Experimental Verification of Dr. Biot's Method of Prediction of Earthquake Stresses in a Building Citation Cawley, Clifford Comer (1933) Experimental Verification of Dr. Biot's Method of Prediction of Earthquake Stresses in a Building. Master's thesis, California Institute of Technology. doi:10.7907/vb95-d527. https://resolver.caltech.edu/CaltechTHESIS:09242024-163130625 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 1933 Record Number: CaltechTHESIS:09242024-163130625 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:09242024-163130625 DOI: 10.7907/vb95-d527 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16756 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 24 Sep 2024 21:28 Last Modified: 24 Sep 2024 21:29 Thesis Files PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

EXPERIMt'NTAL Vf!RIFICATION OF DR. BI<YI"S METHOD OF PREDICTION OF EARTHQUAKE STRESSES I N A BUILDING THESIS BY CLIFFORD C. CAWLEY IN PARTIAL FULFILllfENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA 1933 INDEX PAGE References 1. Fig. 1., Recording Shaking Table 3. Introduction 4. Correction 5. Dr. Biot•s Paper 6. The Recording Shaking Table 12. Method of Calculation 13. Fig. 2. Calculation of Constants for the Six-Story Bent 16. Fig. 3. Curves for Earthquake No. l. 17. Fig. 4. Curves for Earthquake No. 2. 18. Appendix l. Getting the ~ 19. Value Appendix. 2. Biot vs. One-Tenth G 20. Appendix 3. Method of getting Working Curves from Original Records 24. Appendix 4. The 11 West Formula. 11 25. Appendix 5. Experimental Determination of the 11 K1 s 11 Appendix 6. Determination of Speed of Tape 27. 28. Appendix 7. Determination of Natural Free Periods of Bents 29. Appendix 8. Periods of Actual Buildings 35. Appendix 9. Cash Register Tape 35. REFERENCES "Calculation of the stresser occuring in a building during earthquakes, 11 by Dr. M. Biot, Guggenheim Aeronautics Laboratory, California Institute of Technology, Pasadena, Publication No. 24. (Copy included in Thesis.) 11 Theory of Elastic Systems Vibrating under Transient Impulse with an Application to Earthquake-Proof :Buildings." M. Biot, Department of Aeronautics, California Institute of Technology. Reprinted from the Proceedings of the National Academy of Sciences, Vol. 19, No. 2, pp. 262-268, February, 1933. "Reoprt on Shaking Table Reseuch, 11 Vito A. Vanoni, Department of Civil Engineering, California lnstitute of Technology. (1932.) "Transient Oscillations in Elastic Systems, 11 Thesis, M. Biot, California Institute of Technology, Ph.D., 1932. 11 The Effects of Earthquakes on Buildings with a Flexible First Story," R.R. Martel, Ass•ciate Professor of Civil Engineering, California Institute of Tedhnology, Pasadena, Reprinted from the Bulletin of the Seismological Society of America, Vol.19, No. 3, September, 1929. "The Dynamic Behaviour of Some Simple Bents Subjected to Established Simple Harmonic Motion," (Paper No. 490) By R.R. Martel, S. B., Ass. Professor of Civil Engineering, California Institute of Technology. Annales De La Societe Scientifique De Bruxelles, Extra.it, 11 :Propriete generale des systemes elastiques Soumis a impulsion transitoire, Notes de M. Maurice ff Biot. 1932. 11 Design and Construction of an Harmonic Analyzer, 11 Thesis, Jesse W. M. Du Mond, 1916, California Institute of Technology. 11 fhe Overturn of Blocks during EArthquak:es, 11 Dr. Maurice M. Biot, Civil Engineering Office, California Institute of Technology, 1933. 11 1art~ke Damage and Earthquake Insurance," John R. Freeman. (1932.) '3. JIG. 1. RECORDING SH.AXING TABLE., IN THI CIVIL ENGINEERING DEP.lR~T OF THE CALIFORNIA INSTITUTE OF TECHllOLOGY. INTRODUCTION In June, 1932, Dr. Maurice M. Biot, of the California In s titute of Technology. presented a paper at the National Applied Mechanics Meeting entitled "Calculation of the Stresses Ocrmring in a Building during Earthquakes. 11 This paper follows. On the assumption that a building behaves as an elastic system, he arrives at an expression for the motion due to an arbitrary horizontal motion applied at the base. The Recording Shaking Table, Fig. 1., brain-child of lpofessor R.R. Martel, offered a means of carrying out Dr. Biot•s prediction for model bents. Thie Thesis is the summary of this work. A very important correction in Dr. Biot•s paper: 11 K11 is the force necessary to displace two consecutive floors unit horizontal distance relative to each other. 1 NOT J., I) I cs LABORATORY GI,;. t CALCULATION OF THE STRESSES OCCURING Ill A BUILDING DURING llARI'lfQ.~AKES .__ By Dr. 11. Biot, 1 CALI, uiiNIA IN,'TITU1 F_ OF TECHNOLOGY Pasadena, Calif. t1.P~ ~fuN A ~ Th• 110Uon ot • building affected by an earthquake bu tile character ot • tranaient oeoillaUon. J. general method ot analyaie of auch an oeollletlon ie here dneloped and applied to thle epactel caae. The prlnciplea Involved ere the , ... aa tho ■ e proposed by BeaT1a1de tor th• analyaia or transient electric currenta . We cocaider here only abeartng detonnattona. The calouletion 1o carried out completely tor • building of constant maas and r1e-1dity at eTery floor except the tirat. The 1nnuence or en "elastic first floor• 1s investlg11ttad and It la shown thot the oscl llot Ion depende fundamentally on • parameter OC.. Thia parameter le • product of tl'.e number ot upper tloora by the ratio R of the rigidity of the flrat floor to that ot the others. The solution or the problem ror a audden constant acceler8tion leeda to a aimple calculation ot the ettect· of reaonence. n lbr tbe purpoee of opplyiog the theory to actuel eartbqulllce accelerations a general theorem tor elastic ayatema 1a eatsblisbed which g1Tea a Tery eimple method ot celculeting the oaolllatlon amplitudes under a random lmpulae by conalderlnt1 !ta apectrum. Thia motion 1a J-~,i~..~ ~Hf AJ!9e~eo~g;uation ~iy -= 2 --a b~ cH' ,>.T which by putting !j = z(~)e , b~comea 24 d'z -.2 d~.z+""z=O. The general oolutlon 11, z. = A ,os ).~ + S ""' >.t . Cons Ider the boundary c end! tiona OL.L=O X•O ~It :.. .. }J-/JJ!:s-GW bx .l practical method 1a given to colculata the motion ot any type ot buiU- Ing. Tt.ls study wes made at the Coll forn1a Inst! tute ot Tachnol0g7 at the auggeation ot Prof. Th.Ton Kl'lrmftn and the euthor 1111.shea to express hie appreciation ot the cont1nuel interest Prof. Ton xerman hoe taken in ita pro~reae. put R.. ~rat1o ct the rigidity of tile tirat floor to that or the others, and OC.~Rn, n being the number ot theee other noors, tbaae cond1 tl ons toke the form, (1) d:z. =O d~ _4;.+oc.z,,.o ;=1 d~ ~RACTER OF EARTH'(.UAK'E ACCELERATIONS rrom equation (l) B:O and z,.,Ac.c&A~ When an harmonic force aota upon en elastic 11y11tem the oac1llat1on ot the ayatem tends to a ateedy state. In case ot -r eaonence the maximum amplitude la dst,·rmlned by the 1n tarnsl tr1ct1on. But when the elaatlc ayatem 1a affected by a abort wa,-e or a random impulee, t~ steady he.manic motion haa no aurttcient time to danlop and the dDplng has nry little ettact. rrom cond1 tlon ( 2) Atg A = 0:. The roota >..k ot thie equation correspond to the tree oaoillatlon frequencies ot the building. We cbooee certain ve luee ot OC. corresponding to certain eimpla values ot ll and n •• tbllowa: Thie 1a generally • char,cter of eerthqueke acceleration-. It lo theretore intereatinp to 1nveat1r,ate the re!lponee or en undamped building to euch trenai tory torcea. R .la tho 1ntluance ot the ao called •elastic first tloor•· la actuall y nry much diacusaed, we ahell atert the analyaia by oonolderl11£ • building or the following aimple type. Let 1t be ot rectenguler al:epe. The mat 1mportent detometlon 18 ., horizontal'. shear aa abown In tlg. l. Thia would naturally not be true for wry high bulld1nga where the bendlog would ban to be taken 1nto account. J'Urthennore the eheerl ng rigi d1 ty and the maaa ot each tloor are auppoaed to b a conatent 1'rom the second tloor to the top. Only the first floor will be or a different rigidity. Let h be the height ot the building without the flrat floor am x the coordir.iete counted downward.a from tbe top ea origin. we may conai dttr this. p(lrt of the bu1ldlne aa en elestlc contlnuoua b~ whose only poaalble datonnet1on I.• abear (fig. 2). It II ia !ta total meaa, m-•1; will ba the maea per un~ t ngtb. Th• K be torce _tbet number ot uprer floore being n and h,-~ their height and cell is neoeaaery to diepl~ e oonoecutive tloore eo thot the releti lide woula be ec;ual to the unit lani; , t oefr1clent of correepondlng ebeerlng r1gld1tJ of the continuous beeiu 11111 JJ-- . Tl:e rigidity or the flrat floor Is characterlsee by In s 1.111e wsy •• IL a coatrlc!ent G defln j(t). The ground is supposed to mov. horizontall:r with a variable acceleration The equation ot rela t ive motion of the beam la c-~ propagation apeed or a ahear wan %-t.. t:-'t j(~a j 0 9'(T) f=t J,t;=Y The equation of motion becomes, -a2!:J a~z=wz+1(T) All the quantitlee ti,:urlog In this equation are dlaenalonleae. i l l T 1-- 15 l.66 2.50 5 10 l.ll l.e& 3.3~ 5 0.556 0.0s. l.66 The volume or .Ak •• a function of CC. are glv•n In the following table: 0G .Xo u 0.556 0.834 l.ll l,66 2.50 3.33 5.0 10.0 0 0.68 0.00 0.8Q l,03 l.15 1.23 l.32 l,44 oD I ;>...l 7!' 3.31 3.38 3.45 3.58 3.73 :.86 4,.()4, 4.30 q A3 A2 2 'It' 6.31 6.41 6.45 6.53 6,65 6.74 O.Ql 7.22 3K Q,4,8 Q.51 g,54 Q,511 Q.67 Q.75 g,go 10.18 ¥ ;>....4 47[' 12.60 12.62 12.65 12.6Q 12. 76 12.82 12.Q3 13.20 !.f I In tlg,.3 The period Tl< 10 :>-.5 5 " 15.73 15.75 15.77 • 15.80 15.85 15.n 16.0 16.24 ll1t ~1t T T · corresponding to Al< la, TK=u.t. I It la intereotlng to com1)8re the t'undomental period T0 to that T 0 which would occur 1f the build1n;1 would be perfectly rigid trom the a ccond tl.oor to the top, the only alasi1c1 ty being due to th• nrat tloor. lie get, le define the toilowiog notatlona: (fo/;6·0 l T A.I( P..fr~~mi+mj (t) ay I Tl n The rnettcd which a hell be used ia, opplied to mechanical vibret1one, that which 1a well known to Electrical Engineer•, and shows to be ao powerful 1n transient electrical currenta analye1a. le will further eatabll<lh a general theorem, introducing the notion of earthquake ape'ctrum, which e;lns • very adequate method ot treating the problem. California Inat1 tut• ot Tecbnoloey Contributed by the Applied Mechanics Division for praaentatlon at the National Applied Mechan1ce Meeting, New Haven, June 23-25, 1932. All pepers ore subject to Nniaion. Note: Statements and opinions advanced in papers are to be understood aa individual expressions of their authors, and not those of the Society. M ~t~ + Gu=O The ratio of traquenclea fo = ~ c: ~ f.' To 0 °' 0 0.556 0,834 1.11 1.ee 2.50 3.33 5.0 10. 00 T~= 2rr.yf =zrr.~ 1a a tuact1on yoc, f~, l 0.910 0.875 0.840 o.eoo 0.725 0.674 0.5QO l).455 0 of OC. .. .B z ~ 2. Jl)FICED HllldONJC OSCILLATION 9'('T) -= ~ In th1a case iAT we obtai n f 1nslly , tcZDC~ LJS::J£{cos ~IC T-1) • B:• ~utting Bk::: z«~11. ?ly = ~y :t + € D:r 2 ~'t y c 2:(;)el I :>.:)( ' AT~l. whic h by putt1 ~ ).t . end th e equation ot ..,Uon beoomee t =f takea the form ;}+)tz = I The a.>lution ot thia e·quetion oan be expressed aa e aum of the orthogonal :l'unctl o na zircos>-iaatiofyi ng the corres ponding bo..,geneoua equatio n end the g 1ven boundary oondi tions , 0 B,cCOS A"~ (co& ¾T- I) Let ua etudy the cc nvergence of t his s e-ri os a nd compu t e the orde r ot magnitude of the r e a~ . Ea ch o f i tis terma , the fin,t one except ed , eat1at1eo the 1nequal1 ty .,, / x-~.Ab .1:_ f'KzK(cos.:\kT-1) /<£ '\ 4 A.4I< < ('11:1<)4 /\.K Tl:e followi ng 1neqw.l1 t y g1vea on up pe !' l1m1 t tor th e rest Carrying thio upre11 i on i nto the d1ttert nt1al equation IR j< f /AZ!'(c:cs>.kT-1)/<~7r" i. l..lt ~ol'.fA,l#,z d1z, +'>tA· z·] = I I l Telcing into account the 1dent1 t:,, :z. di. ~--XZ· d;z.- l I P .. p The velue ot ~ J..4 I It' we multiply both si des o:I' that equation b1 %1<. and integrate trom O to l with reopect to we 1et, 1 t 1 ~Ai[i--xiJ/2:i21<dE.= jz.,,_dE. I '!'he cond1 t ' on ot orthogonal1 ty A"= .f!,: z. {z z·l d~ = 0t /~It. IO O O I< + i tor 4 p ~t.- :a = lc, 4 .. 1,080 . ~ t◄ s: 0.006 1 gi ,ee, Hence it we take only the toor 1n1t1al termo ot the aerie• er. • /z"- a!,, . • ~ Bl<co• AK~ [co" )..KT-I] ~- k /~~d~ ~-~k thot ).~ 1a f 1von by t he Barnou1ll1 m;mbe r B~ , the error due t o t he ttrmo nogl~c ted will be omeller then The required solution 18, 41.t: ~ )( o. 006 = (X, >< o. 000%5 le aleo oee that th io aer1ee 1a ebeo l utely end uniformly convergen t . Dlaplacemor.t The nl UI ot (T'k m01 be ghen ..,re e:rpl1c1 tly , Oi • 20C. f?>,,,_ K '5:,i" B_ I - :z. - ~ 8k Z 0 The nluea ot Bl< the tollow1~ table:: oc. 8 S, Dn'.CT OP A SUDDEN CO!ffl'ANT ACCl!LllRATICN 0 U,006 0.0M 1.ll 1,116 2,50 3.SS 5,0 lD,O 00 11th the purpo■ e ot finding a more general 1olut1on we ahall UH the preoad111€ reoulte tor building a oolut1on which oorreeponde to T<O 1lTJ• I T >o (F1c;.s) Cona1der the tollow1ng integral in the ocaple:r plane, I 1-(AT a.,n - - ~ d ' ) . . taken alone th• real 1x11 (tig.4), for T..::o the reel port ot i;\T 11 negat1 Te on the halt o1rclo ot 1nt1n1 te rediue A D C. Hence •• mo7 add thio path to the contour ot integration without any etteot on the Te.lue ot the integral. 1or T<O it can then be written , ..l-, (eL"T d)."" .1.....,. r£ eiXT d)..= o 2.?tl L X enebles ua to calculat e the The reel dioplecement 1a . t~ ► U =Joo<,, ~I( zoc l: r':;T"'. ~ • "k "-"IC In thia aeries the values o:I' the coetticiento ~/( tend t o unity when t inoraeaee indet1n1 tely and the con'Hrgence 11 sbaolute .;,d unitorm. f(T)•O 4 The d1111ene 1onleu function mot ion due to a eudden acoeleretion J COliAl, /1(-~AI( k are only tunct1on1 ot OC Bo .,,.,,,. B 0 -~;00890 -0.0116 -0,01{37 -0.0166 -0.0llll -0,0203 -0.0210 -0.0212 -0;01112 ~.336 l,7M 1,425 l.Olll 0.1102 0.802 0,710 0,612 0.520 , on:! are r1,en 1n B■ B• 0 o.uuue112 0.0001170 0.00122 0,00177 0.00234 0.00274 0,00334 0 -0,000138 -0,000208 - 0 . 00')272 -0,000386 -0,000:;.'4 -0,0001182 -0.000lll2 -0.00125 -0,00151 o.oo~ 0,00417 The maximum dioplecement telces place et the top end 1 ta value 1a ginn Tery accurately by, u.= zjJ.a. It 1o p ractically twice the otaticel deformation that would occur under the same aQcel ere t ion .. A 27rl k.DA Jl>r the oaae reoeon, •hon T>O we 11111y add the path AB c and I ;_"';;xT d)..= .L., 2:lt'i ~ _,,. The tun ct1on Z7tl t -X- The total ohear1ng force due to the oheerlne; deror:DBtlon 1o ei"-• d).= I F-f~ =j M~ 1'C&A - ip (TI defined by +.., elXT d)... l / ~; K 1a diocontinuous at the origin wher e 1t jumpe trom O to l and 11 equal to a canotant anywhere elae. ( t1g. ~). The solution correopond1 ng to e ach element ot th1• integ ral 1a >~ ,.81e z ei"T dA. d)=2.oc:...-.,~)·" 2 which, by putting 13K=- ~k = )..1<.f,K Integrat1Cf' term by term 11e get the required sol utio n in the f onn of contour integr als table : _ T I 0 ~= _.zoc.'f..4,sinA McosAK't-1) cc., o ),.:I< f('T?• Z1Ci __ T .. A,zl< ~)( It 1s the total inertia to roe j 0 M multiplied by e d1me no1onleas fun ction r6 ~ = 2«.~ ~ ffi I, ,'\CBI\ ei>-.T )..(}.'-5~,,) I<. The val ues ot the B ~ ~z;. takes the form, - 'b; =~sin.AKt(co.,).K't-l) •• tune t!one ot OC. ere fi ven 1n the follow I nf dA I oc. 0 0.5~6 0.834 1.11 l.6e 2,55 3. 33 5,0 10. 0 .... 1 Bo -oti/ '--) -l,588 -1.388 -l,268 -l,123 -1.036 -0,1187 -0.1137 -0.880 -0,816 I Br 0 0,0294 0.0394 0 , 0473 0.05118 0.0712 0.0774 0.0800 0.01112 0,01105 I B.:t 0 • '1 , uu_.u -0 . 00622 -0.00788 -0.0115 -0.0155 -0,0184 -0.0235 -0.02111 -0.0328 I B., 0 0.00131 0,001118 0.002511 0.00370 0.00526 0.00665 O.OOlll2 0,0127 0.0166 Tm ooettlclento C~ U'9 glnn 1D tho !bllowing table: pNTio u ■ ly..•e can ■ uily ccmap ute the re ■ t ot thia aer1••• le encounter the npree■ ion - - - - '"I" I.ZOZ.05 (StloltJoe Act, Jlatb, TOl, 10 p, 29g), But + I. loZ -- Bence -I 2. 3 - I U oc: -1.» = -,h -h +h = ~ k3-o.04 . The C'ror med ■ by tek:11::1£ only the tour tirst terma 1a amaller then , Tb• maximum poaelble ,..lue ot J!f I ~ "0.04 - o.ooss""- l~lm~2(/&K'''" )."!-zs The tollowtng table g1vea the TP.luee ot thece IB.•in ).ol is',sin l-, I oc. o. 0.556 o.8U 1.11 l,66 2,50 3,33 5.0 1.0 I. o,ggg 0,994 o. 1186 0,965 0,946 0,931 0,908 0,870 0,816 - !s'.sin >-31 s 0 o.ooo.2 0,00081 0,0013! 0,00245 0,00562 0,00825 0,0138 0,0234 0,0328 0 0,00077 0,00017 0,00031 0.00064 0,00132 0,00218 0,00420 0,00875 0,0166 1.00 l,004 l,0Oll 1.00 ·o.99 o.~8 0,98 0,97 0,97 0,95 •• could hew proffd by direct coneiderattou thet the mexlaum ••lue ct the aheoring torce muot be twice th• atatia tol'<!e j 0 M . 'la rind actually ZSjJ'I tbr th11 muimtm1 which 1a a ver7 good approximation. The above calouletion ahowa furthermore that th1a mu:1mum aheerlne; toroe 11 'fery nearly reached during the t1rat tundlllmental oao1llat1on. ~ 4, OSCILLATl0!-5 DUE TO ARBlTRAHY HORI7.0NT<L ACCEURATION Let the horizontal. aocel~ratton ot the ground be 0 The maiimum ab ■ artng force occura between the rirat end the secor.d joM't sin )..0 de Theae can be ea.eily ooaputed trom the following tahle or tig. 7. n_BM 1.11 l,66 2.50 3,33 f(O) a O H (~) ~"tr) ;_:• sin)...~ sin )..;t+ • zoc..,E~ sin i ..~4>.. (1) Ac b < I :>-" s !z~ ~ ~ i n '.>./,cj,"('r)l~ ~ 113.,.j Ji~} ln this ·caae the shearing rorce 1a never higher than j 0M[T/c'0 11i" A- 0 /+ tJ The ... 1uee ot -CI C~.,\n t{6)m0 8<0 "f(8)• •inM 6>o )..J(.('t'-&)sin)..6d9 . '-~+tare plot tod In !lg, 8. In case ot reaonllD.ce •1th an harmo1"'1c ot higher order ,1, I z)..i 1-rJT) <.)!.-)! L IC. takaa the torm., =-fi,n 2 t ( ) U=J 0 o'lsT 1e a sudden hal'IIOnic .tunction ot the tifflfl. It the trnquency ot thia ha?110nic acceleration la one ot the tree oac1llat1on trequenciea ot the building we haTe reaonance; the ampli tudee 1ncrecae 1ndet1n1 tely. Thi PNoedlng formula g1Tea tbe poea1b1l1ty to calculate the rate ot increaae ot the ompUtudee. Put 0, : Noting the Telues ot Bin tt.1a exire ■ 1ion we aee itet it 1a comparatively uall. we oan write, Thie formula WIY be epr,l1Pd to the case wber'° the acceleration f ! co., ).. + ½· ,Z).._ hence, 1 ~{6T)= 2CC~ Ac•~t~~1,inA11,{T-6)f(e)dB. The function 0; , '"IL On the other band, Je[Ur-e)J == -r Bk).11. cos ).Kt sin AK(,:-e) . • • ,. I I I 2 I-r..(T) < :>-,tX: >..;:>...= )!-~. - :>....~-[tJ] 0 Th• real dioplocement 1o o.on Conaider the oaae ot reeonance with the tundemantal hannonic, get the more connnlant ~ (t1J =ft~).!! ,(T-B)d9, dB BeDOa more expllci tl7 c'. 0 0,00622 o,oog,2 0,0123 0,0177 0.0254 0,0324 0,0451 0,071 Upper limit or reaonenoe atreaaaa i their rate ot 1ncre6ae I 0 A.ppU cation to reaonence 0 -0.013g -0.01gg -0.0254 -0,0377 -0.0517 -0,0623 -0.0812 -0,105 -0.129 The laat term NY be naglacted, tor Integntlng b7 parto and no ting that ... hare c~ u 0 0486 0,0066 0,0818 0,107 0.132 0,150 0,172 0.196 0.212 c~•in >... Co -0. -0,540 -0,555 -0.564 -0,578 -0.596 -0.600 -0,91g -0,635 -0,642 o.~56 21s H~r) = j,(T-e)11JJ6> ae . torm c' u -0,340 -0.397 -0.438 -0,4g6 -01544 -0,572 -0,600 -0,629 -0.642 ' °' 0 ~ .. zoc.¾ sin ). the integral c; The mu:imum ohaarlne !broee due to higher harmonic• are j 0 M'tC~ 00 • 0 a~ noor tor the first baraor,ic and hes the Talue ~.o We will calculate the correapondin,g motion ot the building bJ oonaidering the cur.. 'f{T) H aompoaed ct c lntijlte number ot amell Jlall'•• (tlg, 6). Jach ot theae lnc.-ente can be written d'f'• d dT, 117 ualng thm, the tunatlon ' pr.,.iouol7 aentioned we get the •l•entar7 aolu on oorreaponding to the inor•ent at the point 8. Tho tunction ghine the total motion h -0.ocoggo -0,00123 -0.00185 -0.00263 -0,00332 -0,00456 -0.00635 -0.005io j M &,.,j Mc~,:sin A._~c.os )..,.'t, 10.0 '{T-8) d"jl(B)de C 0 --0.=a~a le haTe ebown preT1oue~ that the etreea 1a, term ■: !B~•in A2 1 0 0,0050 o,oog5 0,0144 0, 0254 0,03118 0,0511 o.0666 0,0830 0.0905 c. 0 U,UU22U .0. 0:::11 o.oo~ 0.00578 0.00778 0,0Qgf4 0,0117 0.0145 0,0164 0 u. ,. .. o.6~ 0,633 0,562 0.518 o,4g3 0,-leB 0.440 0.408 - le . C -u.u~47 -0.0197 -0,0237 -0,0299 -0,0356 -0.0387 -0.0425 -0.0.56 -0.0452 Co -•~=• o. 0,556 0.834 l.ll l,66 2,5 3,33 5.0 10.0 If K<1 , hence :.),~).~·m(~'t)COde•tktJ]+~i<>l'"(~T)cos(At~-r)J n\ (~T)= -~c,c.i &:_ cos "-"t 4>iT) K•O ~t( tem. After a certain time the -Yalue ot ~((t') 11 reduced to 1 ta principal The upli tude 1ncreaeea 1ndet1n1 tely. ~ J~ T)= -Ile Put 1e C11,- 1{ cos ')..."'F,, eos )..."T ~ =-ia., The meximm aplitude table plaoe at the top and ita actual Talue .u. .. j}:! crcT = O.t.cic)t 8 5. ACTION 01 JWm!QUAICE ACCELERATIONS DERIVJIP rQ THE Sl'EC'TRtl! 0!' .l SISY.OORAM Moat ot the eiaaogrsna ahow the random char"c t ar or e&Fthquel·e acceleration, with v1r7 otten aeries ot quu1 periodic oacillatiob ■. It la therefore d1tt1cult to identity the action ot an e•rthquake to tt.at or the aimple acceleratio n cunea •tudied till no•. le aight ot courae a pply equation {3) and oaloulat • the dtapla.cement correepot:ding to a g.hen aiemogrmn. But it 1a au.ch aa,re co•enie.m ,o di-ride the problem and to enal.y1e eep ■ rately the elaatic prope rUe~ -:>t the building end the trequency d11 t r1but1on ot the earthquake. We need therefore a generel theorem which we ahell proTe here 1D the 1pecial caae or building oao tllationa. A.a we are more tntareated in the etreeeee we 'lliill not give the corz-.apond1ng equations ot the dieplacementj the method belnc e'Xectly the aeme ea betore we e:et the atreaa ea f'l'om the diaplaoaent bJ a aimple der1-Yat1on. BJ d1trerent1et1on ~t1'=d?« 1 eq_uation {3) may be wr1 tten y ~b i~~ki [s111 Ai. ~).n{e)d9-co• 5 "'fi;\l'f(~d~ (4) r. Put t1 rf: and j (t) = gy (e) f the acce oration T ft ' • t F;(w)= ~ c:os:>..9'f(a)de,. k coswtj(t)dt . • T Sin(w+A)J:] w+.n. rziwl=tlsin 'M"f'(9)df = ~f;~wt j(t)dt. Cone1der13 (: )h: ½ji]J:S):::~e-t)dB j'" ?{ear the maximum H 1 _{. . where j(t) repreae nta an acceleration re_,ord located in the time internl (Ot) we get • .l. J(t) • • F;(w)c.os wt dw+-4 ~(w)s,nw-. dw The expreaaion A (W)= F/(w)t-F,.Z(w) repreeenh what might be oal.led the spectral 1ntena1ty ao a tunct1on of CII. . Coming back to equation ('), we note that~ la tbe res ultant or tree oeoillat!ona of amplitude , i:1~ zoc./91<cosA ... ~ ~ ,1c-z(w)-tF.::,.(w) Alt g,0 Y11 z I( The abear1ng foraa at a oertain 11101Mtnt 1e the aum of the foraea due to . each tree oac!l.let! on ex1at1ng at that IIIOlll&t, Rach of the shearing forcee may be written Mg~ d; . °'A si~).I', E, . 11:t VA (w~ ·t ):I< ~ ) = ~ / s·int~)i,J Thie function ia plotted 1n fig, 10, to different free oac1llat1ona are, The ohearing fora_, oor• re ■ ponding ' I T M (F) Jo sin {W1t-A)x 1. 1t ""T. wK-.n. In caaa of Nsonance •1th the har1DOn1c or order It the corr•pond• 1ng shearing foroe 1 a, f M.,~)f; The attecti TO . . . . bu the ftllle M. Z or ban approximately, F, (w) ~ ~ sin(w-.n.)I t. 'lt W-A MK•M . 2C~•inA1et • The llhear1ng to rce may be put in the torm already obtained b7 a ·,>reTioua method , M 1a the total maaa of the upper tloon . 1h1a reault aan be expi-need 1D e roore complete manner ae tol l o"a : When an aooeleret1on j(t) eota upon o building during a tlllle Inter.al (Ot), the muilllUID total. ehearlllf'. force et the final moment 1o the aum or the sb eor!ng toroea due to eTe:I')' free oacillat!on existing at that 1110111ent . hob or thoae ■ bearing foroea 1e a product of three factor■; Mj.C:C.•in Xic,t t_ f & , OIIERALI ZED l lETR:>D FOR l!OILtlNOS HATillll V.lRIAl!Lll MASS AND RI GIDITY AT TIIE DI!TF.RENI' Jt.001¥! an effect1T8 aacelerat!on The method 1s abaolutel y general and could be uaed for any type of building cona!der1ng both bending and eheer!ng detorm.ation ■, We 11111 reatr1ct ouroel.-ee to the ohear in the oaae of T■r1abla fCX) .and m(x) The equation of 110t1on takes the form , ~ VA(W~ i and a coefficient ;'I- t. The f!rot factor depends only on the elut1c propert1e ■ or tbe bu!ldintr, The aeoond factor which baa tha dlmene!on of an aacelaration depende ee■ ent!elly on the ahapa or epeotrum or the ■ 1a110gram, The th! rd one 1 ■ the relat!Ye lenptb or the eerthqueke, The problem 1a thua dhided in two perts : (r,Cx)~~J• m(x) ~ +m(x)j(t) . It the aocalera~!on 1• an harmonic fllnot1on or t i • t he eolut ion 1a LI, "' !:J Sin W~ , and j-j0 s1n wt, 1) Calculation or the etfeot1Te maaeea tor • ginn building, 2) Calculat! on of the rune t!on A (w) er ■peotrua for a certain nUlllbar of ai ■-ogrme of the region, The use of a lbur1er analyeer •ill make tb!a part of the worlc T81'7 ouy, In Fig, 7 it 1e ahoWD very olearl.J the~ the effect! ve muses deor•aae ff:1'1 rapidly for 'fOlUH of DC ameller than 1, An ad'Jant-.:• or this method 1a the •ct that the affeoti ff aooaleret1on ma:, be derived In a ney ■ illpla ••Y from a g round d1 ■ plactnent record, Lat d(t) be the dieplao-nt. Putl1D(5 ft lb:preeaing the ■ olut!on in teru or the orthoeonal tunot1om ocrreapond1ng to the tree o ■ a1llat1ona and defined b7 the aquat1ona , ~[f(i<)t]+ m(,<)Wt!:JL a:O, we get !:Jc}:A,u• F1iwJ=-rrI co.,w\;d(tJdt, • • hence wZFjd(W)•F,(w) w&Fzd(w)• F2 (w) Putting than , t.d(w)=Fjd(t4+~~(w) l The diapl•o•ent oorreapond1ng to .• sudden acoeleret!on 1s . , • YL ~ '"' j•.r•.....W~wi' f.L.... ~i Finally, ~ .. j.~ ~9~(x)~o.swtt-~ • j(t) • d"(tl • -Jwt.fjJ(W)GosWt aw/w,.fid(w)sin wt dw, ~ and 111tacnt1111l along the ,~ I ~ dh W-W\ d(t)=fid~)coswt dw +j;J(w)s·,n wt dw The acoalerat1on tak•~ tlla form. Yi A("' J,z'l ;z, 111th Ci"" - ; i dh j g..t ~ A, [w~wf]yt= j. Arter ault1pl11111! both a1dee b7 total hel&ht or the bu1l~nc • t Fzd(W)• ~ sin Wt d (t) dt, 118 L.;,L' l The •pl! tud• of each frea o ■ c1l!ation 11 g1nn u the apeotrel 111tenalt7 .0. (w) bJ, a function ot v.,=j.¼!:Ji(x)xYt.(wo, wi we fin.I, the requiraclfoo-mula and the atreea at the ooord1na ta )t due to that oao1llet1o.n b7, 2 ,fA{w) "' w '{Ad(w) Thie enables ua to determine var)' a1mp1J the apeon-a or the aooel&r ► tion record, when we know 121• epeot1'Wll ot tho dioplae...,.,nt i-ecord, "'- (x) ~ =j 0 §... ~. 1( ~ \1 I 1>.>1- w, dx The total maximum etreu 1a tho a111 of all these expreuiona, The ana1Je1s of a aiaa:,r.rem e:1ne a tunot!on ■ peotrel 1nt•a1ty or erreot!Te aacel erat!on f'/A[cJJ h ... ing one or 110:re maxima • • ahown 1n fl&,11, ID a given region there generally exht certain ahareoteriat!o trei;usnciaa or tho 1011 which appar 1n •8111 a1a1DOgraa, '!'beae tre~noie■ 11111 uua~ the mentioned maxim• to be looated 1n the Tio!111t7 ot tb•d nlll•ofW in d1tteront reoordo, It three or1t1oal nlu .. , W1, Wt., W.5, tor th• building are in the region or those lllllilla we ■ 1¥>uld, It poaaibla, place th• between ~r ouh!d;t.•[~;(~~;~;~~r~•;,•(~)~j b7, -~[~sin'),, 1t~+C~sin).z.tWw&)+C~tin>-..,».~] In o;_~er to illuatrate tbie me thod • • f1nl ta o!ne-wne or total length T : j(t)mO j(t)•j.•1nAt will oalaulate tb■ epeotrum ot t<-f t>f -i<t < f Tb• preoed1ng oona1deret1ona show that the problem 1• eolnd when..,er we know the aet of orthogonal tunctiona ~i multiplied by any factor of proportional! t7, i ,a, when •• know the ■ hape of tbo tree oao! llationa, 7rolll !:it we ea■1l7 deduce the frequenoy or the corresponding Wi b:r 11mple enere:, oona1derat1ona , Coneidar the building oac1llat!ng ~!.th that fr•~uenoy. The 1110t1on 1 ■ supposed to be free, When the 1111pll tude 11 au!mum the k1net1o eoergy 1a ·equel to zero and tb■ potential energy 1a ½ff-(><)~fax. On the other hand, th■ potential eneru paaoe ■ through mro when tba etre1n diaapparo ; at that moment the l<inet1o energy 1s maximum and baa the 'Hlua h 'f l • n, (X)W(t. ~i'd )( ' !quoting those two cxrressions we get the value z The second side of the equation ls then entirely known end ~i /p (P/Bzdx is determined by (K) wl = r!,.,cx)yt dx (5) i [f,t 8 + rn:w:, 8 fi (x)J which is inife~endent of any arbitrary constant multiplying ~i y, c(x)=~Al<~ok ) putting this expression beck into the equation, ~AKm• [w,:-w!~9.i. = fl(1<.); One multiplying thet both sides by ~K• C AK= _K_z I fa,(><) 0 wherej~ Cc. (6) I<. If we consider an eleetic strip of unfform thickness h, its moment or ir.ertia will hevo tta value I under the condition that the variable width satisfies the equation, -eh:s - I - _e_ _ A 1Z - - E.mf-w2 - p-m Where A is en arbitrary constant, we may cbooae tor zany scale convenient for we ere only interested in the shope or the function• Yi In order to realize the given boundary con di tiona, the a trip will be repeated s711Detricelly around a point repreoenting the top of the building, ( fig. 11). . The deformetion of the half strip under di trerent loads will give the different tunctiona y, . Only the first deromation 1• atable, so that the othrra will have to be stabilized by a special but very eimple device. we then compute by the energetic method the correaponding valuea or the fre~uenciee. (equation 5). Another method is ennlyticel end might be useful in caee of only slight deviations from the cese or constant rigidity and mess previously investirated. It is known in ato~ic phyoics aa •perturbetion calculua.• m (x) .... m0 +e(,.,), f-1.:f')s f,. + '! (X) I ~L (x) = !:i.t + ~,(x), 1t4 , w~I. =- w~+ LO / m.JJ-. ~•• w:, ere kr.own and e I '(, i 2r, tf are omsll verie- I !Qu et ion ( 6) becomes [(f.+tex~t(9i.(x)+~(x~+fm0+€(x~(w,~+/J&,,+&(x)J ... o Taking into account the identity _g_ l!rLL• ~]+ dx ax rTIO W;, • ..,.LJ. =O • end neglecting the small terms of higher order, ~e finally get This equation bee only fin1 te solut iona it the function of the left side is orthogonal to the charocter1at1c function• of the equation obtained by equating the left side to 7Br0 (1). This condition may be written, ~h J .{ !1,o dx ~(x) 122. 1.,,,'of,b·.$ t (.h Q dx ~to (xil +wf°)_ :,io €(><) +fi m•J. 9,! dx= o. and gives the values at p, I' • 0 (1) Hilbert Courant Chap. V P• 277 0 !:Jc1<. fJx) /,hm, ~."2. (x)dx 0 • Thia is the • ·qustion of buckl1Jl8 un~er a load P of e 1,e,.. or moment or Inertia. where tions, end integrating fr::m O to h we get, l+ k. wf -Wto becomes Putting &<) Expandinc the solution in terms of the orthogonal functions THE CALCULATION OF THE ORI'HOGONAL .EVNCTIONS The ortho~onel tunctiona may be tound by two methods. is eem1 emp1r1c&l end very simple. We note that those tunctiom are detiD6d by equati~n, which by a change of the independent variable Z= (dx Cx)::: Hence end the required orthogonal functions 1:1ey finally be written. U• - ...n - u.+~ CIC. o LO- .)OL £... w,z wZ oK u .;,Ito J( Ii .. J, FIG.6 FIG. I FIG.2 FIG. I I ... , "'• FIG.9 n FIG. 10 ,t. 1----'d=-r"-'e•~"'=''---+-------+- FIG.3 :a C r 0 L--------0--'-.~-=--------~,--"ll FIG.-8 FIG.4 I c.'. o,6 Q.5' If tr.J i ~· I 0.5 0 FIG.5 u CJ.I Biof b-6 I/ I L--/ ~ l..---- V II .., I..,,-"" --- C\ ~ ~ c~ ..-- I ~L.-- :...-- ~ ,. r FIG.7 , " , o{ 111 \L THE RECORDING SHAKING TABLE Fig. l., is a photograph of Professor R.R. Martel's Recording Shaking Table, at the California Institute of Technology. The 11 earthquake 11 is applied at the base of the bent by means of a 11 qua.ke stick. 11 The Q,uake stick is drawn thru a rigid holder, on a level with the base of the bent. One edge of the quake-stick is cut to an irr•gular earthquake curve, and bears against a roller attached to the base of the bent. The bent is forced against the stick by stiff spring supports from underneath. The absolute motion of the 11 ground 11 and of the upper floors is recorded on tapes perforated by electric sparks. A constant-speed motor and rollers draws the tapes thru the bent between the points of spark gaps attached to the floors. The record is a closely spaced series of perforations, which may be traced out in pencil. The different floor records are synchronized by applying the spark before the 11 quake 11 is started. METHOD OF CALCULATION The problem is to talce a record of ground motion, and from this predict, by Dr. Biot•s method, the displacement of the upper floors relative to the base. The check on this prediction is the actual upper floor records obtained. Figure z., shows the calculation of the constants for the six-story bent. For a given bent of say, six stories, the relative defledtion of any floor at any instant T during the earthquake will be given by the sum of six terms of the form j.-l~ Here BK co~ A,.~ Z. ,r/,, 1" .--l. , 'B,.._ ,}..,~ !> , 1 of the bent and earthquake, and ANJ> f (✓.,) • It~ are constants t(J") is The five harmonic terms are all negligible, usually, so that the first term is all that need be calculated. By calculating this term at intervals during the earthquake, a curve of maxi!IlUIIl possible deflections 1s obtained, and from this the highest possible stresses may be calculated. Expression (\.) , Fig. 2., gives the displacement at any instant during the quake, at any point along the height of the building ( defined by ,5 ) for the fundamental mode. The quantity under the radical has a definite value for any instant during the earthquake, and is obtained from the areas of the curves shown at the bottom of Fig. 3. From the table of B •s, page 1 , it can be seen that the calculation for the fundamental frequency is usually the only one that need be ma.de. In the case of this bent, for example, while ']3 1 ~ - .B 0 ~ 0.110 , 0 .o '2. l O , In obtaining the values of f \ ~tJ "'" z.,. lot &t up to any inst aut, the principle of the mechanical harmonic analyzer is used. Cosine and sine curves of frequency lo are drawn at l. and 2. , Fig. 3. Ground motion ordi- ~ s are plotted horizontally from the corresponding points on the cosine and sine curves, taldng signs into account. The area on the cosine curve up to a given point, divided by 21\ ✓0 , is T (G(•J:_"' 2.,v,t &"to for that point, and similarly for 2. Calculations for point A-A, Fig. 3., for example, were as follows: Planimeter value for 1. up to A-A: :::. - (. SfS Multiplying this by 1.11, the planimeter constant, to reduce the reading to square inches; dividing by 1.5 because the cosine and sine curves were laid out with half amplitudes of 1.5 inches instead of unity, for convenience; and dividing by 211' Vo the principle of the harmonic analyzer, = C 05 ( 0) · .'i 2. 0 '2 .0 ( l . q ( ,, from CALCULATION Of CONSTANTS FOR THE s,x-STORY BENT R•I n=S cx:Rn-5 T. • 0.194SEC., SY pPtRl~N'r. _.l_ • J.• 0.IQ4 K• 33.4#, SY EXPERIMrnT. 5oLVIN G ATAN A•O(, .:\,= 1.32 FROM T.- 2 ~;• : t.- & 2 • Q~l.} 00408 B.• 0.710 , FRON TABLE ON PAGE. ; .. I • 1--""fT . FROM PAGE 7., u,.(x) • :j.t~B.co5?..t •2m/4_F(v.) FROM PAGE q_J F(v.)= (2m).,T)' [\G(t)s1n21!l{t+[G(t)cmb!~ Sue-STORY Bf NT THEN , UNIFORM RIGIDITY Tt11?0UGHOUT SUBSTITUTING VIILUES FROM ASOVE, u.= ~. (00408)'(0.710)cos l.32f :211 ok:(cn 0'194 Tf~rtjs1 n211v,tdj~rcosc(T.J,.td{ FIG. 2. (URVf" FOR I I I, A f FIG. 3. CURVES FOB EARTHQUAKE NO. l. CuRVcS Ok [ARHiQUAKE 1·, TIMI:: 1S 1-· RoM HH,I If TO U.fT - - Al3::.oLUTf MOTION ------ -qflATI\/[ MOTION - ·-CAIC MA!\. PQS,\{ll [ REL MOTION 1. I GROUND MQTiON • J'IG . 4. CUB.VIS 10:B. lllBTH~U.UCI 0 . 2. APPENDIX l. GETTING THE ~ VALUE The principle of the Mechanical Harmonic Analyzer is very satisfactorily set forth in the following paper: 11 Design and Construction of an Harmonic Analyzer, 11 Thesis, Jessie W. M. Du Mond, 1916, California Institute of Technology. T To get the value of ~G(t) '> I "' '2." ✓,, ·Ht ' at any time T, the ground motion G0) ordinates are plotted horizontally out from corresponding cos 2. "1f Vo curve points. The area inclosed by these two curves up to the point T, divided by 2. lf '4i , according th the principle of the harmonic analyzer, is the required value. If the half amplitudes of the harmonic curves have been made other than unity, the areas are affected in direct proportion. If the ground displacement ordinate at any point is positive, and the harmonic curve is ascending, the increment in area is positive. If the ground displacement is positive and the harmonic curve is descending, the increment in area is negative. If the ground displacement is negative, and the harmonic curve is ascending, the increment in area is negative. If the ground displacement is negative and the harmonic curve is descending, the increment in area is positive. APFE1l"'DIX 2. BIOT VS. ONE-TENTH G If the profile curve of a bent during an earthquake, as predicted by Dr. Biot•s method of analysis, has the same SHAPE as that obtained by assuming a const ant horizontal acceleration of some fraction of g, then we need not concern ourselves furthur with th.at difficult method, except in so far as it might lead us to altering the value of the fraction of g we are using. The following curves compare the shape of the profile of the bents obtained by the two methods, for points taken at random during an earthquake. The order of Magnitude of the accelerations that the quake stick produces is much greater than one-tenth g, as was found after calculating the displacements of the bent for this acceleration, and comparing with the records actually obtained with the machine. Hence, in the following graphs, the curves by Dr. Biot•s method and the curves by the constant acceleration method have been multiplied by a constant, so that they coincide at the top. This is legitimate for SHAPE considerations. F~ACT\ONAl.. <.:1 CUC\VE' I I l/ I I I I I I I I I/ I 9-STORY UN\fOR M R\G\r.:>ITY THROU6\10UT I/'/ J / I / I / Ifill 6-STO~Y 5-STOR'( FLEXlBLE fu~sT f LooR :u\. APPENDIX 3 METHOD OF GETTING WORKING CURVES FROM ORIGINAL RECORDS It was found not satisfactory to attempt plotting directly from the ground motion tape to the harmonic analyzing curves. Enlarging the tape records with a pantograph was found not satisfactory. The method finally used was to trace the tape perforations out in pencil, the tape beimg laid on a glass with a light shining up from underneath, and then to project these tapes with a magic lantern (grateful acknowledgement to William Beard, Instructor in Technology and Government, and genius in things mechanical) onto a working sheet, on which it was possible to trace them out fairly qt.ick:ly at the proper enlargement. An enlargement of five diameters was used, and to guard against either vertical or horizontal foreshortening, each tape was blocked in, this block enlarged five times with the dividers, and drawn at the desired place on the work-sheet. Then adjustment of the lantern was made so that the block i mage just hit this block already drawn on the work-sheet. APPENDIX 4 THE 11 WESi FORMULA" A r.ecognized method of designing earthquake-resistant buildings is to provide resistance to a horizontal acceleration of some fraction of g, the acceleration due to gravity. This value is arrived at by observing standing and overturned block-like structures in a region just visited by a destructive earthquake. The dimensions of the structures substituted in the 11 \Vest Formula. 11 give a probable upper value for the magJ!Litude of the horizontal acceleration. The West formula may he derived as follows: In Fig. S ., suppose the monument with center of gravity C. G. as shown is given a C.G. ©------1,-...--,,-- \\ FlG.$ horizontal acceleration 11 a 11 to the right, by the ground. If this acceleration is of a certain critical amount, the block will be on the verge of rotating in a counter-clock.wise direction about A. The free body diagram for the block will be as shown in Fig. S . CG . ®--___._ __ }w F. From the principles of Kinetics, with translation occilring, and rotation not occuring: F = -'W C\. s X F. ~ f.'j - 0 = ~-w ~ \'1 C.G. - 0 F,~ -f;_ 6 Fz. = \N . w s from which we have the West formula: ct = b h3 ( \) Qh -Wb APPENDIX 5 EXPERIMENTAL DETERM I NATION OF THE 11 K' s 11 The stiffnesses of the different bent arrangements were de•ermined by applyi ng a horizontal force at the top with a pulley and weights, and measuring the resulting deflection at the top. ~ ,.oo 0.19" \\-= o .08 •• 0.11 " - 3·1.00 0.19 ~ ~ 3.~o ~ K : ~ ::- '3.3 .4(). I 8 The values of K obtained using the expression ,, ~ ~· l .oo 0 .() ~ l2El ~ ~ :::: , estimating E, and calculating I from the dimensions of the steel, were found to be quite far from the experimental. "~. APPENDIX 6 DETERM INATION OF SPEED OF TAPE To get the time scale of the curves, a tape was run thru. the machine for seven minutes, and the length between beginning and ending sparks measured. The resulting calibration was 1 11 = 0.356 sec. The error in observing the time might have been two seconds at both ends, and the tape measurement might have been off a tenth of a foot in the hundred feet, because of the spark record being over an inch long, so tha t the possible error is about 4 - - - - ·lOO + 7 ·1oo \00 APPENDIX 7 DETERM INATION OF NATURAL FREE PERIODS OF BENTS By wedging the bent at different points a variety of bents can be obtained. A bent has as many natural free periods as it has floors. There are three methods of getting these periods: (1) Tape record (2) Equat i ons of Kinetics, knowing the constants of the bent (3) Approximately, from Dr. Biot 1 s equations Bent No. 1. 1st . method: Period -:: 0. \40 sE.c 2nd . method: Simple harmonic motion, . 2..<T Period = - 0,.) LTf i: -: : - - = 21T 3rd. method: cannot be used. -----51.3 -·- - - - 3.2..2.· ,2.·, .io 0.124 sEc - Bent No. 2. m,-. = 0. 2..40 sec:.• 1st. method: Fundamental period First harmonic -; 0 . 0~'1 ssc. 2nd. method: Restoring force 0( displacement, so S. H. M., =- v-> ~~ ':)1"" "->"t and acceleration For maximum displacement u' ~&.. - \~(u1.-\A,) ~ H IA, - h (u.1.-u.,) -- \ - ~ ~, -1 -: "'-1.. ~ - 2- ('> -2.. ~1. - 3 ~ --t l -=- o (» ~ 3 ! 1, - 4 ~ \. S- :J: \. fs -: : + 2. , ls, l <& e,, wy '2.. - -w -:. :::.====- -t- 0 . ) i Z.... 'f'Y\'v2. '-t1 :::: K (u.3-u~) ~ "-)1.. "''" ;::: ~ (u.1. -v.,} - ,~( u~ - u i ) n\ ., ........lo _.,'- \A. .. --1..A.,-\.A3 - 0 -: U , 'LQO wl. -:. \ . S'S" ) 0'~ -:. l.1.. ) u• tV . 5"6 't_ ~ () 5", ) 19Qu. ) il,;-0 2.. 7t I~ I) "Mi;; l•H ~L ?1:1\.,1),!) .;; 15"&).;" f' ri..c;~ t( p.~ \""\\)'"' ' ' - S ~c(),--' !) \-tA"-'"'-l>~ ,~ -= lsc, Ou l... Ti --=- Rr\ -::. 2.. ~. ~ 3 .l,4- 1~ -:. ls,.S4 o . loo ') ~ C ~ - u.olJ, ~ SGc. . 2-= hPr ~ h \~. = 2\•.1s ~-:.l. o(. -:: -- o.z.io Sc<. . '-,, '&' I S° I) -t..'" 3 .2...4. \3 ~ l '"\ • 't ~) • ~ ~ . 1., , I?.. • ) .i 0 I ~ = A~9 _1;73 f \ ct~,. \\- ~n..."" \) >-4 ' ' - -; 7-... "n • 0 •0 <\ 4 ~ 3.~4 = APPENDIX 8 PERIODS OF ACTUAL BUILDINGS In Japan, the fundamental per iods of actual buildings have been determined experimentally by attaching a cable to the top of the building, pulling it over a little with a powerful wench, and suddenly releasing the cal>le. Professor Martel is thinking of putting large, variable speed motors, with eccentric w•ights on the shafts, in actual buildings. APPENDIX 9 CASH REGISTER TAPE The r edording tape used on the machine is National Cash Register, Size C i:a,per. It comes in rolls, is 1.211 wide, costs four cents per roll, is sold in wrapped cylinders of ten rolls for forty cents, and may be bought at any National Cash Register Store.