Dynamic Studies of the Displacements of Buildings Frames Due to Vibrations Citation Jennison, James Henry (1936) Dynamic Studies of the Displacements of Buildings Frames Due to Vibrations. Master's thesis, California Institute of Technology. doi:10.7907/ed6a-3886. https://resolver.caltech.edu/CaltechTHESIS:07282025-150624673 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): Martel, Romeo Raoul Thesis Committee: Unknown, Unknown Defense Date: 1 January 1936 Record Number: CaltechTHESIS:07282025-150624673 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:07282025-150624673 DOI: 10.7907/ed6a-3886 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17561 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 31 Jul 2025 17:42 Last Modified: 31 Jul 2025 18:20 Thesis Files PDF - Final Version See Usage Policy. 38MB Repository Staff Only: item control page

DYNAMIC STUDIES OF THE DISPLACEMENTS OF BUILDING FRAMES DUE TO VIBRATIONS Thesis by James H. Jennison In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California. 1936 Preface Inasmuch as there are a nun I~er o f sen f1rate studies included in this paper, it ha s been d 1ffic,.1lt to organize tl1e ?'esult, s into a W-91 l u nl fled reuort. However, the materia1 cont ri ined herei n can b e t3 1 vided into three main divisio n s, first, the de s criutlon s of two theoretical analyses, anct t heir annlic a tion to the Bank' of America ?u1lding , Sa n .Tose, California, described in Chapters I to VI ; second, t he tests illade using a shaking table to nroduce transient vi bra tirrn s, describe d i n Chapters VI I to X; a nd thtrd, the tosts of si mple bents subjected to e s tablis he d simple harmnntc ground motions, descrl t ed 1n Chapter XI. All of t he work must b e considered to be of theoretic a l L :roortance only, in that many simn lificattons are made, a nd the apn11cat1on of the results to desi g n of earthqu ake resist a nt structures 1s a rather re mote po s sibility. Howe ver, it is felt th a t t he results are interestlng in the light of vibra tional theory, and the t the work has some 1nportance in illustrating the nature a nd effects of vibrations si ~ilar to the e a rt hquakes t□ which structures are often subJected. Sincere appreciation 1s hereby exnressed to Prof. R. 'R. . ~:.artel for his interest and mRny va luable suggestions. Grateful R.c tmo wledgement ,_s a lso raad.e to -1- 1!r. Paul Kartzke, who assist r': d in the early stages of tbe work, and to ~ r. Le Van Griffis who gave va luable aid in carrying out a. number of the t~sts. -2- CONTENTS Part I r The 1~estergaard Analyei s 1 II Appl1cat1on of the Westergaa.r c'l Analysis to the Bank of America Building, Sa.n .rose, Cali f . III The Biot Analy sis 6 17 IV Application o f the Bi o t ;'\nalysis to the Banlt of .America Bu1ld1ng, San i ose , Cal i fornia. V Comparison o f the Blot and Westergaard Analyses and Conclusions Regarding Them v~ 23 30 Application of the Westergaard Analysis to Hypothetical Buildings 32 Part II VII Description of the Shaking 'I'able Used in Analysia o f Transi ent Vibrations 48 VIII Comparison of t he Behavi or o f One 3tory s.nd Three Story Bents of the Same Fundamental 51 Period I X Bumping Between Bu ilding s of Diff e rent Fundamental X Pe1•1ods 66 _; ffects of Tension Diagonals on One Story and. 11 hree 6tOl''Y Bente 82 Part I II X! Behavior o f Simple Bents SubJected to Eatabllshed Simple Harmonic Ground 2fot1on 110 Chapter I The Westergaard Analysis This analysis of the motion of tall building s under the influence of an earthquak e shock. is p resented in an article by Prof. H. :it. Westergaard, in t he Eng ineer1.Q3 News- Record of November 30, 1933, under the title, Earthquake- Shock Tranam1se1on in Tal l Bu11d1nge. The ideal building considered ie one which has uni form weight and st1 ffneee throughout the hei g ht of the building. Moreover, the deformation cons idered is that due to shear only. Thus the bu1ld1ng may be con - sidered aa a vert1c a.1 beam in which deforma tion in shear only is considered. Since the actual building deforms primarily in shear (that is, with horizontal displ a cement of one floor relative to another, producing flexure tn the columns), and only to a very small extent in bending , producing leng thening and shortening of the columns, the assumption that the bu11d1ng deforms only in shear is justifiable. It 1s further assumed tha t t he bu1ld1ng and its loading are sufficiently symmetrical to produce little or no tendency to rotate. Prof. Westergaard points out that an approximate analysis derived on the assumption t hat deformation is due entirely to shear does not apnly in case t he shoc k 1 e so sudden tha t stresses will rise a nd fall during the short tt me t hat 1. t tali:es for a wave of deformation to travel t hroug h one or t wn stories. However, t he ~aore si g nific a nt e a rt hqu alrn s develop less rapi dly. The dyn amics o f s he a r defo rmation l s si Jmle, ' . a.nd t he equations of motion ha ve the s ame form a s t,hose of !'3. wa ter hammer, or o f long itudina l sound wave s i.n a rod. 'T he horizonta l motion of the ground may be resolved into t wo components parallel to the two sets o f vertic a l f r e.me s . wa ve f rom t i-1e 1'he shock then tra vel s a s a she a r· ground to the top of the bu:l ldlng , 1s then reflected do wnward, t ra vels to the ground , a nd i s a~a t n reflected upward. The ma thematic a l presenta tion o f t he t heory ls a s fol lo,vs: The follo wing notation 1s used : t : ti me. x = vertical distance, '90B1t1ve upw '1 r d . y ~ horizontal deflection, functio n- of x and t, post ti ve towa rd the ri g ht. s = horizontal shear at noi nt x. k • tota~t1ffness of the columns. w = weight of t he building per unit o f hei g ht. g = accelera. t1on o f gra vity. -- velocity of s he a r wave . h - hei g ht of t he buildi ng . V m l -- fundamenta l neriod o f t he bl'!lilding . -2- The relation betwe en the shea.r s, stif fnes s k, and sloue ~ sx may be written a s s = k Using t.he dynamic rel a tion that force equals mas s ti mes a ccelera t\on a ppl ied to a differenti al of hei g ht, we obtai n the equation Combining these two equations, A solution o f this differential equation is y : f (t :t ~) V where The phys:lcal meaning of this e quation is that the magnitude of the deflection remains the s ame, but :lt has been moved along the ti me axi s by an amount .! V which is t he tt me :lt takes for the wa ve to tra vel from t he bottom of the building to the point x. When the wave r e ache s t he top of t he building , a new wa,ve i s reflected do .vn ward . The correct boundary co ndition a t the too is that t he shear t s zero. fro m the r el a tion s =k ½; the shear a t any point x ls f' (t-L ) V Now Equating the total shear at the top to zero, it is found that the deflection at the top ls doubled, a.nd the new descending wave is of the form f( t + X V 2h ) 'Hien t his wa v e r ca.ches the ground ano t her wave is re f l e cted, and the boundary c ondit i on in t his c a s e i s tha t t he new deflection wave cancels the d e flection due to the d o1Jmward wa ve, or y 3 :-Y2 . Tl1erefore, X + 2h) f·c•· t - -V If the g round 1s actually in motion at the ti me of this reflection, tbe actual ground motion must be add ed to the reflected wave. It may be of interest to note t ha t the shears depend u pon the der1vi.t1ve f', or t he velocity of the ground, as P rof. Westergaard has p ointed out. The VAlue of v may be obta ined by t he u s e of the equation or by using the relation V : 4h T in c as e the period of t he building is l:cnown. The analysis c a n be applied to buildings whi c h have offsets by st ~ply considering a certain p ercen t age of t l1e wt1 v e to be re f lec t ed at t he offset, and the remainder to travel to the top where 1t will be reflected. Damping can also be considered by multiplying the amplitude of the wave by a damping factor at each reflection. Thus, the motion would eventually die out d.ue to damping and incomplete reflections at the ground until the building returned to rest. A graphical ap1:,11cat1on of the analye1s resul te in curvee representing displacements of the particular flonr considered, plotted age.inst time. -5- Chap ter II Application o f the Westergaard Analysis to t he Bank of , mer1c a. :-3u1 l ding , 3an .J ose , Cali forn ia. It wa s thought tha t the applic ation of this analysis to an a ctual building under the influence of a.n actual earthqua ke woul d bE~ of particular intere st. Acce lerogra.-0 11 records obt ained by the TJn i ted States Co ast and Goodetic Survey in tl:Je Ban~ of /uneri ca Butlding at San ,J o se , Gali fornia , during the western Nevada ee.rthquake of J une 25 , 1933, were nvaila.ble . These records gave a ulot of vertical, long itudtnal , a ncl t ransver s e cor.1ponen ts of accel erati on at the thirteenth floor of the buildi. ng and a t the ba sement. The most vioJ. e nt mott on was in a longitudinal direction (parallel to Santa Clara Stre et) , and only t hi s component was analyzed . Before proc eeding t0 a descri ption o f t he analysis , it may be well tn de s cribe t he bu ild \ng briefly . '1'he stru cture ha s thj rteen stories and a tower . f:n inspection of the drawings of ~i gure 1 , which g ive the raai n d:lmenstons of thA buildtne; , wi ll sho w tha t the building was not as symoetrical as ml g bt ha.V A be en desired for a si.m.nle appli.c at ion of the ",r,stergaard t heory . In a dd : tion on t he e a st . t0 the offset, L one story bu 1ldtng abuts " The fra me le steel, flrep ro o ed wit h con c rete ; wal l e 1 concrete wit h fac e brick ; nar tition s, lat h and -6 - -.J I r U- fr:'. (/) f- ti) 5~4 l3 FLOORS [Q] TOWER . I ~ - PLAN 84' - I _J ___ __j I FLOOR I FLOOR. AND ME.Z . 2 FLOORS 138 ' IU ....J uJ > < g_ cl V") c::::::; I CLARA ST t. 1 I_ ,.. SANTA t--, "t I N I It) I ~ ! t-- rf) 1/) FIGURE ---t - 1 I / I H EAST ELEVATION ~------_. ., ______ TANK. FLOOR- TOWER. I l!) 1 t-- SCALE. r" ~ so· C\J 0 -t ll) r1") l'-- it) H - :-+ N t-- II) \!) 0 plaster; flo ors, concrete; f oundations , concrete footings o n pi les; soil, unconsolidated sand and clay (old creek bed) . " 1 In ord er to o b t ain curves of displacement plo tted agf1 ins t ti:ne f rom the accelerogr·aph recc:toe, a ~raphic a l met hod o f double integ r a tion was used . The ac celeration curves had previously b fH:m enlarged to abou t fifteen times their original FJize 1 n order · to g ive curve s of convenifmt size f or t he g raphlca 1 i n te gration . Since the s tronge t motion o c curred dur. ng t he f irst twenty seconds o f the qu ake, and the motion began to damp out after that, it was dee med sufficient to analyz e the motion during the f irst t wenty se conds only . The f undamental period of tbe building parallel to Sa n t a Clara s treet, as de termined by the United Gta tes Coast a nd Geode tic 'Survey using a vibrator and vibration meter , wa s 1 . 33 seconds; And the ano l _ye is was c ar ried out for t he recorded e a rt hqu ake using t h1s fund amental period . This analy s i s i s s hown i n ~i g . 2, and a co mnarison of the curve of thirt e enth floor deflection obta ined by d ouble integ r a tion of the thirteenth floor accelerograph record a nd the de flectio n curve obti ane d by the 'Ye etere;aa r d anal y s ts shows g ood a.gr e pment l U. 9. r: oast and Ge odetic Su rvey: P reli mtnar.v Reuort, Cali fornia Se1s:nolog:lcal :P r ogram, ? orced Vibra tion .:res_1_§. - 8- u.J 0 u.J u.. _J uu.J i= 0 z <I) z "o <D 0 13TH MOTION -------··- ---·- • - TIME - SECONDS GROUND FLOOR COMPONENT TRANSMITTED OF OF OF SECONDS THROUGHOUT FIGURE 2 NO DAMPING ACTUAL ANALYSIS . TO GIVE BY ANALYSIS WAVE PREDICTED 15 "J MOTION . , , \ ,,T,JII\ RECORD . GROUND BY INTEGRATION OF ACCELEROGRAPH TO TRANSMITTED FLOOR 10 1.33 WESTERGAARD ADDED PERIOD OBTAINED THIRTEENTH MOTION FUNDAMENTAL MOTION WAVE . MOTION COMPONENT ABSOLUTE ACTUAL 5 - ~ ' /\ '' '_,' I '- \ \_/ \, JAMES H. JENNISON \,, ' ~-, , : \ // I'\ . .._ as to amp litude of motion . However, it c a n readily be seen t hat the t wo curves beco me dlstinctly out of phase 1n the course of the violent motion . It was thou~ht at first tha t reflection at the offs e t m1e;ht account for the d1fflculty, but an eet1 mate of t he percentag e of the wave reflected at that point led to the conclusion tha t th1e W3 S not the case. An exam1natton of the original accelerog raph record s howed tha t the period was apparently nots constant. By measuring g roups of wa ves of different amplitudes, 1t wa s found that the period was ahout 1.35 second s for ema.11 amplitudes , e.nd that the period increased to about 1.65 seconds for the largest a mplitudes recorded. The ma,ximum amn11 tude of the thirteenth floor motion during th1s quak.e was esti.:ne.ted to be about one inch, whereas the vi bra tor gave a maxi mum a rapll tude of only about 0.0017 inches. Thus, in accord with 'J . ~:: . c . o . s . figures, t he period of 1. 33 seconds 1 fl no doubt correct for ve ry small ampl itudes , but i.t does not seem to apply for t he more violent iuo tion. I t is difficult to expla.in why the period should increase with amplitude of vibration, but the work o f Prof. c. ~. Inglis on vibrations in bridg es seems to suggest a poes1 ble explana,tion. 1 1 Inr,lis, c. :l .; .Y.!b rati one in ,:3r\df.§:~; T~ ~H, ructural _];ng in~, Ju ly, 1935 , Val . 'UII ~v o . 7, D.29 Z~. -10- Inglis found th::, t short span rail way bridges exhibited a cha nge 1n pertod of vibration when the spring friction of the loco mot1 ve spring s wa s overco,ae, thus producing a pronounced benefit in the eli mination of prolonged reson an ce. In the case of a t Rll building, it s e e ms poss! ble thi3t there may be a g rea t number of very minute cracke, esu ec1ally where brick facin g is used , as in the c a se of the Bank of America Building . F'or mild vibrations, t he friction between these surfa ces would be sufficient to prevent relative motion, and the entire building would vibra te a s though constructed of an entirely continuous ma.terl a l. However, as vibra tion becomes more violent, the maximum force that can be developed by friction a long the cra cks or jo\nte is excee ded, and a sli ght amount of sli p may occur along a great number of extr e mely small cracks. Sli o o f eome of t he joints 1n the steel frame mi g ht also possibly occur. Since the force s involved would v~ry w1 t h d\ fferent cro.c k s, the c hange of stiffness would be gradu al; but,as more and more slip occurs, the buil ding would become more l imber; or, ln ot her words, t he period would be leng thened . • - As the vi bration subsides, the forces involved g r a dua lly ree.c h the point where friction can !)revent slip along the cracks, a nd the period decreases again. This e xplanation is merely suggested a s a p ossible reason f or the a npar ent c hang e i n period. It should not be construed to be a dogmatic opinion , but merely an explanation of an observed effect which seems to be reasonable. 1~oreover, 1t should be pointed out tha t the movements along the cracka,as well as the cracks themselves, are thought of ae being of extremely sruall magnitude. The analysis was carried out again, using the period of 1. 33 seconds for small ampl1 tudes, a.nd changing the period to 1.65 seconds when the a mplitude of vibration became large. Th1s brought the motion derived from the analysis nicely into phase with the motion recorded during the actual quake, but the a mplitudes of the strong motion were much too great, as c a n be seen from an examination of Fig. 3. Comparison of Fig. 2 and Fi g . 3 gives an idea of the 1mnortance of resonance. In these first two analyses, there was no dampi ng considered. However, a study of the original acceler- ograph record s howed tha t considerable damping preva iled 1n the actual structure. The damping apparently reduced the amplitude about 10% during each complete cycle of vibration, and co~sequently a third analysis was made 1n which a damping factor was employed. Each time that the wave was reflected, 1te amplitude was reduced ·by 2.5%. The periods used were 1.33 seconds for s mall vibrations, and 1.65 seconds for large vibra tions. in the previous analysis, shown 1 n Fi g . 3., t hls was -12- As w 0 0 MOTION PERIOD 1.33 SECONDS OBTAINED BY 10 I· , ' ... ., \ J I ... ..,,' = ', - ,' - - - - - ·- - - -- - - - - - - - - - TIME - SECONDS 5 C OMPONENT TRANSMITTED OF OF MOTION TO WAV E PREDICTED TRANSMITTED FLOOR 3 DAMPING FIGURE NO OF TO GIVE PERIOD A CTUAL GROUND ACCELEROGRAPH FUNDAMENTAL BY ANALYSIS . INTEGRATION WESTERGAARD ANALYSIS ADDED THIRTEENTH OF MOTION WAVE . MOTION COMPONENT ABSOLUTE ACTUAL FUN DA MENTAL MOTION . RECORD. 15 1. 6 5 I I\ ' \ ,' ....... ' , ,', \ \ J.t.MES H . JENNISON SECONDS ' ' UJ ''' / ~ \.; \ IL GROUND FLOOR ' I \, (' ' w u J i= 0 z (/) '°z b 13 TH -- n e cessary in order to maintain n:roper phase relationship. The results of this analysis , w'lth damplng cons i dered, are 1 n good agreement wi th the actua l motion, as shown in Jtig. 14. In this connection one mu st consider that there 1s considerable chance of inaccuracy where so much graphical work ls involved. A rathe r sma,11 original record was enlarg ed to about 15 times its ori ginal size. Double integration of this record by graphical methods results ln some unce r tainty as to both t he ground motion and the thirteenth floor motton. It was particularly difficult to locate t he axis of the ori g ina l record with sufficient accuracy to prevent an accumulating error from showing up as a permanent displacement of the building. Then the !Vestergaard. Analysis, 1 tself a graphical method , introduces some inaccuracy. Fina lly, we a re aesum1ng uni form st1 ffnee,s and uni. form weight per unit of height throughout the entire height of the building fro m b a sement to thirteenth floor. Uoreover, the analysis takes a ccount of ehe?r deformation only, whereas the deflection curve s determined b y the -:Jn1ted States Coast and Geodetic Survey indicate that t here 1s also a sli g ht amount of hending which c an b 0 detected in the fundamental moti on of emR.11 a !Ilpli tude. In the l ight of these considerations, the results obtained in the t hird analysis, whe re a: damp ing factor was used, a gree re markably well -14- uJ FLOOR - 0 MOTION COMPONENT -- -- - OF OF MOTION · \' TO \J / ·' WAVE 2.5 4 AT E.ACH FIGURE PER CE.NT ,... : TO \ _/ I OF REFLECTION GIVE ,- ACTUAL \ PERIOD '.. . . ' ' GROUND ACCELEROGRAPH FUNDAMENTAL ' BY ANALYSIS . INTEGRATION PREDICTED BY TRANSMITTED FLOOR OBTAINED 10 WESTERGAARD ANALYSIS ADDED DAMPED MOTION / t;-. I\ j', I /\ i \ I \I I I \ / \ THIRTEENTH OF I IC- \j I\ IS 1.65 • , MOTION . RECORD . = , ,' I :\ JAMES H . JENNISON SECONDS , , \ \ L-<"hq~-l.Y/f::\< V , I~~z.....::::: ·;_,,YP-,-,t.. ,, ~ I. \~ '1 - L =,,p- SECONDS WAVE . MOTION ABSOLUTE TRANSMITTED - 1.33 5 COMPONENT ----·--- ACTUAL PERIOD - - - - - - - - - - - - TIME - SECONDS FUNDAMENTAL GROUND ~ /~~~~L ~ ~~ IL _j u w i'.= 0 z 1/J z 0ID 13 Tl:i ~ with the a ctual motion as obtained from the accelerograph. -16- Chapter I I I The Blot Analysis The theory developed by Dr. Y . B1ot while at the California Institute of Technology wa.a first presented at the National Apnlied Mechanics Meeting in June, 1932, and 1t has since been presented in two papers by ~. Blot, namely (1) Theory 2f Slast1~ Systems Vibrating Under Transient Impulse Wit!,l An Annl1cat1on to Earthquake-Proof ?u1ld1ngs, and ( 2) 'fheory of Vibrat~.on of Buildings During Earthguake. Unfortunately, the mathematics of the derivation ls somewhat complica ted, 2nd the physical e1gn1ficanoe of the terms 1n tl:le final equations is obscure. Consequently, the full derivation will not be presented here, but only the results and the methods used in their application. The Biot Analysis ie based on the fact that the motion of a building during an earthquake has the character of a transient osc1llat1on. Applying the principles used by Heav1elde 1n the analysis of transient electric currents, Dr. Biot, considering shearing forces only, makes an analysis of the vibrations in a bu1ldlng. In the case of forced vibrations, an osc1llat1ng system tends to aporoach a steady state of vibration with the same period as the applied force; and, in the case of resonance, the maximum amplitude 1s limited only by the -17- internal friction . However, when the system is under the influen ce of :transient vibrations, the steady sta te harmonic oscill a tion doee not have time to develop ; and it is t he respon se of s ys tems to such transient i mpulses t hat Dr . B1ot considers . The analysi s :ls based on the fact t hat any vibration in an elastic, undamped system may be considered a s a superposition of harmonics. The analysis disrega rds d amp i ng . The t heory, as developed by Dr . Biot, considers four separate t ypes of vibration sepa r a tely, and then combines the results in a general theorem. The four tynes of motion are as follows: 1. Free oscilla tion s . 2. ~orced harmonic oscillations . ; -. Sudden constant accelerations . l 1. Oscilla tions due to arbitrary hor1zo r.t a l a c celeratirwns . The genera l t heorem derive d ha s the form TJ where k ( X) u1t ( x ) s deflection relat1 ve to the g round of a floor at a distance x fro m t he roof. Kk - constant dep endi ng on the properties of t he building . F( ; k) ~ functio n invol ving constants and the eart hquake motion . This equat ion can be written with a p p ropriate s ubscri p ts for the fundamental and the various harmonics . - 1 8- F'o r an analysis considering vibrations i n the fundamental period, the above equation becomes U0 (x) = K0 F{ ; 0 ) The consta nt K0 can be found from the relati on K0 : 2 7t" 0 j 0 t 2 B0 cos (.A 0 e:_) l The terms in the above e quat ion c an be eva luated as fo llows ; T0 - fundamental per•iod of the building , 1n sec o nds per oec1llation , a e found by exper1me.mt . ; o = the fre quency of the building i n the fundamental mode of vi b r ation , in oscillat ions per second . ,1\ 0 "\ /\ O : • 1 To the va.lue obtained from t he relation \ o :ex tan l'- where where n : number of stories above the first . R. - ratio of the rigtdi ty of the first f loor to t hat of the others . The va lue o f R. may be f?und e xper1menta11y by deflecting t he floors by apnlying a l at er~.1 force. Then d n R - dl whe re dn = deflection of any fl oor above the second rel ati ve to the flo0r benea th it. d1 = d e f lec tion o f second flo or rel a tive to ground f loor wit h the eame lateral force applied . - 19- If t he columns have very consta nt c r os s s e ctions, the va lue of R m9,y be found fro m the equation 'r'I Ln 3 - r,; - ~- L 1.:;i where Ln : he ight of e a c h story above the first . L1 - heig ht of first story . t 0 is f ound fr om the follo wing equa tion . t _ To )-o 0 - 2 7r B0 1e found from the e qua ti on 2 rx/3 0 Bo• - - - /1. olJ, where COS~o l + o<.C OS ;).. 0 A- 02 X h where x :: di sta nce from t he top of the building to the point at which t he deflecti on 1s desired . and h = height of the building a bove the fir st story . These dimensions and the deflecti on s of the bui lding are illustra ted 1n Fi gure 5i i/t1 r ,____________. Fi gure 5. - 20- The function F( V0 ) can be eva luated from the equation t 1 s t he indep endent variable, tl me; and G(t ) is a function of ti me expressi ng the g round motion in terms oft . The only remaining d1fficu1ty lies in evaluating the integrals under the radical sign. F'or any given ground motion, this integration c a n be carried out graphi c ally as follows: The sine and cosine funct ion s, when integ r a ted , become cosine and sine functi.ons re s pec ti ve ly . These curve s are nlotted, and the curve of the groun d motion 1s p lotted at ri ght angles to them, giving due consi dera tion to the si gns of the area s. The sign conve n tion used ls t ha t t he a reas are positive when the ordinate of the ground motion and t he slope of the si ne or c o sine cur ve have like si gn s . The positive areas are p lotte d to the left of t he stne or cosine curve . The areas a.re negat ive when the ordina te of the ground- motion and the slope of the s i ne or cosine curve have unlike signs, a nd the negative a reas are p lo t ted to t he ri ght o f the sine or co sine curve . The cumulative area , with due reg a rd to si gn , may t hen be me a sured with a nla.ni meter . At frequ ent 1 n terva.l s. t he cumulative alg ebraic s um. o f the areas is noted., and the deflection i s calculated a t these points. The result 1s an envelope w:tth1n which the motion of the building , cons idering only vibrations i n - 21 - the fundamental, should remain. If it 1s desired to consider the harmonics, the subscripts 1, 2, 3, etc., for the first, second, and third harmonics,respectively, can be used with the above formulae to calculate new constants . The graphical integration can be carried out for the harmonics in a manner similar to the 1ntegrat1on for the fundamental. The deflections obtained are then added to the corresponding deflection in the fundamental mode . The assumptions underlying the B1ot analysis should be kept clearly in m1nd. They are as follows: 1. The building must be of fairly regular shape. 2. The most important deformation must be due to horizontal shear. 3. The shearing rigidity and mass of each story above the first must be constant. 4. Damping is neglected. -22- Chapter I V A':Jplica.t1on of the Biot Analysis to t he Bank of America. Bu ilding, San ,Jose, Cali fornta.. The records used for this a.nalys1s were the eame ae those described in Chapter I I , in connection with t he 1~estergaard ana lysis. The a nalysis of the fundamental raot1on was first c a rried out cons1der1ng the fundamental period to have a con s tant value o f 1.33 seconds. The calculations shown on the following page are ba·s ed upon a fundamental period of 1. 33 seconds. The graphical integration is shown in Figure 6, and the results of the integration and the va lues ca lculated for the envelope are shown 1n the table on page 26. The envelope obtained 1s plotted at t he top o f Fi g . 6 ae a dotted line; and an examination shows that the envelopa contracts due to the motion getting out of phase, wM.ch occure at the same time as this difficulty was observed 1n the Westergaard analysis. To eli minate th1 e, t.he period was changed to 1.6 5 seconds when the motion became violent, and the remainder of t he analysis wae repeated. It was found upon investigation that a change in fundamental period alone did not affect the constant 1n the final equation U O : 0. 582 V Al 2 + A2 2 The results of the g raphical integration, a nd t he new -23- CALCULATIC)N CC)N.STANT.S C)f- FUNDAMENTALT., =- l.3.3 5t.c.. .'\ Yo"' • I To:::. ( U 5 . C.G.S. Foe..CE.O V1BR.ATION I 1.33:: 0.752. I I Jo:: T.'Z. :o - 1.= 1. ~3 0 PE~ 05<.Jl.LATIONS STUDY) 51:.C. 0 . 5~5 U. 5 .C.G.S. Di::FLE.CTION FROM ([STI MAT~D CuR.VE.) o. : : Rn= 24 ;\ o t-a n .A. " ex = 24 ')..o = ). 508 RADIANS to= Ao.=. To l . 33 2 ir X 1.508 0. :: z 'T(' .31.9 _c.;;;_o_5_A_o.c___ _ = o.O<o 25 f3~=- I+ I + OC COS '>--o ,\!B.,:: = 2.x 24 X: 0 .03'15 = 0 . 349 Zo<.4 ')..; Ko:::. = 0.0.375 24 X O. OG,2 5 ). 508 2 I. 5084 2. 7'" ✓,, j 0 t~ Bo C.05 ( Ao 't_) FoR. I~ rH t_ = ~ =- i'z. Ft..ooR.., Cos C Ao~):: O. 993 Ko= 2. 1r .x. o.752 x. o. 5<D5 x o . 319'x 0.349 x o.9g.3=0.093g F(Y.)= (2-rr'(,10)2./(/TG(t) SIN 2.1ri>o tdt/·+if~(t)cos21rl., tdt/o O F (v'.,)= 39.4 /(AREA #l)z. + U 0 _- Ko E. ,,.., ' u .;~ F(Yo) Y 2. 7f Vo P :: 0 . 9.38 :i<. (AR..E.A -uz) 2 39 .4 { A,2. + Az.2 _;___....:.....:.J_....___ : X 7"(" x 0. X - 1- - _ _ _ _ ___;__ ) t = E.>JLA~GE. Mf.~·T JI. I. 5 2 7 5 Q. 5 8 2 /A I 2- t A'Z. z 2 I . I I 3<, FACfOR_ . y = AMPLITUDE. OF Sna:. ANO Cos1NE. Cu~vE.~. P = PLANJM.E.T.EIL FACTOR. ,o Rf.DUCE.. ARE.A - 24 - 'To SQ. IN. ?undamen t al ~ o de of Vi br ati on Ar eas and Envel oue Values -P ertod Point Al = 1. 7. '2_ A.2 sec . Uo Period = 1.65 sec . Point uo Al A2 0 0 0 0 1 2 3 4 5 6 7 8 9 10 -1 .15 +O. O5 +0 . 58 +O. 38 - 0 . 06 - 0 . 19 - 0 . 10 - 0 . 30 0 . 105 0 . 178 ') . 3147 0 . 222 0 . 039 15 .i- 1 . 17 /3, +0 . 61 +1.65 0 . 116 0 . 310 0 . 232 0 . 071 0 11 12 13 14 - 0. 53 - 0. 43 - 0 . 07 +0 . 01 - 0 . 24 - 0 . 18 - 0.30 - 0 . 05 - '1 . 2? +0 . 01 +O. O3 4 1 . 06 +O. 06 - 0 . 05 - 0 . 10 - ().06 - 0. 37 0. 831 +·") . 9li 0 . 977 +l.68 +1 . 08 ... 0 . 27 - 1. 39 17 - 0 . 81 - 0 . 18 - 'J . 19 1. o9 0 . 638 0 . 192 20 - 0 . 03 - 0 . 09 +0 . 73 +0 . 38 +0 . 55 - 0. 55 . 320 O. 62 31 +0 . 9') +O. 79 +1 . 07 -+ l. 34 +1.06 +1 . 02 +l . 25 - 1.h6 - 1 . 52 - 1.55 - 1.41 - 1. 33 - 1 . 18 - 1.06 - 1. 11. -1 . 01 f g m 0 . 257 + 0 . 25 - 1. '.)5 -1. 46 d 0 . 1+32 ') . 646 r, . 035 15 16 21 22 23 24 25 26 27 28 29 30 C h i j k 1 - 0 . 72 -1 . 07 - 1 . 34 - 1 . 50 18 19 b O.e-85 0 . 95 0 . 991 IJ . 961 1 . 04 0. 94 0 . 99 1.04 0 . 872 0.881 o. 934 - 25- - 1 . 68 - 2 . 15 - 2. 33 + 2 - 32 +2-.. 73 - 2 . 15 - 1.81 +3. 32 +3. 18 - 1. 35 +3 . 07 +3 . 17 +; . 17 +: . 25 +; . 57 +:;.. 55 + 3. )-13 - 0 . 90 -0 . 98 - 0 . 93 - 0 . 90 - 1.11 - 1 . 27 - 1.17 - 0 - 95 0 . 981 1 . ;O 1 . 67 1.84 1 . 90 2 . 09 1.92 1 . 87 1 . 92 1 . 92 2 . 00 2 . 20 2 . 17 2 . 07 •- MOTION -- . - -- ·- - --- - · -- · OF THIRTEENTH GROUND .---....... ,,.--.-.,_ ------ /--...... MOTION I I 0 TIME - SECONDS o -0 .10 -o3o --0.22 •0.01 10 .-a.OJ .. o .o• ... a .oe --OD.5 -0.10 --o.oe -O.37 -1.01 -0.12 -1..34 FUNDAMENTAL FUNDAMENTAL PERIOD INTEGRATION SAME = 1. 33 AS -1..50 20 -139 PERIOD s -0.11 --o. I a 1.33 SECONDS -0.19 --o..55 -1.01 -1.•• -t.•• -1..52 -1.5.5 -1 .41 -1.33 -1.11 -1.01 -LIi SECONDS ABOVE . -1.34 -1.68 -2 .1.5 -Z.33 FUNDAMENTAL 0 +0.11 SECOND MODE PERIOD BIOT = 0 . 38 SECONDS ANALYSIS NO DAMPING JAMU M JENNISON FIGURE. 6 values of the envelope using a period of 1.65 seconds for large a mplitudes, are shown in the table on page 25, and the resulting emvelope ie plotted in Figure 6. It 1e obvious fro m the results (dot-dash line !ig . 6) that thie gives more nea rly the true condition. In their summary of the periods of the Bank of America Building parallel to Santa Clara Street 1n the va rious modes of vibration, t he U. s . Coa st a nd Geodetic Survey gi.ve the following periods. Fundamental 1.33 sec. First Harmonic 0.38 Second Harmonic 0.24 'fhird Harmonic 0.18 The B1ot analysis was applied for the first harmonic a,s well as f or the fundamental. The calculations for the constants are as shown on page 28. The graphical integration is shown at the bottom of Fi gure 6, and the results are tabulated on page 29. The deflections 1n the first harmonic when added to the deflections 1n the fundamental give the envelope shown as a full line 1n Fi g . 6. It is apparent that the envelope a g rees quite well with the actual thirteenth floo r motion during the first part of the quake. Moreover, since damp ing has been neglected 1n this analysis, one would expect the envelope to indic a te larg er deflections tha n actually occurred, the effect becoming noticeable only after the -27- CALCULATJC)N - J Y, = - \ T, : - l 0 3~ C)f- CC)NSTANTS H.b.RMC)N IC - FIRST = 2 G,3 n = I 2. R" 2 ~ = 24 ~ tan A 1 = 24 1 i\ 1 = 259°20' = 4. 53 RAo. t,=- T, A,-= 27t 0 . 38X4.S3 2.'7i = 0274 /3, = _c_o_s_>--_,_ _ = l + 0<. cos 0.185 + 24 X 0. f&S .>..., = 0. I 5 2. 4. 53 2 )..~ 2 X 24x0.152 4-5~ G05 4 = Q.Ol?:3 / z.5~"20 1 I 'Z. F<fi) = 39.4 { A/· + A./ u I : K. F(?) p = o '. 138 i<.39.4 )( 1.114 x4. /A, f:. y z;rv, E. = E.t;LAR.G£.M£.~T ~ =AMPL1-rvDE- Of = o. 49 /A.1.. + A-Z· + A{ FACTOR. . 511-n.. ANo ·p = PLANIMEH.R. FACTOR. u1 2 1 -'''------ Cos1Nr:. CuR.vE.s . To Ri:.ovct, Art:..r.A TO s~ TN lapse o f a considerable tl'.ne interval. ''f hus, the results give exactly the condition which one might exnect. First Har moni c Areas and Envelope Values Al A2 0 +0.02 0 0 1-0.ll U1 () ... 0.13 0.0098 0.0982 0.101 0 .05 82 +0.1 8 0.0945 -0.20 -t-0.23 -0.15 -0.11 +0.24 0 .149 0.139 0 .1 57 o. 16 :-5 -0.17 -9.14 +0.05 -0.07 -0.03 -0.04 -0.05 -0. Ol+ -0.10 +0 .03 - 0 .4 3 +0.02 -0.23 -0.13 ... 0.15 ..-o. 22 -t-0-30 +0.33 +0.415 ... o. l+2 ..-o.47 -t0.38 ..-o.44 +0.50 ~o.41 0 .109 0.227 o. 205 0.235 0 .187 0.302 0.245 -0.05 +0. 50 0.230 o. 263 0.215 0 .272 o. 260 0 .286 0.272 0.277 0 .250 0.2li6 -0.05 -t- 0 .51 0.251 -0.09 -0.22 - 0.23 - 0.23 -0.17 -0.08 -0.02 -t0. 52 ~0.4 3 +0. 51 +0 .48 -t 0. 51 +0.53 ,t 0 . :-6 +0.51 -29- Chapter V Comparison of the E1ot and Westergaard Analyses and Conclusions Regarding Them The results of the application of these two methods of analysis to an actual building 1n an actual earthquake show that, for a case where the ground motion has a period of the same order of magnitude ae the fundamental period of the building, the results of both analyses check the recorded motion of the building. In fact, for practical purposes, the results are equally good. In both cases, it 1s necessary to know the fundamental period of the building to a deg ree of accuracy which might be rather difficult to obtain in a case where the actual 'behavior of the bu1ld1ng was unknown. The ract that the fundamental assumptions regarding the symmetry and homogeneity of the building are only apurox1mately correct d1d not seem to cause much trouble 1n either analysis. The Westergaard analysis has several d1et1nct advantages. First of all, the principle underlying it le simple and tbe physical e1gn1flca.nee of each step 1s apparent as the work progresses. Moreover, the result, be1ng a. curve of motion rather than an envelope, g1ves a more comprehensible representation of the behavior of the building. Then too, the graphical work ca n be carried out much more rapidly than the graphical integration 1n the B1ot analysis ca.n be performed; and no special instrument 1s needed for the Westergaard analysle, whereas a planimeter 1s required for the Blot analysis. Damping can be easily considered 1n the Westergaard analysis, whereas the Blot analysis neglects damping. However, since the envelope can be reduced by a certain percentage at the plotted points and the cumulative effects carried on from point to point, the effect of damping on the B1ot analysis can be quickly approximated. One or the most encouraging things brought out by comparison of the two analyses is the fact that the t wo analyses produce approximately the same results when the same conditions and assumptions are i mposed; so that one 1s confident that both theories are about equally valid. -31- Chapter VI Application of the Weate1igaard Analysis to Hypothetical Buildings Analyeee of theoretical building s of various periods were made with the ground motion assumed, first, ae one hump of a normal sine wave. and, second, as a half cycle of an off-set .s ine wave. The analyses are shown 1n Figures 7 to 19; and Figure 20 shows the curves of maximum relative displacement plotted a gainst the period ratio. In observing that for a period rat1.o ot one half or greater the maximum amplitude or relative displacement is constant at 3, one sbould bear 1n mind that all of these analyses are for a ground motion consisting of a single hump. The results indicate that the gradual start and finish of the hump of the off-set wave give a considerably less violent shoek when the period of the bu1ld1ng 1s small compared to t hat of the ground motion. For both types of ground motion, however, the maximum amplitude of motion was obta.1ned for the cases where the fundamental period of the building was one half, or more, the period of the ground motion. or Moreover, as the period of the building was made a smaller fraction of the period of the ground motion, the ma ximum displacement relative to the ground d1m1n1ehes rapidly. as 1e shown 1n Figure 20; but the decrease was somewhat more rapid 1n -32- the case of the off-set sine wave than 1n the case of the normal sine curve. The smoother curves shown i n Figures 14 to 19 as compared to Figures 7 to 13 would also indica te much lower a ccelerati ons resulting from the off-set sine wa ve than would be caused by the normal sine curve. For period ratios of one half or greater, the maximum deflections are the same for both c a s es , and so the difference 1n the magnitude of a ccelerations becomes the outstanding effect. Thu s, both fro m the standpoint of deflections, and from the ste,ndpoint o f accelerations, t he gradual curve of the off-set s1ne curve ie benef1cial. -33- V I I - - - - - - - - - -Ii•• - - - - - Transmitted Wave. Motion Due to Reflect1on. - - - - - Total Mot1on. PERIOD OF BUILDING l¼ TIMES PERIOD OF GROUND MOTION Fig. 7 -34- I I I I I I I I I I I I I I I - - - - - Transmitted Wave. - - - - Motion Due to Reflection. - -- - - Total Motion. PERIOD OF BUILDING EQUAL TO PERIOD OF GROUND MOTION F1g. -35- 8 V - - - - - Transmitted Wave. - - - - - Motion Due to Reflection. - - - - - Total Motton. PERIOD OF BUILDING THREE-FOURTHS PERIOD OF GROUND MOTION Fig. 9 I I I - -- - Transmitted Wave. - - - - Motion Due to Reflection. Total Motion . PERIOD OF BUILDING ONE-HALF PERIOD OF GROUND MOTION Flg 10 -37- - - - - Transmitted Wave. - - - - Mot1on Due to Reflection. Total Motion. PERIOD OF BUILDING ONE-FOURTH PERIOD OF GROUND MOTION Fig. 11 -38- Tranemitted Wave. Motion Due to Reflection. Total Motion. PERIOD OF BUILDING ONE-EIGHTH PERIOD OF GROUND F1g. 12 -39- Transmitted Wave. - - - - Motion Due to Reflectton. - - - - Total Motion. PERIOD OF BUILDING ONE-SIXTEENTH PERIOD OF GROUND MOTIOK. F1g. 13 -40- Transmitted Wave. - - - - - Motion Due to Reflection. Total Motion. PERIOD OF BUILDING 1 ¼ TIMES PERI OD OF GROUND MOTION F1g. 14 -41- - - - - Transmitted Wave . - - - - Motion Due to Reflection. Total Moti0n. PERIOD OF BUILDING EQUAL TO PERIOD OF GROUND MOTION Fig. 15 -42- I I ) I - - -- Transmitted Wave. - - - - Motion Due to Reflection. - - - - Total Moti c,n. PERIOD OF BUILDING THREE-FOURTHS PERIOD OF GROUND MOTION F1g. 16 -43- - - - - - Transmitted Wave. - - - - Motion Due to Reflect\on. - -- - Total Motion. PERIOD OF BUILDING ONE-HALF PERIOD OF GROUND MOTION Fig. 17 -44- - - - - - Transmitted Wave . - - - - - Mo t ion Due to Reflection. - - - - - Total Mo t i on. PERIOD OF BUILDING ONE-FOURTH PERIOD OF GROUND MOTION Flg. 18 -45- --+- I • - - t I t-----1-------l-L - - _ J - I __, ~_.:_ - - -j -- -1 - r~+~.:._ _ . L_ • 4 .:__tt_: ~ • ---+-1 ir+ I, • I . I r·~r-r· · ;-•, '"' 1 t- · t----+i----+----+---+. ·-- -- i - - - - + , 1 - - , - 1 - ~........ I -rl---'-1 : ~~~~· ~·-.f ,1;-frf • L - -~ - _ .i - -:· ' ;-·1-; .: L .• t•~t ., !r · ,. • I t . ' ~ ·t'. -~ -~ -I- ~r-r.--:--~L.I I ~~~-~~1 ............ l t I ·-t J I : I _,___ •...,1_, ! __ , 1 I t • , i l lI • ~ -1 t- + - , ) I • I -- -+- · f ~--1~· ! :~ ! lI _, ---,+tj t ~ I t li I 1 I t - I - -----+-·- ·-+- l , :~11t-,-J - 47 - PART II Chapter VII De eri t1on o 5hak1n Tab 9 Uaed 1n . n lys1s of Trane1ent '!ibr t1ons 't'he sha 1ng table us d tor the studies of transient vibrations 1-s 23. hown 1n 1 ur e 21, 22, and e com_lete a,..,paratue 1s shown n gure 21, Figure 21 while 1 urea 22 and 23 how the main ort1on of the he bent ( ) 1 mounted U'Oon a base betwe n r1g d columns whieh su ort ta es apparatus 1n closer detail. (T) upon which the record or ot1on 1e seeur d. ot1on of the b ee. or ground, 1 the quake stick(~) which 1 by th 1ghts ( ) and The rt dg_e r duced by releasing nulled alo y t m or cord quake st1ek has -48- Th the tr ck ( ) nd ulleys. n 1rr e ular outline, and roller tt ched to th baee pr see ag inst this lrregul r surf ce, c ue1?18 an o otll ting of the baee t e quake tick 1 ot1on dr wn pet the roller. omewhat diff rent ground motions can be obtained by us1n d1ff rent we1 hts to there 1 ull the qua tic, nc c.o neiderable fr1 ot1on 1n the system, thus eaus1ng 1ncre sed weight t give gr t r velocity. 8 Figure 23 1gure 22 he record ls obt 1ned on the t pes (T) by e of as ark g ~ att ehed to each floor or the bent. ho tapes re drawn through t es rk ga at const nt sped by the synchronous m tor( .. ), the speed being uch that one inch on the tape equal -49- 0.358 seconds when the motor 1s operated on 50 cycle alternating cur~ent. The spark punchee minute holes in the tapes , which trace the curve of motion for ea.ch floor. The portion of the tape containing the record 1s placed between cover glasses and inserted in a slide machine adaptecl to gi ve tht:? desired enlargement. The curve is thus projected upon a sheet of paper as a series of light dota,and the record 1s traced with a pencil. Different degrees of enlargement can be obta1ned fol" va:r.1ous purposes, but, for most of the work, an enlargeiuent of two times was f ound convenient. -50- Chapter ompar1 eon or the Three Story ent o II hav1or of' n tory the S e ~unda ent l nd er od A one story b nt was construct d which had the on e fund ental er1od s story bent be1ng of thee or th three tory. base, s1de by Both wer thre st ry bent, the e height as on ounted on the s aide, as shown in Ftg. 24, floor e nd record 1gure 24 were obt 1ned for the motion of the top f the one story bent •and the three floors and roo.f or the thre tory bent, fr a var1 ty of ground mot1 ns. bad eriod of 0.323 econds. -51- th bent The results of the teeta are given in Figures 25 to 34, the red 11ne 1nd1ca t1ng the motion of the one story bent, and the blue line the motion oft.he three story bent. For the purpose of comparison, the roof motion of the one story bent was traced with the roof curve of the three story and also with the second floor motion. A study of these records reveals that the motion of the one story bent gives an average curve ror the v1brat1on of the three story; 1n other words, the motion of the one story bent 1s of the same form as the roof of the three story bent would have tf harmonics did not 1nfluenee the motion. This ie brought out partic- ularly by Figures 25 and 26, which show the motion w1th a long,gentle ground vibra tion giving resonance with the fundamental, and Figures 29 t o 31, which show the effect of resonance with the harmonic on the three story bent. It should also be noted that, for a ground motion such as that of Figure 28, there ws,e a much more violent motion of the three story bent than occurred in the case of the one story bu1ld1ng where no harmonics were possible. An approximate estimate of the maximum velocities and accelerations was made, and the results are tabulated on the following page. It can readily be eeen tha-t both the accelerations and the velocities of the floors of -52- the three story bent were much greater than the oorrespond1ng values for the one story bent. Maximum Acceleration Ft/sec2 Rec. No. mix. Sustained Velocity Ft/eec. 2 Rec. No; One Story Bent 11 155 0 .58 155 Three Story Bent 57 155 3.1 149 Ground M.otion · 205 155 3.3 155 ~ he J estergaard analysis was apolled to Record No. 14-9 of Fig. 30, and the analysis and its results are given 1n Fi gure 35. A quake which caused t he three sto~y building to vibrate in the ha rmonics wa s purposely selected, and it c a n readily be eeen that the mo tion predicted by the an.alys1s checks the motion of the t hree sto'ry building fairly well. The overlapping of the denection waves takes care of the harmonics. However, there are s mall oectllatione 1n the ana.lye1s curve that d1d not appear in t he actual motion due to the a'bsence of harmonic s of hi g her orders tha t the second harmonic. Moreover, the record for the one story bent,w1th no ha r mon1es,is much different from the curve p redicted by the :eetergaard analysis. In t h1 s cmu1ect1on, it 1 s well to bear 1n mind t hat the We stergae.rd analysts assumes uniform d1 s tr1but1on of the wei g ht o f the build ing throughout the height . • -53- %hie assumption 1s violated to quite a degree by the three story bent, with the mB,es concentrated at three pe1nte; and, of course, the one story bent , with the mass concentrated at the top, 1s entirely outside of the range to which the Westergaard analysis 1s applicable. The results seem to indicate that three stories is about the lowest number of stories to wh1ch the Westergaard analysis ean safely be applied. -54- 1/ ( I V Roof Third Floor Second Flo or Ground Motion - - Both Bents. Three Story Bent. One Story Bent -- Roof Motion. Ground motion caused by pulling quake stick through slowly by hand. Record No. 157 6-18-35 Enlarged 2X Fig. -55- 25 I V Roof Third Fl'tl>or / S cond Floor ....... Ground Motion -- Both Bents. ------ Three Story Bent. One Story Bent -- Roof Motion. Ground motion caused by pulling quake stick t hrough sl owly by hand . Record No. 158 6-18-35Enlarged 2X: Fig. 26 -56- Roof • fV" Third Floor Second Flo0r Ground Motion -- Both Bents. Three Story Bent. One Story Bent -- Roof Motion. Quake Stick No. 1 Pulled by Two Pounds Weight. Record No. 143 6-15-35 Enlarged 2X Fig. 27 -57- Third Floor Second Floor Ground Motion -- Both Bents. Three Story Bent. One Story Bent -- Roof Motion. Quake Stick No. 1 Pulled by Three Pounds eight. Record No. 145 6-15-;5 Enlarged 2X F1g. 28 -58- /\ Roof Third Floor Second Floor Ground Motion -- Both Bents. - - - - Three Story Bent. - - - - One Story Bent Roof Motion. Quake Stick No. 1 Pulled by Four Pounds ei g ht. Record No. 147 6-15-35 Enlarged 2X Fig. 29 -59- Roof V Third Floor Ground Motion -- Both Bents. Three Story Bent. One Story Bent-Roof Motion. Quake Stick No. 1 Pulled by Five Pounds Weight. Record No. 149 6-15-35. Enlarged 2X F1g. 30 -60- Roof Third Floor Ground Motion -- Both Bents. Three Story Bent. One Story Bent -- Roof Motion. Quake Stick No. 1 Pulled by Six Pounds Weight. Record No. 151 6-15-35. Enlarged 2X Ftg. 31 -61- J ,J Third Floor / Second Floor Ground Motion -- Both Bents. Three Story Bent. One Story Bent -- R0of Motion. Quake Stick No. 1 Pulled Through Rapidly By Hand. Record No. 155 5-18-35. Enlarged 2X Fig. 32 -62- / I Roof Third Fl Second Floor Ground Motion -- Both Bents. Three Story Bent. One Story Bent -- Roof Motion. Quake Stiel{ No. 1 Pulled Through Very Rapidly By Hand. Record No. 156 6-18-35 Enlarged 2X Fig. 33 -63- Third Floor I / Second Floor Ground Motion -- Both Bents. Three Story Bent. - - - - - One Story Bent -- R □ of ~ot1on. Ground Motion Caused by Displacement and Sudden Reloase of Base. Rec£rd No. 160 6-18-35. En arged 2X Fig. 34 -64- --- 7 ·v /1 I I \ \ I II \ I L'~. i \ \ \ I \ \ ,~ - 3Rnnoo \ I I I I I r-1 \ (\ I I I \ ·v'Wf\ , r r/) 1' z. 1' 0 - f IL) Lu .J I IL I uJ 0 ---2.ND •lOOR. \ I \ \ I -~-1--t - -- 1 I f ' 1 - 0 I - 0.25 --· - - ---+-- - - 1 , -· - 0 .5 TIME.- 5ECONDS · - -· t 0 . 75 - - - - - - GROUND --- - - - + - - - ,.o - - - - ACTUAL MOTION FROM -SHAKING TABLE RECORD - - - - - TOTAL DEFLECTION PRE.DIC TED BY ANALYSIS . TRAN5MITTED WAVE. MOTION ADDE..D TO TRANSMITTED WAVE. TO GIVE. ACTUAL GROUND MOTION FUNDAMENTAL PE.RI 0 D OF BOTH ONE. THR.L f 5"TORY BENTS ::: 0 . 32.3 .SEC . WE..STE.RGAARD ANALY.SIS FIGURE. -c;; s- 35 ANO Cbapter I umping Bet ean Building Different Descr1pt1on of Te ndamental er1ods s: 51nce the build ng ar of 1n bu 1n s d1str1ete u1lt against one anoth r, 1n . any caa s, without an 1nt rven1ng s ce, it 1e interest! g to cont nl te the eff ct th t bu 1ng between the bulld1ng ay have on the1r ;1th thl ot1ons. tests w re and thr in ind, a ade on the shaking tabl story building. erles u 1ng a one story The to bents 1 ure 36, on the ba e, end to end, so th t th would bun tog th r when v brat1on occurred. Test were made for two ditf rent F1gur -66- !6 shown 1n ere mounted t1ffne e of the one story building, and 1n ea.ch <u1se the motion was studied for the following fcur cond1tione: 1. No bumping between buildings. 2. Buildings toueh1ng when at rest. 3- Bu1ld1ngs separated o. 1 rt when e,t reet. 4. Two bu1ld1ngs tied together eo that they are forced to vibrate together. Ser1es 1, lr1guree 37 to 41, shows the results of the tests for a fundamental period of . . 321 seconds for the three story building and 0.275 seconds for the one story building. Serl es 2, Fi gure s 42 to 46, shows a similar series of curves for a fundamental period of o. 321.; seconde for the three story bu1ld\ng, and a period o f 0.258 seconds for t he one story bu11d:1ng. Conclus1one~ The primary effec t of bumping seem~ to be a change o f the average period of each building , bringing the effect1ve periods closer together. Consequently, there can be no generalization as to whether bumping w1ll reduce or increase the ampl itude of vibration. The result depends entirely upon the degree of resonance with the quake under cons1 de rat1on. In d1scuse1ng the curves gi ven 1. n Figures 37 to 4.iS , the effects upon the three story building will fi rst be considered, and then the effects upon the one story building will be discussed. -67- In the teats of Series 1, Figures 37 to 41, the three story. bent dominated the motion . As might be expected, F1g. 37 ehows that, for vibration approximately in the fundament al of the three story bent, ~ the one story bu11d1ng ht<1d li ttla effect on the three story. ~owever , F1gure 38 ehowa that ty1ng the two bu1ld1ngs together gives resonance to harmonic vibration for the quake of tha t test; so tha t tying the buildings would be worse than either bumping or free vibration. F1g . 39 also shows detr!mental results from tying the build1nge, but tn this case the curve for bu ping under the condition with the buildings touching when at rest ahowa much smaller amplitudes of v1brat1on for the three story building, than were prevale nt when the building was free to vibrate without bumping. F1g. 41 shows a sl1ght benefit from tying, over free vibration, but not ae small amplitudes ae in the c a ses where bumping took place. ~eries two with stiffer columns 1n the one story bent, where t he vibration o f the three story bent 1s due primarily to the fundamental (Fi g . 42), gives a more pronounced effect from bumping than was observed 1n Ser1es 1; however, the maximum a . pl1tudes are not greatly effected. ·rne other curves of Series 2 are somewhat similar to the corre sponding curves of Serie s 1, and lead to the same conclusion; namely, that the best condition ( free, bumping, or tied) depends upon the -68- ground motion and no general eta.tement can be made as to which 1s best . Now, let ue consider the motion of the one story bent. The same conolua1on s eems to apply to 1t. F1guree 37 a.nd 42 g 1 ve the mo st violent vi bra ti on for the condition with no bumping. ?1guree 38 to 41 and 43 to 46 g~.ve the largest amp11 tudea for the .c ond1 t1on where the bu1ldJ.nga are tied together. Figures 38 and 43 show less amplitude of vibration when bumping occurs t han when the building vibrates freely , while ~i gures 39, 40 , 41, 44, 45 , and .!16 give , for cases whe n the buildings bump , amplitudes wh:tch are either greater than, or are of about the s ame magn1.tude as , u·,e amplitude s of vi bration when no bumping takes place. Therefore , benefit or detr1ment depends upon the quake ; in most cases, however, the building that would have extremely violent vibration 1f osci llating alone has 1 ts amplitude of vi brat1on. sorue whe,t reduced at the exnense of the other building, ~htch euffers an increase in amplitude. In t he case o f an a ctual building, the battering action against an adjacent structur e may have harmful results. The r:ipid v1 bratlons due to bumping a r e es~ec1ally noticeable in tbe curves for tbe t hr ee story bu11d1ng Figur es 39, 40, 41, and 44. These shocks due to 'bumping are probably more serious than the appearance of the curves would indicate. -6 9- SERIES 1 Fundamental Periods One Story Bu:tldl.ng 0.275 seconds Three Sto ry Bui. lding 0.321. seconds Combined Pen·i od 0 .312 seconds -70- Roof 3rd Floor 2nd Floor "'-./;- One Story Building a v :;_5 ..::e>: ' 5 Ground Motion Not Bumping.--Record NO. 259. Bumping; Touchlng when 1 at rest.--Record No. 264. Bumpi ng; Separated 0.1 ' when at rest. --Record 275. Two Buildings tied together.-- Record No. 270. Enlarged 2X. 8-29-35 Quake Stick No. 1 Pulled by Two Pounds F1g. 37 -71- eight 3rd Floor One Story Building Ground ;,f otion - - - - - Not Bumping;-- Record No. 260. Bumping; Touching When at Rest.--Record No. 265. Bumping; Senarated 0.1" When at Rest.--Record 276. Two Buildings Tied Together.--Record No. 271. Enlarged 2X 8-29-35. Qualm Stick No. 1 Pulled by Three Pounds Weight Fig. 38 -72- 3rd Floor One Story Ground Motion - - - - - Not Bumning.- - Record No . 261. - - - - - Bumping; Touching When at Reet/--Rec ord No. 266. - - - - - Bumping; Separated 0 .. 1'' When at Rest. __ qecord 277. - - - - - Two Buildings Tied Together.--Record No. 272. Enlarged 2X. 8-29-35 Quake Stick No. 1 Pulled by Four Pounds Weight Fig. 39 -73- Roof 3rd Floor 2nd Floor One Story Bldg. $ rvV\( ; :.zs::;;,ftiii3 :a; Ground Motion Not Bumping.--Record No . 252. - - - - Bumping : Touching When at Rest.--Record No. 267. Bumping; Separated 0.1'' When at Rest.--Record 278. Two Buildings Tied Together.--Record No. ·273. Enlarged 2X. 8-29-35. Quake Stick No. 1 Pulled by Five Pounds Weigl-it F1g. 4 0 -74- Roof 3rd Floor 2nd ~loor One Story Building Ground Motion - - - - Not Bumping.--Record No. 263. - - - - Bumping; Touching ~hen at Rest. --Record No . 258. - - - - Bumping; SeDarated 0.1~ When at qest.--Record 279. - - - - Two Buildings Tied Together,--Record No. 274. Enlarged 2X. 8-29-35. ~uake Stick No. 1 Fulled by Six Pounds Weight Fig. 41 -75- SSRI ES 2 ~undamental ? eriode One Story Bu1ld1ng 0. 258 second.a Three Story Bu1ld1ng 0.324 seconds. Combined Period 0.284 seconds -76- 3rd Floor Ground Motion - ----Not Bumping. -- Record No. 296. - - - - Bumping; Touching When at Rest.-- Record No. 2eo. - - - - Bumning; Separated 0.1'' iVhen at Rest.-- Record 291. - - - - Two Buildings Tied Together.-- Record No. 285. Enlarged 2X • Quake Stick No. 1 Pulled by Two Pounds Weight. Fig . 42 -77- 3rd Floor One Story Building Ground Motion Not Bumping.-- Record No. 297. Bumping; Touching When at Rest.--Record No. 281. - - - - - Bumping; Separated 0.1~ , hen at Rest.-- Record 292. - - - - - Two Buildings Tied Together.-- Record No. 286. Enlarged 2X. ~uake Stick No. 1 Pulled by Three Pounds Height. Ftg. 43 -78- Roof 3rd Floor One Story Building Ground Motion - - - - N o t Bumping; Record No . 298. - - - - Bumping; Touching When at Rest.-- Record No. 282. - - - - Bumping; Separated 0.1~ When at Rest . -- Record 293. - - - - Two Bfiildinge Tied Together.-- Record No . 287. Enlarged 2X ~uake Stick No. 1 Pulled by Four Pounds Weight. Fig. 44 -79- Roof ~rd Floor 2nd Floor Ground Motion Not Bumping.-- Record No. 299. Bumping; Touching ~hen at ~est.-- Record No. 283. - - - - Bumping; Separated 0.1'" 'When at Rest.-- ?{ec ord 29~- - - - Two Buildings Tied Together.--Record No. 288. Enlarged 2X Quake Stick No. 1 ·p ulled by Fl ve Pounds Weight. Fig. 45 -80- Roof 3rd Floor 2nd Floor Ground M0t1on Not Bumping.-- Record No. 300. Bumping; Touching 1Vhen at Rest.--Record No. 284. Bumping; Separated 0.1·1• When at ~est. -- Reco.cd 295, - - - - Two Buildings Tied Together.-- Record No. 289. Enlarged 2X ~uake Stick No.l Pulled by Six rounds Fig·. 46 -81- ~eight. hater X ffect of Tension Di gonal ne and br e try on ent Dig nal hr c1ng copsed of rubb r as hon in 1gur 47 used or a tr1 eries oft t . th n re oved from the lower fl or, nd . 1gure 47 D1agon le wer the b h vi r of th_ bu11d1bg studied w th d1ag nals tn the upoer to fl ors onl. ln th u er• The cond1t1on floor on y w -82- leo tr ed. tth d1 Two onal et of diagonal springs were also used; first, long, light springs, and second, •·s hort. st1 ff -s prings were attached as diagonal bracing. Various init1a.l tens1one were used. Tbe rubber strips ~d long spr1·n gs were g1 ven about 1. 07 oz.. 1n1 t1al tene1on for the first teete ., anrl 3.68 oz . and 10. 5 oz., respeeti vely, for a succeeding serte·s of trials. The short snring s wet'e used first with no 1n1t1al tension, and t hen with 1n1t.1al t.ens1on~ Fi gures 48 to 67 show the results of these tests. As can readily be seen, there was 11ttle effect other than a changed period resulting 1n a. slight phase difference and a e.11ghtly different response to resonance. Although there was fa.1.rly good damping obtained ua\mg rubber strips as diagonals, the short tlme of the transient vibration was insufficient to real}?.e any benefit from the damping . It may be observed tha t , 1n order to use tene1on diagonals ef-fectively aa brac:1ng , there should be a large ·inerea. ee in tension, wlth small defo naa.t1on , rather than a very gradual incr•ea.ee, as was the case for the rubber diagonals and als o for the epr-lngs used. Figures 58 to 72 show ma.xi.mum amp11 tude of free v1 brat ion plotted against time. In other \lords , the curve~ represent the envelope of vibration as the einuao1da.l osc111at1on, resulting from releasing t he bent from a d1splaeed position, 1s damped out. Figs. 68, 69__, and 70 s.bow t ha t the rubber diagonals were eapa.ble of absorbing a considerable amount of energy , al though a longer period of time thB,n that required for a ••qualte 0 on the shi king table was required to produce not eeable effects . Figu:t>es 71 and 72 show that short stiff spr1ng e , especially when looiHij have some damping effect, but that th libng li.i::1oor ~rings produced very 11ttL~ dampi ng . The study, taken a.ea whole, tllustratea the fact that, where t r e.nsient vib1"ations are 1?1vol'lred, resonance controls the v1brat1on , and damping eha racter1st1cs have little influenc e . V / J Three Story Bldg.--Roof Three Story Bldg.-- 3rd.Flo0r. Three Story Bldg.-- 2nd. Floor. One Story Bldg.-- Roof. Ground Motion. - - - - - Rubber Diagonals at All Floors; 1.07 oz. Tension.-- Rec. 184B. - - - - -Rubber Diagonals at 2nd and 3rd Floors; 1.07 oz. Tension:189B. Rubber Diagonals at 3rd Flo0r; 1.07 oz. Tension.-- Rec. 196B. - - - - - No Diagonals.-- Record No. 201A. Enlarged 2X 7-5-35 Quake Stick No. 1 Pulled by 2 Pounds Weight. Flg. 48 -85- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Wloor Three Story Bldg.-- 2nd Floor. One Story Bldg.-- Roof ot on - Rubber Diagonals at All Floors; 1.07 oz. Tensi~n.--Rec. 185A. - - - - Rubber Diagonals at 2nd & 3rd Floors; 1.07 oz. Tension.-190B. Rubber Diagonals at 3rd Floor; 1.07 oz. Tenston.-- ~ec. 197B. - - - - - No Diagonals.-- Record No. 202B. Enlarged 2X 7-5-35 Quake Stick Nor 1 Pulled by 3 Pounds Weight. F1g. 49 -86- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor. Three Story Bldg.-- 2nd Flo~r One Story Bldg.-- ~oof. Ground Motion. - - - - - Rubber Diagonals at All Floors; 1.07 oz. Tension.-- Rec.185A. Rubber Diagonals at 2nd & 3rd Floors: 1.07 oz. Tension.-191A. Rubber Diagonals at 3rd Floor; 1.07 oz. Tension.-- Rec. 198A. - - - - -No Diagonals.-- Record No. 203A. Enlarged 2X 7-5-35 Quake Stick No. 1 Pulled by 4 Pounds Weight. Fig. 50 Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor Three Story Bldg.-- 2nd Floor One Story Bldg.-- Roof Ground Motion - - -- - Rubber Diagonals at All Floors; 1.07 oz. Tension.-- Rec. 187A. Rubber Diagonals at 2nd & 3rd Floors; 1.07 oz. Tension.-192A. Rubber Diagonals at 3rd Floor; 1.07 oz. Tension.-- Rec. 199A. - - No Diagonals.-- Record No. 204A. Enlarged 2X 7-5-35 Quake Stick No. 1 Pulled by 5 Founds Weight. Fig. 51 Three Story Bldg.--Roof Three Story Bldg.-- 3rd Flo0r Three Story Bldg.-- 2nd Fl0or One Story Bldg.-- Roof Ground tbtion - - - - - - Rubber Diagonals at All Floors; 1.07 oz. Tension.-- Rec. 188 A. Rubber Diagonals at 2nd & 3rd Floors; 1.07 oz. Tension. -193A. liubber Diagonals at 3rd Floor; l. 1)7 oz. Tension. -- Rec. 200B. - - - - - -No Dia~onals.-- Record No. 205A. Enlarged 2X 7-5-35 Quake Stick No. 1 Pulled by 6 Pounds Weight. Fig. 52 -89- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor ·rhree Story Bldg. -- 2nd Floor One Story Bldg.-- Roof Ground Motion Rubber Diagonals; 1.87 oz. Rubber Diagonals; :;.58 oz. Rubber Diagonals; 10.5 oz. No Diagonals.-- Record No. Enlarged 2X '.renslon. -- Record 134B. 'l'ension.-- Record 208A. Tension.-- Record 217A. 2O1A. 7-8-35 Q,uake Stick No. l Pulled by 2 Pounds '\'eight. Fig. 53 -90- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor Three Story Bldg.-- 2nd Floor One Story Bldg.-- Roof Ground Motion - - - - - Rubber Diagonals; 1.07 oz. Tension.-- Re~ord 185B. - - - - - Rubber Diagonals; 3.68 oz. Tension. -- Record 209A. - - - - - Rubber Diagonals; l:>. 5 oz. Tension. -- Record 218A. - - - - - No Diagonals. --Record No. 202A. Enlar~ed 2X 7-8-~5 Quake Stick No. 1 Pulled by 3 Pounds Weight. Fig. 54 -91- Three Story Bldg.-- R~of Three Story Bldg.-- 3rd Floor Three Story Bldg.-- 2nd Floor One Story Bldg.-- ~oof Ground Motion - - - - - qubber D1agon~ls; 1.07 oz. - - - - - Rubber Diagonals; 3.68 oz. - - - - - qubber Diagonals; 10.5 oz. - - - - - No Diagonals.-- Record No. Enlarged 2X Tension.-- Record 185A. Tension.-- Record 210A. Tension.-- Record 219A. 203A. 7-8-35 1uake Stick No. 1 Pulled by 4 Pounds Weight. Fig. 55 M92- Three Story Bldg.-- Roof. Three Story Bldg.-- 3rd Floor Three Story Bldg.-- 2nd Floor One Story Bldg.-- Roof Ground Motion - - - - - Rubber Diagonals; 1.07 oz. - - - - - Rubber Diagonals; 3.68 oz. - - - - Rubber Dt~onals; 10.5 oz. No Diagonals.-- Record No. Enlarged 2X Tension.-- Rec~rd 187A. Tension.-- Record 211 A. Tension.-- Record 220A. 204A. 7-8-35 "ualrn Stick No. 1 Pulled by 5 Pounds Weight. Fig. -93- 56 Three Story Bldg .-- Roof Three Story Floor Three Story Bldg.-- 2nd Floor One Story Bldg.-- Roof Ground Motion - - - - -Rubber Diagonals; 1.07 oz. Tension-- Record 188B. - - - - -Rubber Diagonals; 3.68 oz. Tension-- Record 212A. Rubber Diagonals; 10.5 oz. Tension~- Record 221A. - - - - No Diagonals-- Record No. 205B. Enla~ged 2X. 7-8-35 Quake Stick No. 1 Pulled by 6 Pounds Weight. Fig. 57 -94- Three Story Bldg. --Roof Three Story Bldg.-- 3rd Floor I Three Story Bldg.-- 2nd Floor One Story Bldg.-- Roof Ground Motion - - - Long Springs; 1.07 oz Tenslon.--Record No. 224A. - - - Long Snrlngs; 3,68 oz. Tension.-- Record No. 229A. - - - No Diagonals; Record No. 236A. Enlarged 2X. 7-10-35 Q,uake Stick No. 1 Pulled by 2 Pounds Weight. Fig. 58 -95- Three Story Bldg.--Roof Three Story Bldg.-- 3rd Floor Three Story Bldg.-- 2nd Floor One Story Bldg.--Roof Ground Motion - - - -Long Springs; 1.07 oz. Tension.-- Record No. 225A. - - - Long Springs; 3.68 oz. Tension.-- Record No. 230B. - - - No Diagonals; Record No. 237A. Enlarged 2X. 7-10-35 Quake Sttck No. 1 Pulled by 3 Pounds Weight. Fig. 59 -%- Three Story Bldg.-- Roof. Three Story Bldg.-- 3rd Floor. 2nd Floor. One Story Bldg.-- Roof Ground Motion. - - - - Long Springs; 1.07 oz. Tension.-- Record No. 226. - - - - Long Springs; 3.68 oz. TensiQn.-- Record No. 231. No Diagonals; Record No. 238. Enlarged 2X. 7-10-35. Quake Stick No.l Pulled by 4 Pounds Neight. Fig. 60 -97- Three Story Bldg.-- Roof. Three Story Bldg.-- 3rd ~loor. I Three Story Bldg.-- ~nd Floor One Story Bldg.--R0of. Ground Motion. - --Long Springe; 1.07 oz. 1ens1on.-- Record No. 227. - - -Long Snrings; 3.68 oz. Tension.-- Record No. 232. - - - No Diagonals; Record No. 239. Enlarged 2X. 7-10-35. Quake Sttck No. 1 Pulled by 5 Pounds ~!eight. Fig. 61 -98- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor Three 2nd Floor. One Story Bldg.-- Roof Ground Motion. - - - - Long SDringe; 1.07 oz. Tension.-- Record No. 228. - - - - Long Springs; 3.68 oz. Tension.-- Record No. 233. - - - - No Dlagonals; Record No. 240. Enlarged 2X. 7-1O-35Q,uake St1clc No. 1 Fulled by 6 Pounds Weight. Fig. 62 -99- I Three Story Bldg.-- Roof J Three Story Bldg.-- 3rd Floor j Thre~ Story Bldg.-- 2nd Floor. One Story Bldg.-- Roof /J\ ) Ground Motion. - - -Short Stiff Springs; Loose.-- Record No. 243A. - - -Short Stiff Springs; Stretched.-- Record No. 250B. No Diagonals.-- Record No. 236A. Enlarged 2X. 7-15-35 Quake Stiel{ No. 1 Pulled by 2 Pounds Weight. Fig. 63 -100- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd. Floor. Three Story Bldg.-- jnd Floor. One Story Bldg. Roof J Ground Moti on - - - Short Stiff Springs; Loose.-- Record No. 244. - - - Short Stiff Springs; Stretched.-- Record No. 251. No Diagonals.-- Record No. 237. Enlarged 2X. 7-16-35. ~uake Stick No. 1 Pulled by 3 Pounds Weight. Fig. 64 -101- Three Story Bldg.-- Roof Three Stnry Bldg.-- 3rd Floor. I Three Story Bldg.-- 2nd Floor One Story Bldg.-- Roof rv Ground Moti0n. Short Stiff Springs; Loose.-- Record No. 245. - - - Short Stiff Springs; Stretched.-- Record No. 252. - - - No Diagonals.-- Record No. 238. Enlarged 2X. 7-16-35. Quake Stick No. 1 Pulled by 4 Pounds Weight. Fig. 65 -102- Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor. Three Story Bldg.-- 2nd Floor. One Story Bldg.-- Roof. Ground Motion Short Stiff SDrings; Loose.-- ~ecord No. 246. - - - Short Stiff Springs; Stretched.-- Record No. 253. - - -No Diagonals.-- Record No. 239. Enlarged 2X. 7-15-35. Quake Stick No. 1 Pulled by 5 Pounds Weight. Fig. 66 -103- \ V Three Story Bldg.-- Roof Three Story Bldg.-- 3rd Floor Three Story Bldg.-- 2nd Floor rx One Story Bldg.-- Roof Ground Motion - - -Short Stiff Snr1ngs; Loose.-- Record No~ 247. - - - Short Stiff Sprtngs; Stretched.-- Record No. 254B. - - - No Diagonals.-- Record No. 240 . Enlarged 2X. 7-15-35. Quake Stick No. 1 Pulled by 6 Pounds Fig. 67 -104- eight. -~~~-~[-- f ~l --·-/··:1-'.-l---~'-:-~i ~ : I _I _ I ~ - ; - - ··, v-:~- {c'. -i- R __:_ I _____;_I · :••'! II : : .S -·-,--. I i/ i; - - . • . · . : ·' 1 -:--i---,I ... . ,, , I . I •• •. , . ~ 1 · . .• !:. I ... . ; ~ :· j- - . ' 1 ~ . -. ' ·- · ; ~- --, . : '·· . ·--' '. I ~ . i- :-. t "' - ~ ~ ,!; '. ~ _.• ~ : -~,~~r- 1--•-·· • <:) \, \g , . I • · .: E1J ~ ~:::l~:- • --~'-' l >- __ ,._ • ' ---• • ! . : ~-: :: :":T- -1 ' . • · • -: ,. : :! • ;:: l r· I - ___._-4_ -- · ·: Ii , - ··-· I' . . : .~·~, ---~~ :: "-· ' I I l. u..1 - ~ -~~~-L- :~ , +-rtr: l' " I' ~,. ..::J~q-~1-: i-.;+,.;.J, : i'I· ·. , I I,. •, .l:-~.• ~;:: :::: j L i --t· -+---- . ' ' I' •• I ! - ! ;/~-: .. '>: -i- --!: • ~~l-ki· r , . t lI "'i : l I I ' ' f. • i=t·~=i~.t--~.,......:p;~~i~i~~.J_.;.J~W.Sµ.:J.~ I •·· I f · -l · ·. I :' I 1-~,~-7,'i'F· •r+Jl·. .• , ,.. :•. ! J:4- . !.(, , T •A •,• I' C '·• r+·, .J i ·i; i I:··: f :-··· "' ' 'I · ,' · ,. w . , --1I ·J·· q;_ T l, 11 't' ... ,' ! : - -·:·:·: 1 I , , . r~t-·· >··· .L., I : [_ -:'__-- ~t·_ ; _-...... -i-_-_.l, ": :' <liI " ' , i'. ·:· ~~++Lll"' '1 · lI l• -: ··.i.· ,_ ,..1. +1 r-:.~T"'!'+~~~I1~1Y--rt L • _•;-;-r :t __ ., . ' l --'-- : 1 i I , ' ' . l . . __...J_' ,· - - ':- - -J I, . -l •·' •• • ' 'll , .I I ' . 1 • • t: I 1-1---- -----,· ---·'1·-· __ I.· .. I - • _ __,__I • .. JiI .._, ....I . . . :I,I ._; .,, . L~: 't • •• • , + -1+ ·.i-! .. ,' - = ' I I '. : _i._ _ _ __i_____ _ 1 : f I: .)<- - i - -·-·.;.l_ _....._ ...,: _ _l-_:_,J _'_-l-_ ____...;.....j..._ ......_ :•41 I , : ; :, - ; I • :-::: ~ ~ · : : . . '. I . . 1. . . _ , _- - "· ! ;t~ ( > • • 1 1::1: ; :1: ·i,;; , ; ; n '._ l :r 1_,L lVi; ~ 1 ' [ [ +-_ _ _ _--+-· - J·- - •, !~_ -·--:---t-.....,..4---l--:-f-,-- t --t--+-+-.z.....if-t_:;!=.:;..:...::;.~:__;::i..,.~ -i-:::::r::;f--.:=+==+.:.:,..+ i-= ~=.:;.;+ 1-i.::;.;:.:t+:1 f : !j:_: : : • , i · : :·· - 1 - •., 1.1 ! • --- . . .;itt n: PART III Chapter XI Behavior of S1mple Bents Subjected to Established Simple Harmonic Ground :,1ot1on Description of Apparatus: The apparatus used in the study of bents under the influence of established et mple harmonic ground motion 1s shown in Figure 73. The base upon wh1ch the bent wiis mounted was caused to oscillate with slmnle harmonic motion by means of a motor- driven crank and connecting rod mechanism. The speed of the motor was adjusted by means of a rheostat, and the speed of tl:ie crank wae measured by a. tachometer. The speed was held constant at a given value until the mot1on of the bent achieved a steady eta.te condition, and then the amplitude of the top floor motion was measured by means of a mlcrometer screw. Two types of bents were used. Both were two story frames with st•el columns and rigid g1rders, of the d1mene1one shown ln Figure 7l}- A . 9owever, one bent had columns consisting of flat steel strips , as shown in the sectional view Figure 74-B, while the other had columns eone1$t1ng of secttons of flexible steel tane wt th the convex side outward, as shown · 1n Figure 74-c. -110- flGUR.E. 73 G" - - I - II 11 SECTION ~ - ~ A-A - - -- A r --- -- - n ] [ \S) C. I, - A. - ELE VATION FIGURE. 74 SE.CTI ON A-A . The object was to study the effects of buckling in the columns with curved cross section, as compared to the behavior of the ordinary bent .wtth flat columns. e eults and Conclue1ons: Simultaneous observations of the amplitude of vibration of the top of the bent and of the speed of rotat ion of the cran~ were taken . From this data, the :t"at1os of ampl1 tude of the top o f the bent to A amplitude of ground motion (A), and natural period to period of ground motion, 0 'f ( - ) , were calculated . To The results were plotted to give the curves shown 1n F1gure s 75 •. and 76 . Fi gure 75 shows the resonance curve for the ordinary two story bent with flat steel columns ( F1g . 74- B) . The peak at a value of T/ 1: 0 of 1 ls the eond1t1on of resenance,.,w1 th the maximum ampl1 tude det,erm1n9d. by the internal friction of the system,. while the peak at T/ T0 equal to 2.6 1s due to the condition of resonance wi th the harmonic. Tbe ratlo of the natural fundamental par-iod to the period of the ground motion at the peak due to harmonic vibration 1s the same as the ratio of the f'unda ental period to the harmonic pe:riod. - 112- Thie value - ii It- A AMPLITUDE OF TOP OF = Ao AMPLITUDE. OF '·I:: , 1.,.' ' : ' ~ I~ - ~ r.T :. • . l , ) '' ' ' I ' I I i-: ---I ' - • i , . :.µ., •...... r➔ i -,j- ' f - -;-,.-;.t,,.._.. -'-'-~-_,_ - -_ .. ,--1+ - ;.. - ~-'----!-----l-'---l-~-h•--I-- • •- + ' - --1-~ i·•·· . ·1··.- . ·· -· ·· • ·- ::-- 1,~-: ••• l ~_:_:· - ·. t-~·-:. . . : -:_: • ~- . - : _ ·- - • · · ·· -;· •• - - ~-. -- . ~ ; - ~ BE.NT GROUND MOTION . '--l--,..µ.-~~.....- ......,--1.;.~-1-~-1---1-----h rt,; -- L - H -1 r·t J~j:i . ,-'. - ! ., ! . ~ -1:t . :!; ~! ':·;+D- i~; - i...:. . 1.: :.L ;1. l '. :f_;-j" -t-:1+- " s1SH~*mitl [ L:I }; : ~ ~rn;jrfi::-1~4,4~JJ]t8:i ~~~~~, :QF1t:i~~~~T ~~:~:r~#~\cfH.\.r~LlN•~HtGLl·~ ;. , -J j,:·: 7 ~ ~ m~1 --,--·::.:: r::,1,1u:Htwn,rtJi·~1,:'.i w+ , ~ ;p,rnm Lrn r,rn,1~ ¼-' t J:gf \Fk::t ' . ,+f . •·- • i :.-. ,-,· ,·; a - i ~ - -·-•-,- : ~. ,_ ::j ! ;·,,:+::;.L J.:bj: ~.• I,ci. ;Li ~~ -Ff _):~::t-'.-iL:~~:~:-i :· :;. '. :~r: l)~?fift ffi r H . ~;.:2 :;:1:i:: :· :±fr• •· •jif'lj'"' 1"" . ·r; \. .: :.: ;.. I • • •I • .. ':,jJi 'f"~~1{'ir') i~ .+1,:/-.~:~;~ 1 ~::-;.~ .:.;,.:~~~~~~.t•=~ti~~---:..lri "tt :i:rr77:t s=r :77:f t :~-: ·-+l . '.] :~ :.;l ~_ :l H~~ ~ :'._ I -4---: --~- ") .> . •1•• t , . ,·I i· . . ·r· • r.!::::~ :~:.u • ::.:::::~: r.: :.':.· . -:-~)- .t-t.;·~- .:~ ::: . ·::;.; :· ~~ :~-"- ~-:~: ::.t : i: ;: i:· - l-;- •· • ~~: L~ -,· ..!~ -- - . -f! 1 . ' . ·- :. ~ f"]':::-f---~7-~~~:~~;-i;_-:~•.·.-.~=--; ·• · •··· ·~· '.d:·;t -· · · ~·· t· ··H , .; p., ;-'·T~!: ;r ···· '·· •··\:l:r•• :·::·1:.;Lr, ::: ±.. I ;;Ulr·:: : ·r: ~--f* · j· - , ~ ~ ,--~J...- cc••,c,>·'''"C'CL"' • '~d:\ 1 1 ·;· ·1;··+ ,·•- i : :T ·1.. 1 ·~·· fi· ·--:t;-,,' · · 1···:<-·'r,- ·:-:~:: .•··y ::· >:T_ ::· · : r,~ 1:: : :: r-·,.··•·r-·•.t· ·: ':. :•·H . :' ;r · -, t:i::): L::1 :'."·d~:ff1:1T~GFTU(:T :Ti:-~~; ::! t:FL: ::: i:; • · · .T '} • • • ~-?F> -:-~( :: :: i y: T :: - :;-~- - ~1 ;: ••• • can be calcula:ted using a. theo:Nltical equa tion, for bents with identical stories and floors, presented by t~eri t P . White in his thesis. where U) - 1 angular frequeney . L - number of stories . n = order of vi bratl on. ( 1 for f\.tnd a.mental , 2 for fir•st harmonic, 3 fol:> second harmonic, etc.) = stiffness of pair of columns. - W2 TI T 0 - - - • Then s1.n ;;( zxz ,. g.x?, 1 -+ 1 ) . uJ l 'fl/ - ff-I~o 54° -- sin 18° -a1n 2 . 61 It can be rea.dily seen tha.t tbe exper1mentoJ. p eak occurs at the proper r atio of T/ T0 as predicted by this theoretical calculation. Figure 76 eh.owe three ourves of ·the ea.me type for the bent with curved steel tape columns ( E'igure 7l~- C). ,, 1, th the anroli tude of the ground mot1on equal to 0 . 0125 inches, the ordi nary type of resonance curve 1.a obtained. The accelerations were sufficiently small tha t praet1ca.11y i10 buckling l Wh1 te, l~er1 t P ., A11~ly!,t_c~! .§~:J!diea .Q.f ~ Dynami 9 ~eeoonse of Certain § trueturea !,_Q Assumed Ground !.!Iovements. Thesis for Degree of Doctor of '?h11osophy Californi a Institute of Tech~ -114- 1935 . . j _ ...... .J~:_:; .1 •• -! ·- ·• • 1- ·7·· · : · -1 •+,-'"-!'+•-,-+,--..+..._-,-,-'-'+:... ,_• _.,-,-i+.l ;_;; T'" ·r•:lrT: -L;:; .. i ~ _: ·: f ~:_; ; ·~~ ' ..., ••¥--',T · · 11. . .!..; ... ·j-: .. . . ··~:-1 J· ~L::~;l · •- •-4- - ·4_ ·-•-.'. j + ·-· l ' \:Y=·:r£Jrt :/ ~ ~'30n.l 17d 1/\1\t .::!O dOJ. -do '30 (lJ.17d V\l\f dO J.N:3 9 :: -I 15- occurred, and 1 t can be eeen that a, curve of a shape simila r to Figure 75 for the flat columns was obtained. The peak at T/ T0 of one is due to resonance with the fundamental , wherea s the pealt at T/ T 0 equal to l. 77 1s caused by the harmonic. The reason for the lower rati o of T/T 0 than for the bent w1 th fl9.t columns 8 for resonance wl th the hq.r mon1e, i s that there wa f a, flutte ring type of buckling at the high speede necessary for e. ratio of T/ 1' 0 of 1. 77 , and a consequent redu ction i n st1 '.ffness . !U th eccentr1c1 ty of the cr~,nJt ad.1usted to g1 ve an a.mr,11 tude of ground motion ( A0 ) of o. l£7 inches,. the ground motion was auff1c1ently violent that buekling of the columns took ulace, to a greater or leas degree ~ for all rattos o:r T/T 0 greater than 0.85 . When buckling takes place, the columns g,re suddenly reduced tn et1ffnees to a marlted degree, and the p er! o,1 of the bent is made muc h longer. Th1s is the reason f or the peak betng d1. spla oed to the left ( T/'r0 = o. 85 ) . ~:ar1 ous degrees of buckl !ne: 1-),n d reAonance comb1 ne t o gl ve the jagged curves shown, etnce the columns were snapping and buckli ng intermittently . It should be mentioned that t t1e moment required to c ause buck11ne- of a convex column 1n one d1.rect1on 1a much leea than the moment required to buckle the column in the other d1rect1on. surr1c,.ent, buckling occurred at the h1gh cra::1l{ speeds to - 116- destroy the effect of' the harmoni c enti rely . The curve with the ampli tude of the ground moti on e qual to 0 .125 i nc hes s hows the ef f ects o f t he vtolent buckling. that took place a f tf.:r T.,_ / ..I' 0 exceeded 0 . 73. The jagged nature of the curve again is a n indica t ion of the vari ous degrees nf buclt11ng . A comparison o f the t hree curves of F'lgure 76 ehow-e that the buckling was very effe ctive ln reduc ing the :.;;,nplitude o f vibr ati on relative to the i ntensity o f tae ground v"lbration. The curvee indicate tha t t he rel~1tive amplitude (A/ A0 ) decreases as th~ amr,li tude of the ground motion (A 0 ) inc rea s es . Another ef fect tha t c an be observ· ..d 1s the a bift o · the pea.ks toward the l eft, due to decrea s1 ng the sti f fne ss when buckl ing occura. The greater the extent of buckling , tbe greater wi l l be t he decrease in the average sti. ffnee s , wi th a con sequen t ;novement o f the peak of the curve toward t he left . In order to apply the system of 'tuck.ling members to an ~1.ctual atruoture , for t he puroose of reducing l':l..mpl i tude o f vlbra:· i on in eurt hqual..:es , i t would , of cour·se, be necessa.ry t o support t ho structure by other members that would nt)t be s eriou sly effected by the motion of' the str·ucture . The buckling members would not be used to support the load at all , but -117 - merely to g ive t he sudden changes 1n s ti f f ne s s r equired to ur e vent e xc essi ve vi b r a ti on due t o r esonance. - 118 -