\io:na is co~ut.ed,. we find iha.t 1b.is tak es oam ot tho short oaolllatl.ons at t,Jle ti.rat and th.e n dies ou:t aa the building 'bagin:a to mare 1:n tile 1'l.D '1:i.bu.s a coapleto study or ttle motaent C8J1 be mde and ea evelopet obtained. It must be l"8m , hared, turaever, that tbts la ,a eu...""Te whieeh goes tbNm.€$1 the points ot z:ia:llm\D rel.a- t.1Te 41splaeement snd doe,s not glve the actual Jf!)VEmm t ot the butldiJlg,. Im example 1,12 , 1>1.ates #3 and ff 4 show two ifest.-gaa.rd anal,pea ror t h.e eetual motions o:f the butltUneot 'the dotlections e&am to be hip. The me.gn.1tudes 'llrese disel"epa nd.eu am dlle,: i'1Ht to 1naccu.rae1u tn plotting and seoond to the fut that ao ~ing t aetor was employed tn the anal.yeis. It ts to lhe dvantagc ot -this method that the r~au.lts obta1oo4 -a re in \ha. tc:rm ot the e.ctus.l movanmt of the building. Tho ua e of the Biot equations nor i s the e m r , hi et i l 1 nt Hgre_ti on ~ is nat c11 f.fi cult, c tin.-::;. ·.:11 e n e t h. o d , huV10Ve.r , 1.s loni and rc ::;,u iros o. f;OO<l d~ .l of tiI:te t o use, especinlly if a nur.lber of ditteren t building'!l a re to b e analyzoo , since c. new set of constants i s :requirm1 for e.::cli case . l'he use 01.· thE: ,,est t31."gR>:l.T d r.i:efuotl is mor1! rcpid. , bu:t; i s e.ot ti s sc hol a1•ly. The chief .. i :.fficul y is i n ~:>L'tu i ni n~; t he correct vo.lua for t , a p robltn wh i . . ,:r· 11 I ot be t .Jken up 1.n thi s pa.par. !u conclu:~on it. . •. y b~ s:lid that hot h «Jlt~1ors have f'ormuleted ri r n ctici::.1 mc thoo.e ot :ruialyoi s ,i\ic11 ht:vo b~;n a1t- perL"J.0!1 t n.11 y ve rH'l ed .