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TRANSCRIPT
Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1954
A study of the effect of vertical loads on the dynamic
response of structures.
Stephens, Henry Edward
University of Illinois
http://hdl.handle.net/10945/14652
'^
mm
ii
I
Library
U. S. Naval Postgraduate
Monterey, California
i
A STUDY OF THE EFFECT OF VERTICAL LOADS ONTHE DYNAMIC RESPONSE OF STRUCTURES
BY
HENRY EDWARD STEPHENS
B.S., U. S. Naval Academy, 1944
M.S., Rensselaer Polytechnic Institute, 1948
THESIS
SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY IN ENGINEERING
IN THE GRADUATE COLLEGE OF THE
UNIVERSITY OF ILLINOIS, 1954
URBANA, ILLINOIS
THesis;
1.
T.ABL3 OF 0CWT.HW3
usT or HayRSS !I. irrmomxiTtm
1.1 SURKSsry ••......•... • 1
1.2 !iMMttoletur« •• ••• • 9
AfTISR YIHD 1.^ u
t«l Oltjsot of c.«otloi) • t
S.2 Miit«Brlsl aad leoiMmt BttUitlon 6
3»3 I?00lat«Me of ri{Lned»i;&8e Fmae •••••••••••.••• 6
t«4 SflslatsaieA of Flxed«B&Be FraaiA $
S»5 tiae of thtt llesS6i«daee CiarrM •«• • 10
2.6 CosperiaoQ of ?Xf«et of Vertical Load nHth fipps^«i«5t-te
Vothod • 18
iTi. «rr?CT or tehttcj,?!. ldab o^ i^-M^Sia FK^rutmcr of nBn:Ti-:^
3.1 ?r«qQaRe]r and PerioS • 13
IV. f!OftT^^^'^ -L ?.?stzf,^<m or yrnifcrm^ .mm oi of Tf:» :::xG?noK
4«I Form &t H»0istanee ••••••••••••.••••••••• 14
4.S Slop« 9f RMtiflttmoft :7urTe« • IS
V. Tn" ' ?L *"1^. ''T -1'; ot OF T^P' STR'XJ?UT?K
5.1 Fora of Resist •• 1®
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7.4 Cnrr»otlon to tb« ipproxlMste ><!quiTal«n.t ^JKllbrn
T:Qsl9t£^io« iH»thiOd •••••• 89
Till* jwr'w nv i9^rs0f OF 7*tru" a^-'PF mm oc op sTm.TSTms
8*1 seop» of Seetioe •• «••••••• 8S
8*2 Amiljrtioal Coaputatl on fbr Sttp Pul0« tt
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X«S DlsetissioA •••••••• •••••• '^S
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INITIAL nr3H.ACi:"flK!T
n«X S«op« oed COHpotetioas •••••• If
n«8 DiseassKnii • 88
APFRHDTX in. nofm^- 40
VITA • 73
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ifiawiyic, «?«i«Rreh ^^Tof 989or of struotural ^x^liMwrlag, ualv«rslty of llIinoi8«
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1 ^t7»s»»SStralfi Com for TiMl riestio lf«tnri^l • • . 40
S H Ts. 8 CuTTs ft>r Ito«l ?Ia«tlo t^6t«riel wd B trp» motion • • 40
3 H/H^ TS. r/y« 1^ P a»i o6 lq[ia»I toO 40
4 Pio!»i»B«M YrmM ••••*••., •• 41
5 Fix»d->t'S»9 FxaiMt • 41
6 lestio ]ar««l«trao« Paths aa^ 7l4d4 Loot for ot of .truetuiNi
«^MiBt«B»« 8vm1 too 4S
7 Pcflietfiinec Fetlui tKtr r • 2 '^ • •••••• 4S
8 "^eslstawe Petks for T » l/gnr • 44
9 T«rl8tloQ of 8 «itki dWBg** i& ? la ?lKfltlo 'n&xm* •• 46
10 : t«^ ntnotlon TerlfitloQ of 7«rt!o«l to«d .•• 45
11 TransitloQ Botwtoa !^<»slstaaoo ?«th« P» ^^tuip Fmxotlon 4i
It cocopaHvcm erf intflrRctioa ^^^it^thod ttitlt ippr<nt!.7»t« .^thodt*» V^» P„ 4f
13 Coisp&riBOii of Iat«raotlnQ 'ethod with ''PFrffrlagst© Mftthod
for P« • 1/S Por • ^14 Tdealiznd mAc^ •• 9/e^ DiansrtA fbr ?Qrk B«s4flMiAg li«t«rial • • 49
18 5tr«8«-3tralB Curvs ft>r i^ork KardMlng !tet«rl«l • 49
U VA^, vs. 0/0« !»«lfitl(»t for 0^ • 0.2} StKmii^ BffMt ofAxlel Ix>»d on "ectloQ •••• • 50
17 n'/K^ vs. y/y© ft>r I4«all2«d H emotion oad o^ of H/fe, «». 9/9^SqiMl to O.K. T/ry » 1.0 imS S.O SI
18 oC "^esigXMtltKi •••• 5i
19 Toim of 3tru8turo ^—i&tvm^ ibM a£ ^ C SS
iO ";qulT^<«nt '?alform <>«siBt«noo Ibr Ta^yf ...• 98
21 l^alvalant tAaifoxn Hoslateeoo fbr y{g<7f •• S8
88 ?rror in ^j^/B^Tvom ^pproxiswt • ^:quiv«l«ot Uaif&rm??«ii8t«aoo v«^od tbr y^^yf ••••••• 98
88 Lo«d ^"iftlao SIteyo* ••.....••• 54
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t4 Velocity, (y/ye} > R«<ialr«a «t t^ end y|^ ft»r yo • /f • • • • 54
tt Straoture Peapoiuw CtrTea Ibr ?r«eK9 >i f(>r 3t«p ulse •••••• 55
S# !?truetar« Peapoasci CtorTos fOr Fraais > for Sti^ Pul«« •••••• 56
tf otruoteir« Response Ovrtmn for Frwae A far iKltlsl Peek Piilee • • 87
M Strtietare {?esfK>oee Ctiorves for Trmm A tbr TartnlBal Feek iniXce • 58
S9 '^truoture Re«]Muee Curves for Prcnw B for Stiq? Poise 59
SO structure ^eaponse Curtee for Tvene B A>r Inltiel Pwak f^ee • • 00
31 oo«!poelte v^truoture f^eepoaee Curves fbr Fnme A for Step,Initifil oad Termio&l ?«ek Pulees • SI
3S Coeipoelte HtruelKire Response Ctarvee for PrsHHi B t»r Bt«p,Tnltial cHEid Ter^lsml Vmk Fuleee •••• ••• Si
3S Proportl onallty Betweea Pulse Types fbr n'j^ • aad Tr * yf • • • 43
34 f«:{uiT6l«»e Beteeea Tnltial Peetk aoA Stei^ I^ee fbry^i
• yf • • $3
35 ^l/^e ^^ Inltiel mA Teraiaal I'eek Paleee liqpreefMid es5S of Ai/}i^ tor Step Palee ••• ..••• 64
36 Approximite lC ConTerelon Feetor to c^oaetruot ^lA^^ "vs. tj^/T^
OoTTefl for laltl nl sad TerfnlRftl reak Polaes fbr yg, • y^ 0s1b8
^l/^e ^* ^l/'^o ^^ ''^^ rulott .••••• 69
37 ^>l«m lio4<iI for fippoedUee landll 66
n Uoeut ^'' ^ ^^ pA * 1«02, 1.00 end 0.98 • 67
91 Tlieee Plene t>iaisr«ne fbr Tnitl«I Cre<ritoe6iie«i« p/q • 1.00ana 1«C£ ••• ••• 69
40 Ti-Tie to r^eflwet, u^^j^ »od ttn0^% 7»» Initlel Crooked&ese . • • • 6t
41 rhaso Isiae DlagrasMt fbr mniftlly ''tralgjtt Elwstle Colis»ii
OlTon 'Metarbenee tt^ 70
*^ ^Wx ^« "l ^^ TnitleUy ti-**.!/** ^iBetlo Column f-lTen
TnltUl rj^-slricenant tt^ .••••••••••••••••••• 71
43 ^ to i:)lsplftoe «• Tnltlel Olsplaoe.'neat for Initiallytrai#4tt Oolttgn 7t
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In tfc« »«*!» t*xt » study t» laado of th« effect of v«rt'eel ttsKi
9B4 4«*!4 lo«4« on %h9 horiacooiftl rftar-vr^p;** of 8,lr»i;le 4«ar^r«« of fr««d(m fr«»a€»B. Ta
f:'n*;idSx«3r I fesd :i fe p4aa«d«-«aS ODlwan Is- aluaulsted by n siagis t3««!T«« of fr<M»-*
4q« aekUl wad •• atu4y 18 -"»d9 of th« tff^ot of lnlti'>T ero^knismmm «o5 laitlsl
Tb •olflsg for tho rsspoaso of th« si:. :-' -i^^r*® of fro«»da.^ fr^irje, using th»
aeoelorfttion ntt!B«rtcs<l ^gothod, tfei» horisontfel ©e<j«l©r«tlor4 of t:'© floor uRsas i»
tekttt *'» th» '11ff«r«»ae« b«tiwi«n th*» horl^r.t?! lr>?».1 -nr! t^e h«>ri«oatGl r(s8l»terjO«
dlYlded by th^ -nwo^ of thf? pyst^a, Tor si-^.nllfie-ieia:; the syst^c Ss t»3r.fta with
th# affsB coac»ntr'^.t«'3 «t th«s flcor IbtbI ?yjt3 Xlm boTtlifontf^ I m-xt^rnnl for«e telcen
to j«€t jt.t t^,# floor lavel, ?o d«t«rr<5!n«. the r^spcsrsK^ ft la tba« n*ie«9«*-'ry to
-'.n-i- tL« -.oriy-otatvl r«dtst»nC9 wea verttORl lo«d8f ?--r9 irxolucle^.
Mditloo of the v«rti3sl Io«4« will «b«Bg« the structure y^orlrotstel realet-
ftnd« fc«o«>u8« of b-.Jth **r* 1nflr'=*ft8e In tho oolomi JK>sM«t» dan to tb^? r/co^ntrlcit:,' of
the TertlCPl I..!:i'3 c-o"! by f-s r-xl;>l l»«',d or^ t>'* calu-.n '^f^'Ctlor. onuair^i inelssiJe
eetlon to be^ln with !©«« eeetioa -iKJS^nt, Ta «a ©liatiaf, 8p:?roa:l>-«et© oiffthod for
>!^llowlni? for the v«rt5 cul losA tb# eff-iet of the tidltionry.l m^^^nt i« allowed
for, b'it not tr.e effect, nf t!".» bxIsI lof-.d on t^.-- sootJoo .iaelrsetie r>cttor;. mr-
ther, the v?»rtieftl lof^da, wtether live or der^d loed, w1.ll e<jae« ^ ct'^nge in -cho
nfttursil froqaeocy Ifi t^a <&lGst-c ?ana;»« "^ms ^fi'^s 's iftprssderft only an thp* d'.»«;d
lo^'d; but the -^ TnctlYe fj^irTiE eaastssjtt 1.. ?j9;;>«nct«r.t on th<!8 totrd rfrt\crA 10ft<i,
t.-:cl»dlaif live load.
It-e foro of th«5 horiaoatel reeistnaoo l' ahovn for:
i, ^"h** o«*»e -i"; ere « o^ tr?» structure ro^ii-tt-'\.af '. 7'!, v,. • ,. ro.
t—rt'i'wa '•? >c
c m9lS0*^ •!'
-•©n*?
fcC3 >IKlCf*WllOO^B
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-•n ^ -1*
^1^ Hm fi*'
*"*»a03 J
^» ten c
-fr^i ,n
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t i»7ii^«a
> •trr
a.
£• The ».4»« i*ir« Of of tb« structure FMil»tft«««, n vn, j. le not x«ro.
.?, '^^» oes« »fe«r« a of tn* doloom B««tiOfi aa»«Bt-eurvfitar« r«t^.tJ in ia
Tot ci'«« (l^ eboT« It 10 poftCi^l« to d«t«»r«»ln« thi> eollftpew dtefl^etlon (th»
,#fl<*et!on »t iriileh Ray !ncr«es« Sa the <l©i&4 lo«d with only a*»«d lo«d fiotfii^^r «111
Oisua* tha «itf ;f;*ur-i! t? e^ntl auo to d«»fl®ct to fsiltar©) aod tti© »u7rl3£oet»l r©«l9t--
oac*. And for oes« (£ It is «pesy ts 4*t»TO5lR«B if tha 35truot*»« h«« a colls^*
d<ifl<»otlon «?« deflnc^rf eloir», fen'? If ss its 7ibXu«, Tt l« «1«q •fcay to iaterwlfse
tr» f'^rtr rvf frs re»»lat:;acs for thi» ci^so, -ut fbr (s«*fj ( ?S^- t^--^ rsl'tl^n l3 ?ho'^m
r.<5t to fe« «o #!s©ple. For 0imb« (5) «o?3!e c.rlt«ri* ''^uat b* tt&tt^-bMffh^^ fv?r a rf-l---
tlon betwi^ftn yertJc^l Xoed ivn^ cat to d©t»r;slae «'«» th » »truct«r*? ^11 h^^v* % «ol-
le!>«ie 5®fl«»etloa. '•, r«l»'tlOB is foujs4 ia that if r^ oc , fv^r ovs«i {:>':, no f^ll-
ur<? 4«»fl*^otlon «xi»t»« If no co.lla'Dis** ^efl^ettos, f^, _ •<>.1:, »:'4i'^s so"0 other
fallar* cpit»ri& rn^st b^ »©t»blieh»d«
iTof. ":,'. -twip^rk, ^alT»r»ity of TIUdoIw, i ff.& .I'^v;: > -.-^ ^n sp'^'.roTiXsfit^
•qalVftX<9r;t aclfDrra r^sflistifscai method fi*T d«t«r3iinli^? th'?> locd-tl! • relstlOR ^i^Ai
will proitic© ir^ ^n«rlr:ur. d^fl-cti^a ecioal %-' ss^.-.-m pr«4^t«TOi.n«d in-law wh®n oc of tb«
stTTMiuTf* r':'sl?}t»j;noo is ttot ;eor-', fcut kjj eoDstnftt, if tb*? yield « rd .n^xlour '?«-
fl<»0ti3aa, ii*«a put ta tbe propfl^r tiprrcs, ?*r« aaed Ir t?;!:--' «>tcpr««s1<>ii it c»--f» b«
tMMi4 t.c-j a>taput« th* r«^er^CiS'? 1«t;lul.l t..<? th?' «ff»»et of the v#rt.ie?]t ln-?j. ?. oori-
//rli^o-.. of lh«» r>'-?ult» of tM ?^ .-.,;•,-..: t-- r*»lPtl5S has be«u ^ -.ith '^'.iare ex^et
y^-l'i'-? fo«>y:(! for tbft ra&xlfiia'^': defl<?etl.tn ^r^url t'^ %h,» «o11?:tj»» 4«fl«5CtlOD.
•itfit!9tio»l oorr^ct!-T: ffietoor for ijs*? si.t«-> the ip,ppr»tl!r-«t« ftqulvftliy-iat uniform
r«»9tst-3ne« is reco-' •: :''"" f^r tri^ r^n^?* of tj_/T'o b9twna^n '^•l *rtd l.O,
'i/hm tin© <3d|»«Q<S«»8t loRcl.? 5?.r« ?prlie<l to « structure th« sfc- p« of tb» pttlMi
will influanoa tb* rewpoas©. In th^ rsa«« of tj^/To !«"» th»& 0,1 tt« varttcsl
ti-^© lft::^nd«;^t lo*»,d 'f » little eff«ot od tlli.*? 'rorlS';^nt«l r»«poo«©. Por pul3©« of
«niwtl «i«oont of iipspuls© t'-:n i-iitlal p««k r>«lR*! will rrroduc© « 3:r#«^t©r scaunt of
.3
JO *,
^-W: a^Oi T
MriB,r*-.
5# cvt
asxi?
6 •ffi
s.
«j>profie^«8 thet of th<» lnitl«l pm-ik- ^;::i f^r I«»rgc«r vrIu©* of ^x/'^q the tsrwitJftl
'^aw^ r«5t>oa8« sx'ProftCh^B thfi^t of tKe ist«p pttle»» * {^nsrs^il tr«r-d, tsdftpw'^dSant of
*C' f* i>r:r>;irit of V!-rtto«l lo«d» Oi " b* found for t''w renroari--^ cjf U^« t»r^ln*,l rsu^
pula*! It lEf POBslblr to find ©n e:;ulTRl©nt R^o^' puls's for the Of»»« ory-"*yf«
If rn t'Xi'sl lo«3 0t»p ;5ulse, equel t^ or r;r!»?u.-nr v ; n t'^,** /jrltieftl buckliG-"^,
lopd, tis ^J)pll«1 to ft piar*?td-©ad oolu?na which Ij&s isltlel err>ok»>:l.ri«as tfc« sraouat
of t\;« Inltl^il «roo?r«dRos« tflll taflu®:;©© th© resixviig*. T^je fr«f'tGr tfe« initial
^rOQi6«NSa«»B t>,« lifr^j^tsrr t«1I1 Ij«> txi« ^e.xj.rus? ^9fl<»ctlos find th« l»»» tin* raiulrewi
to r»$efa «»3iisiafB 4«;fl«!ettoa. Tkia 1» iB.T'*stIgnt»d In wpyS'idlx T,
~n i»p.ip«n31x TI sa inT«»ti.!!^;tlorj l$s ^m49 of th« «ff»ct of the S6;j'.lt-ad« of <Mi
ialtip.i dlflpl^Cfti^'^ni .?iir5»j; t'> i.n l;;.lti©]Lly strsiflrt (r>rann «hea a s-t«p piil«<»,
«qa»I or rr«nt#r in r.rjgjiltu<Je thias th» ctrltiORl buoi:iin.=!? l^ed, Ik applied. rh»8«
plp,a« Siair«tB», ploti? of tln« to ree«b nfsxts'.a-'s d«fl©etlon va» Inltlnl ?3l8plnc«-
i^.ent ^r-'J (j^e^jsltu-fe of .r.»ixlr;»j?', don<i-ctiafl vu, l.;ltl»l dl»?if-C«.'a.«er<t: ure lis^d to
s^T-'sr xh» ^ffipot of tij« Saitifil dlST'lfcCfmant .on tfcc r«8|x)as» of th# aoltiKti,
T: "^orirontftl forces nnr colaaan,
H, 5tBtlc feorl7/>ntsl fore©, per coIu-Ti, thct provtu«»« yl^l^in^ with no vsrMcfti
'•« ''t^tlo t-.orl2oritftl fnro«, t>«r eoXur.R, et yield of t>n e^Gii^ftldnt unif^rja
r*slfitfvne«,
Itg "Iprlnf^ forc«, r-er e:>lufrn» piftsr yS^ld, no vi!»rtle&l lo»d»
^Ir '-P'^^^'P f<ircf» u?) U» >'l«i>lii, ©rticf^j. lo«d eetlnt'?.
kg^ BprSftg force Rft»r yi*?l.i, vertical loe5 oetlng.
L L«a«th of eolai^.
•ji.raR* :i .fj" J[*fllyJW** *
tc
7 »/tf ^l9l9iot
'' v.r
flf »«^5 -
viii •»tiafiTAotri» 1'.
1©
•ft «r-
t^g , -T-'.^ -
••r;'/i
/.'» Jd'Uu'rtB -^
•»
! h r«
oji'Ion 1«1
S.l
T*ati«
%9t
!»n £»#•
XL
1L ^*€m»&t n% top 3f pin&e>1oen^ oola^n, ttmt •% top mud botta-:) nf flx(i<|-¥«ui«
U^ Sttotioa noaoot et yl«ldln«, ao ckxl&l loa4.
P^ Slftatic llsslt exlial load, per oolu^ua,
P 7*rtJo?(l OP ftxlel lo*d noting;, p©r eolueam*
t]^ Dur«tlon of lo«d[ pttXs«.
T ifBtiiral p»id9d wltb Y«rt!oel load.
Tq Matoral p«plod id tbout «rtieal Xoed.
u y/L/B, 99fpmaAi%— I luitf XI«
a du/dtf app«ndlx»8 I aad II.
• •u d ta/<S"t • ftp^es^lxaa T «»d n*
a^ y^j/e, Pv>Jon3ix®s T and TI (laitlel eroolaidi»«i "I
.
y Lataral d1«plsc«r>i«nt.
y^ Diffar€Hiti«l dlsplaaaegwDt bstwaac top moA tettMb of frst^e oolioltlol o^t9r daflaetion far pi.neied-^atf laoMl <>f »psM>a41i:«8 T euad 71.
y^ Yld:' d^flactlon, no vartical load.
^mr ^^^^ (teflootloa, with TartlOftl load.
yjg -lexl'BU'^i d«fl»etlon.
y^ '^.ollepaa dafleotlon; thet defl0Ctlon nt wbidi tha Tertlenl Idttt «S31 «•«•thm struetura to ovntlcu* to daflaet without horlxoAtal load*
y^ Tl*ld daflaetion of approximate aqulTftlant ual-Pora raalsteaoa.
y|^ Valoolty of top of oolttaa at ti.-n© of yialdiag.
f\t\ Velocity of tor) of eoluma nt «ad of loed pulaa.
Tfee ratio tjetween tha slope after yialdiisg to the elope before yielding of
od atmotural real^tnive, H va. y, «jnd M ve. 6 eurvea «bea tba avrvaa iiave
•oaataat alope before f^tnd ofter yielding.
rf V "'/^(jr : f i& the fr-etion of P^p thtst th«» Tortiosl load is eqiuel to.
ff-^ The frwotloa of V^ t>.et the deed load 1» equal to.
^X T^** fraotloa of P^^ thet the tlise dependeat vertleitl load 1» equ^l to.
.»
.9t
or, Th« fr««tlofi of P-« thct Hm •fwapi of th» tli»« d«p«JB4«nt r^rtlcs?! lo^d ia*^ •<]u«l to. ^
6 ftotlon rot 't 1013.
©^ :'«otlOB rotn.tlon nt yield, no «!xi»l lad.
c^^ Tleld strMia, uQl-»xlr<I stst* of Htross.
A^ lilt* at«reisit borii»3nti',l tl^« d«p«!ad«£it lo^^d.
G)^ Wattupftl fr«QttBney, ao Y^rtic&l lo«4«
£.?« 1l»%Wfcl frequency, ^th v«rtlcftl loed.
fh* 4«flili;D«tlOB of T Is tb^m ••• «a <r Axoept thet it refsrs to F^ InatMdof
cr'
>5b
.A
CD
^-
IT. bop:•'""
" - ^^- -- "-rmz mocw.
: . . T3
S.I pbj«et of etttlon
In tl^Ss a»otlo» the ralntloa of horlajsatsl lof^d, vwtieal lo^d sad llsplso©-
BBMit Is 8tudif>4 ts i^ov th« interfotiori b«tw9es thmsm for tb« oaso v^ere c^ot the
•tru«tuartd r*»«lstfsaeo Is x«ro. Thl?5 la dati« for bath fixed «;nd plaaa»a-V"^'S«
•Inpile «tr>ry frenea. tite girHvr iascfint of la«rtl^5 ia ts^.lcea to be very lerge iB
e<*ai«»rieon to ^e moavnt of laexrtlf^ of the eolams* The %ximX lo»d ulll 2aex«f«ee
the eii4 eooMBBte by na ^aount »<^ii«»l to PyQ, Tbla inorafiii© ia moment aa»t b« al-
lowed for, sfi fl?eH »s th«i r^uot5oa In m»fnltu<le of 5^. at yi«ldSog due to tke
«ixial lot«d. Tbrougbout this report T&luee ivlll be voxiced out In t«Piui of io«>
dlTi^luel oolutin lor'ia «s«d properties.
"^he oolu-na T^teri*! is tekea to be nn 14e»l plsotlo ob« {etxMies-strain
curT« 3ho»a flirure 1». 3 th*i eolwaa aeotlon tn h type ®®etIon is ^tseuursedi sa4
th* ben*la<? rxIs is tcikrea perp«ndi©ul«ir to th» wofc. For this type awetioa «si4l
ideel plaetle laatitrls^l tbe %', »• § relstSon for tite eeetion will be of the type
ehowft ia figure B na T^/m^ vs. 6/6^^. This curve eesi be l<ieelize«« to tn^o streight
linee s^e nlwo »homi figure 8. *flth the tbIu© of H^ ohosen, tho cnirve opa be con-
verted to di3ie«i8loaIe«6 fom. Tf «my sllcwfcos i« to be mede for the effoet of
iaereeee in yl<?ld point b<i*e8tt«e of epeed of lo^^ln^r* or etreia r'^t«. It would be
refl**ete<l In value of ^^, tbue produolr.*? n oSi? nfe ia the eotaal .'. vs. e relet ion.
ikny elXoweiMie fr>r Boa-perfsct section wojld be ««de in elioiee of V-.^, pr-^oticnl
ehoiee of ^^ in to t«)kd U ^e tLe igiCMBezit nfe^ea laelaetio eotioa just pi^netrf^tes
through the flaagee. 1^« etnictore horl7.ont«l reelstnsee, no vertlerl load, for
thle ideelized "^'^A^^ vs. ©/©« curve ia efeomi la figure 3.
2.3 f^e^lgt^Dca of Vlane^»B»>e yr«y.
For the plnned-beee frs't©, ehown fifjuwi 4» end for the elestio eondltioa the
sBOOttit, I', »t eny aaotioa will be:
.»
bom bmol ^•itntn til
flf €»1W"( TS**^ <»^ r.t rf*
tai
9At 9f erl^
-al •»
't « ,"^ *#»
t-.i-I*.'^ ri;
f f$m «« , T<rt Amp
cifi'r Js-3«ai33; arte iJja'siQ is^^i as ec o? avil^t ei l^ticts--^ mvttlot •/?"
ft«« b«U)»' o» 9iii e .1 eii.r r^ MiciCi •tut
Bfia jicl^«R 9ev -roT .dew 9i£t oJ
»v »«>&•< ^ ..
»o 9101^1 mt t r» TOO,
*'jif9n at ti
'•p ft* BO- r'Ani I
oMMwoXis tai
J,lM>S ^'^
».
M = -M^+ Py . W. ^ _EJ d'y/^^, ,2.3.1,
TlJis Ictfi&a tc rn expr»»«lon for the d«l'l«»ctlon r-t •*ny point «« follt>if«:
7=^/ JJn" r^/N jp" ^ - -/an aTp L c o5 >rp^ X ^- -^AN\r^(l _ x\ (2.3.2)
L«t ^ p«r colu-an b* d«flaed as F- v"'or»'r. ,^, where r^^~- rr'E 1/ ^ (p«,r eoluaa)
for th« plnned-bfla« frnso. Then the deflection y^ et x»L 1«:
It Of a be 3«»en fpoa (J*. 3. 3) that ftsr a oonstaat ? the deflection Is i». linear
function of K.
nth thf) Idealised •BO/oeat-euryRttrre curve, figure 2, the relfitlOB
vAio PA'a " i defines the yl©ld eoadltloa» aod also ma«t be estisfied after
yielding hs^s occurred mi tlse deflection increased beyosd tfce yield deflection,
n elRSto-pleatic jRiiplyaia, ^^th e hlng?^ forrziof, at the top of the column, is
used, decreK-ae In deflection pft^r yielding h^a noeurred vrtll be elj^atlc.
Tbea at yield:
€
fbMi: . \(l-T)
Ke^<)v^TT -^ ^ '^ {S.3.6)
»br the elnetic rejoge H - k^, «her« k^ i« ttL« r«diM«4 «prlng oooetent because
•f the exid losd. « previously noted, fofr a eoaiitant ?, the Fr vs. y rel'stion
la lineer. Hence Itj^ is co.istRat for thw constant r.
l^^ 2 > (^/r T^ ) (S.3.S)
By • ({repliledl oooipRrleoa this cf?n be aho'.vn to reduco to:
,v
(I.e. a)
imr
s\^
!'-> .jt,ai r ,-^
u . ,„q -v ^M - = M
1 vlcTT
v.- .^.^-.3y-]f)j.\vH -
n«K'
. ? --^ A ! 1 - , f . ,*i ?E.i*M» ii ttft tni* (C. "««^» "^ ""^ *I
'*-^-^'* JH- 5l^(T-\) ^
Ssi:
? ,
« *A*? * , ^.
nil b»*i-i;occ .>f.f-.j.-.
cJewl* c
f.'c.^rr^ii .si %a4ft<r9«ft A .5«»j»
:i>i B»rfr
^ %.\^' ^
(>) T, -TV v.aV %_ ^"^VI
'V ^A/S-\-
?.'
K„-i^(l-08.
<y TT ZFqantlona f'^-.^.S) and (S.O,S) for K^^y &na y^j. i^lll ttien iaflne the yl«lfi
eoa?1itloa» yfhim tiximl load is laalud«d. Tor tfcla iRforr^stlon to b« r«fldlly ask-
able It 'suat be pos«!bl« to aaeic© n plot fioss vrbiah t!:i« yiisia flwfl^otlon nrA r©-
Jatwacft for eny soloaa ead axlel load Ofen b«» ti^«>n. The r«letio?5 of tbe^e de-
o«ad on tb© r»lp.tiv« r&lu«3 of ?^ »ad F^y. Thus « siristlon in tfe« rstlo of
'^ • ^er/^e *^-'-^ osus© « eb«a^« 1b th» yl^ld d«fleetion aad ylold reaiBtstna*
maipriitudas. To r0?^rp.3«nt tile T^rl^tioa th« yiald loci for ^llffapent v-r.lMan of
th« TV, r«tio fere plotted In dX<a«n»loal©«8 forss ia fi^ur« 6. The r^eisteyoa psth
froRi th« origin will b« tdoa?, «- «tr«i|Eht 1I^^®, th® alop9 of 'vbidl^ is dependant
only on tl*« xfiH5iiitu4« ->f f . "'hn lct«r»»ot !•:>£» of thl« r©si«t»inco pftts ???ltb. tJia
dlffdroGt yi«ld loci ^111 dafluo tt.« ylald point ft>v a»eb ^ retio. ftsr
ylaldlag tb« alopa will b«mM sa^tltra. For tfca plaaed-b©.©* coluam y^'^^/Ztl/l^
«md H^i • ^«/ti« lo RdditloR to yli?ri(j loci ftir aararal rctlos of r/, raslst^noa
IDatba for saTsrsl v?.d«a« of y «r« alao a}*^o«m in fif?:ur» 6*
Tha «xrjroaslon for y^j, fe©aor»8 iad«t«risia«»t« «8 r-*i • '^o d«ttiifltB.e y^j^
for 'Y • 1 r:nr^. r/y^l us»^ th«> fts I lowing expreasions
^^'^ -rl^P I /3El/[5 }
•^r r/ff^i r-j y^^ Is si so y^-
TO daf ','•»« *5".?i elj«»tic r>orti>n of tb« reet'?t?nae i?'.irT© nil Xhtst It 1« tt«MMH>
•ary to 4? la to ohooaa tte r/y 1 3u» twrrssi^nai ia^ t. th« ooI'jke prop«»rti«»8
,
and than follow tha f r«a5st:-in« pmth, eorr«»rondln*> to the loed, out to tba
ylald looQft. Tte portl-^r !^f tba y p«tth batvaan the origia «ad ylald looua will
b« i? *s sleatic r«Bl3t^«C6t j>sth«
?or Taluaa of n^#y i 1 tha yl-^ld looi «t tha oriole v^lll be p-^rEllal to tb«
*f roalatsaoa peth eorrs^pondin?^ to t = 1 . Tfean T^l^ (y ^l)/''» <^be llsiltiag
(^j\ 13 5>\ j (-^ ^^'^•^^-^f^'^ ^
' -^ ' ''^
*• •«{ ^«ita ^: "W.' i A.
rf^ - rt f '
It bJ ' ^^
l^^v-l- Y^" '
.'.
.
%dM taon
.. T niif S» \l»
i4.-:^t tacrwt^."
^.1^ «ii*telX
lani &liitv 0^ ^ • .H *«a
T •'^-'••v ?> "ijif
ic
•oa
if. . 21 inoT
Xitv at:
9.
*?rtic»5l lW5d for oo posltlTe horlsBoatftl r#fi<8t^ttc«. ~ts«l«[i%i# action will tbim
bitgin et y - end y^ • 0. For this o«s»b for t -l t y^p Vf "^H^ iilop« of th*
n»retlv« r«ttl£teae« p^.tb for thl» o«i«« «ift»r yl aiding is tho «»s« ea kg^ fouatf
for th* ««rs« Y for nnr otb«r vt?la9 of t/^ . This «••« is lliaatr«t©<! in figur«
7.
?fb«n ir/y^l th« ylftid loci <3o not start from tho 9rlisi« •• 4idiRi is fl^^ur*
• and •xpr«fi»«4 by equetion (£.3.9). The llwltln^! lo&d for im> positiT* horJxoo-
tBl re8i8t«!ia0» !• y =1 • For r~l t^« hori»»ntRl r«»«istrQee is oatil
U i IZ ( I - T) la ra«^eh«<32 »*j«r« the hor1»!>ntsl r«siBt'^ii«« tbea bis-
c3Rj#« ae*ffttive. T>vl« oabi^ 1« illuBtrwtadi in flj?are 6.
k% tJi» oollaps* deflect108 1^/Sf^ P/p^ • 1.
*Rd: D o ^ .-^f.^x/Ki/ KknMf not to exc««4 1)
/He /Pe -'^ TTT-^ V /itI/L / (£.3.10)
Thm re«l»t??noe p«th batWMn tk« jrldLd polut ss4 tte collapse d«fl«etlon will
«l«e b« «tr*l#'.t lia«; th© slop* b'»iJ:;C d»t«wsined by T .
Tfc© spring oonatR.nt, kj^^, up to th» yi«l»3 point i» i^ - JEI/'^ii-y) . 'rhe
wpTinF. oon«t*ot ki.^y fe«tK»fl(«n tha yi*»id polat ««a?! th© oollftj/ff* deflection la
— KiR
Tfc« v'j1u« of y^^, '?gjy and y^p tite^o p^ut Into dln^msionloa* for* »Hit
>r?'7r 2
/He 2 -fAM CTtt-CrTr z
""OP e «^ f1x«»(i-b»»« frr"n», i&oirB fiierur© 5, thm «OAiiot, is, «t Mgr •••tioja is:
M = -Mp 1- Hx + P;^ = - EI d'v/jx^
•Sie »olutiori •>/• W;l» equ'iti-c • x^ vr.® »ub»tituti..Mi u- *'''^-*« ^2^<i
•f
«|#Jt fit* nottn» 9tt99l»&T .99
u> *• ^ • n f« ( S
i:;V -- 1= V
(t - O &t -,
Iwc
O
<«»B» Of fm T bit»T^
(iMaiV^)^^y^-i- ^%^\^^"t»/
«r . ^-.),^^3t.,,;. r « > /-.• ^1 r«
:•* •ec' :1
r»T cir/^ «l<*il1«jttf)«-'
«« •rJT
-<J^ — .^ .,5£T
-•f-'V ffifT
(1I.C.B)s>i (-p '0 ^ w
'TT T ,^^tr x>
J«l «•!!••« t
^xb\v'k> 13 - "= vS ^ > M -. ^V,- . is^
10.
P^^ = n^£I/ ^ wiri lerf d to tfc« followlas «xTr««itM !*» mm v«l»« of tfc» i»-
fl««tion y^ «t X • L:
vl^ld:^ in tii9 •ftiMt «s fter th» plna*»d-b»a« frg-j3«, -owtv^^r, it !» not^d th^t •for
th« •«««• ooltusa c«etloxi «n-1 l«ni^t>- the» t»Iw« of t"^^ for th» flx»4-b«s« fr««« la
four t^s«9 th^t for t>j« r'!cn®<^-bs'se. .a el«v3to-pl«»tle enaly«l(B 1» »i«d wltb
^l<»9tlo M nr*"^« fja^aT^flfJ to form s^l'^ultwiitetMily mt th« top ^nA fcottci-; of [email protected]«
^'ollovini? tr«! is^jr.*! r«s30tiin«! ?>«« it»r that ?iaij«d-fe»8« frsjsw the following <«qufftiaa8
for th« fls«>;1-t'08i» fre^a »r» RirriTskl et:
^-n /2/if-, 'i M..= Jli«(^Vl) y-- /!:;k u.^M u/L')
>-qartl'>n« C'r',4.3) sr«» of tb» aa.'-..^ f^f^r^r «..e tl;<* C!ir'rT«f»prn.^tng oa»» for tfe0
r>i«n«d-b»»«i» frv-:7.«», T!!«r»forf> ti« r«sJ uV'- jus** :tu7-v A-h tt« fiJijMll «adl
pirm«d-«n<S«'? frf'!"i««i, «toc In tt-fr di.«r>Bi:>r.l*»''v« f'>rn, srsi l^;© «n7!se ?»fi^ fi^ur** 6
©ey »l»o b» ti«ei for tt© fix9d<-hp»« fpR«ti«
8«5 l!s« of '"^•'i-t^.-!0!? -^nrvf^s ^or Detersslniiy? '^.iwlataae© t&r Sa« in :'tmpt«^nMW«MMr'>«v . . - .. . ^ If II 1 rrwr- in-M n-f 11 ii r ,-i r
-,••-•-,
— , 1 ..p..,
..,
. > . . .--..i.,.. . ...^^_e_
Typicftl f*-r:Ut*ts ->r .-a fo^'<J>4
• * •*"* lA~ ^t^* lll«»tr»t«<l in
jai
\0'n-.
[l_TL|V^Af^-%]^H--.\:
'94 |MI>
•4} •!
••i « A <j
/ \,M\ !il:dL . u f »\^>^^\ ClrlL- = HW\I3^' )-n3kMA+ ^(-^-n ^^^ U^ ) -^^v^A^.^ "'^
Li2Lz_ =.,>(^%,«N^V)
^S.'
(T -n ^^ V, __Jij-Al r
11.
th« loed P, ioeluiSlnj; dottd lo^^l^ r«ataift« •oosttnt th« r«0ist«ae« will
fallow ?«loai^ thn np^ro^i'lRtft y r®3iKtf.fi«« p^itb. *'ow^i»T©r, If " l» « v^^rylag qoflR-
;iy th^s r«;'-i3t.'-nea -will usdargo a tr^naltlan btttwoea th© «Jff«r*nt pt^tha, but
not ttlwAjrt In A dl!r«ot ^iRa«rr. Th« possible i^Btfeoda or ^enfje f^re covered in th»
follot^lnn es«9S.
Zn.e9 T » truoturfs a'v'R yi«ld«d*
? een T?.Ty in «*/ Bwaifi«r, lncr»a9« or 4«or«sa«, sii(i ^t a gi7«n y th* re-
slsteaow for «s« in the nogjerie^l proo»'1ur« "sdll 1»» tfe^st oorrfiB^OBdlag to th«
y«lue of -y «t the riven y. This ia illuatr«kt«4^ ie fii?r«r« 9,
C«»« TI - ;tr»icturo yiigldai ; ;i'i tfeers is ft a«cy»e»i( in i- oscurrlag «• • sttffixaetion.
L«t F v»ry •« showB la fl«?ttr« 10, ^n t)j« dys*»lo •ttsijn^S th* cbPUf:© ia P
tNfc«! p3.«e« at di»plec«?»ent yj^ ^t »•- rlvwi ladtoni of ti-^j*. I a th« aw6s«rlosl
ftolation rtf th!9 dynamic probl«a i*©n " ©hf. ii|?®is , ch«a«{«6 in 0!»rv«ture snd 1efl«o-
tif>a do ijot t«k» t>l60« until Sf>r?i« l^ter tl 3«,« Tharefore becyu-s© »f tb)& tb;e 8l«-
maat there v^lll b« no Itifitstnt^nftous ebtmji>« i& ioomemt end ^« ncM8«mt f.% tb« ooltma
ftwS will b« aasintjined *t i » (1 - Tj^I t-^ until afttet %•,, 1^ m»iat«lK th« saoaaa*
«t this Ttslu»» B Mill hftTO to Instii^atsmaoueily incroase to ao^.<» i^fd^m Hg fit tj^*
Thus: ,, / r/>i N ? n ''' 1 ^'^^ lo^^ hefornt t*
VLy= r*'"'^'' ' '^z'^^i If 2 fexr imi «*ft»r t.^ (g.5.1)
/"e i '2 ye -^yj^
• y et tj
nils v»ltM of Hg will fe« 1*58 then th»» H of tho ro petfe for the dl«fl**etion
y. at tli« l«Ns4 step. ?ro'n tfcls vplua of Hp nt y^^ th* atyuetur*? ^^411 ««t elsstio-
fiXly until tha yg j«itb is lnt?>r«»cta<l, i.'i.i -t tJ.a al^stio slopa of r^, »nfi will
•Cain Ito plfistic »han tha O'g oijtvb is lnt»r*??jated. t. 9;p»?pfeio*^l '"sathod for ob-
taioini; th« vf»lu© of }%p is shown In fi/'^o?® 11. P.owav«r, in the cijt??«rio^l aolu-
tJon, ir th*» d«flaetl OR at the aadl of tfaa firat 4i--.« Interval ©ftar tfaa load
chr^ng* is much larfaar ttitu taa dafl«ction nt tha lo?»t2 dioag®, tha s^ell ecsoimt of
alestla ttdtian !a*y b« aa^lactad »a it would th«i5 latrodnee llttla orr<5r. If the
•Iftstle feti<»a lo to ba naf;laetad, at tj^ 9X> V'5ir*,ieelly fro» t^^a Ti^ to tha Yg
M
mAi at tovtfroo
9<}S o^
ffms^
91"$ fm90»r
J »..,• -.'»-.
•SLMtfi ^ 9b<^tf»
I'^mns*
. s^'fi o *; nl?»c ri«?
• fe^bi -<>«'«; ~ T s:^;.:'
•xirr B'^s
<-> .O r? '.f < » f «
n ml '^
'H OB •
-MilOb baa «i. at Mftttiil* ••••«'«<• - fi»J^ »t»Nntt; ««*» to aottuif'it
5 ftB He??
£ 9<S
^ t- -'^ o.
,t 'mrtm man ,« (x*^-
.i.iAifc ©•e-lct?et f o* ftv
r p ^^
aotm>^ti^^
[-^^-^^'^]%uV.\;^
I r-
>-K»l ^^
liJ
^r mtU •$ ^r mdi i»n •<; on ati'
12.
ovarrm,
Ctm9 ITI - "tructure jrlelded end F Inoreaslog.
In this oes« '^^^ p/?^ • X will b« »Blat'5ln»/l during th« o]iftag«. ab ? la-
ovRMMf P/P^ incr«<n»«« end therefore 'A'l, nu^ deorsRao. To BceorapllsJr tb« d»-
er«AS«, T? etust d«cr«»ft»« to tfce imlu« earrespond In*? to tbw n«w riilu© of r et th»
pertioular d«fl»etion« TSi«r*for«, ea ^^' ln«p«aiaea, »^»«th«r es at«p fuaotioo or by
©onatRXitly ''iryicff, go (towsisaftril ^ireotly to th« approprlt^te o' copt* <Kt tc« par-
ticular deflection (flgttr?i 11). >^«c tfe» Y rtfsistciaoe petb p«!«as»» it» feiliir«
4«fl<9etl3o th© horl?>or«tol rwalatsriee will no loni?«r b« posit!v«, \>nX a^gatlra.
C«<i« TT • 'traotiara yl«ld«! end ? d»ora»slng by contlaoous variation,
hm wtBS »ho«rn in fi^i^re 11 for ifee st«p function, it tek9« ery little ed-
dltlcnal <i«flnct1on ?sfter s l^rge d»iST«tn.»m in P to fully <i«T«lo]5 Xtn additional
#l»»t5e reslatnac* end bav» th« 8tr^iOtur« e^in go plastic. ^a««<jaMktly for th«
BUf»«rio?"l ?rccedar« ^,«r» P 1« contiii«ioa»ly d»Bre)!!8ta|| ^ directly to th© r«»l«t-
sao* peth at tfe« pwirticttlisir d«flectlOB in » afiniser siallcr to fiifiHr* 9 to d«ter-
aii3i« the horl7.ont»l r«slst«iio«.
2,6 Camparlson of ~ff«ct of Vortical I.Ofsd with ;^^7roTci"4it« *'^,«tbod
In r©feranc« 1 «& wpproxl^tiato !tMifb<9d i«« f;iv«iQ for ellowin^ for th« vortical
lo«d an e fTMje. ?j3l» ««thO(5 oonslsts of rwduclns? the rjjorasat carrying oapfiOlty
of vhm unloRd^d frw8i« (no tr«rtlC8l losd) by the nMiiiit of tba :«5Q«Qt cr«fit«?d by
the ycrtic«!i lo«d «n3 deflection. Tr;# pr^ceodln^ r?»»tfcod, d«voiop»d for allowing
for th« T^rticnl load, differe fror. tb« R-Rproxl'nato <a«thod In thf^t It fjllowa also
for « d«er«oiic irt jl'^ld iKoemint da« to th«> &xUl lood on tl!« ool«sm section. The
0lo';e of rh« ««l»ctio portion of t*-.« wwitetr.noe -^fs foun<^ to Y-a l«>6e th&n thr-st of
tee appro! 1:'»7^te method for the ,i;:- f» ^-xi&l load.
k oosapnrlson of the two isettode l« *o«rn In fi??rar«« 12 end 13. The differ-
•neti between the two isetbode 1« eleo detsendwnt on t>"« ^ilaa of the V'-y ratio.
mat 9u X
©1.
•awjar
•jvm * T
%A» -.
1% 9ii oi \.
aft 9t
iMkCiMnte
i»J{
»•& o
9tost
•% .3K^ M ^
si»aiF r uiT'^r*:
«1J anV'
^4 Wtli*n* ^MN^
n?tt «f' a
-"Wl^tt •!
•c-^
Jfi
\
IS.
711. ^T^---.3* '? T^:.fT'Ci.L LOST- CfJ? ^^n^miX
Tf Vtm aMi»« of th« *rird«r ft»4 floor 19«€ t9 %«lreai to b« l«r|^ In «o»i^rl9<»ii
to th<i ooIttMft twli^tt, and/or ft nartior. of th« oolium w»l^tt Includad in th« «riy*
4«r iwiiiJit, the nstuTRl fr«iu«n«y of Tlbr»»tloa will b«: co = y^K/m «
'8 previously noted k^^ • Sm./l? (1 - 'V ) for plrvne^-ead fr<!Wi« «Bi IHKl/t^ (1-7)
for f1x«H3»«»n<Ji fr«'a«. T^*^ enissn will b« 'C+'f^^^lA^ L^ for pina«d-b«is« sad
%^+Tf^EJ/ql} Tor f:x«(l-ba«e. Th'Sk th-* rif-tur»l fr^^- tit; aw for both ?>lan»d »ad
flx»d-ba»«« %?ill b«:
CO ^ JTF^TT^^TV (5.1.1)
In tfca ©leBtie nsnge, «fc«a llv«t Twrt^cna lofttf la aAtfiitf, %h» trtsmnof at vi1w«»
tloK ?fHl be:
V TT^L r„,.
«o.r> /'2s('--y.„..,^ (sa.i)
Tt« fr^Quency not allow!. n>-| for r«t.iu©tlrja \n k (this Ifetor b«j»g used S8 ft najlti-
r>ll«r) la:
Tf t, i? expr««fl»d la t«ra» «f ?! {!/4>o)» ^^w*'-
/T /2 77 ^
iife«r« T» Is p*»r-.^c* )f atriKstura not Isoludl iv, offeot of afiy Y»rtloaX load.
(2.1.5)
-iflur
}t9tmmpmi% t,t
fn\)ip t:a>
=-0^
»^*\
sN^ ^t\^"-
14.
XT# F' -^ oL or fnt
4,1 Forci of "«si»tfsGC8
Instead of s flet top vfyi^ «• 6/©^^ dllaiErMi es reprss^ntfsd l»y flpuro 2, l«t
t ,: • ?<3tion t/ vs. r»lstior. b«i eucb thtt It ly better ffiT«n by » v/m^ v«. 9/6^
rwl'^tloa sho«rQ by figure 14. "^Is typo \?/?jj ». 6/e^ die^rs® might rftpr«s«Bt
:
(1) An i(5»«i Izettioa of th« eetuel 'f^ V8. 6 dttgrtm.
(S) "^b© " TO. f) mrT3» far «c 5d«:»«li2«d H 8«ct.1o?i of o -ftfit**!*!*! with, b stress-
strain ourv« flueS: cs fl^ar© 15.
^<lp to «xtd ineli^iag th^ yi«ld point, tb« fb?$a of tb« T»&iu%mamm will bo tbo
spj»« »9 boforo.
BoyoiiA ^o yl«>ld polat tfco jnwietftaco i^'ill be of ?x aifforei.'t foris. The ceeo of
th« ldesll?,«d B aeotlon is «fcn»«n to llluatrpto th« g«fi«r«l. fbrtn of the feorlKo©-
tftl r«s!siRn«« swrvos «sft#ir ?l«li11ng;. Th« ^'A'^ a* ©/S-. dip:«:racr. for ss-ucfc fin
ldOfilS««d aocti'jn, with e »siu« of oL • O.S, Is ofoowa is finire 16. it Is aot«<5
th«t for letpgo ral.ii^? of S/©^ tho asxiel l©r4 on the l^!dl?.«d auction does not
•ffwet tho '^M^ »• ®/*^(9 rolf>tlon, Ho»ev>»r, for «& actual a«otion th«ro -sMOuld bo
• ftttoll differenoo, 'ir.d tte cui've for the sctunl 8«»cti on would lio bolo^ eni to
tlso ri|^.t of thia id««li30i5 cur TO. But r.e 9/Q^ laoroeaoa, tbo two approjioh e««b
othor.
•Pft A<w tiio offoot of \.\i% «rtJ«'-.I load, and »l»o tt© effect ii't the rf^tio
WtPMia F^ aad P^j^, tbo «mhi of tks flxod-booo idoQlisod F soctioa hna boon «^»rk-
•d out for oL O.S. nf bas boon t^wn «« 0, 0.1, 0.3 isad 0»3« For oach of tbofto
-7 velues. "T^ spfko takoa «« 1 cad 2. To obtain tb© r«l?^ti ^g botwaen H/H^j, and y/y^
ft nttaoriool proeedtiro, e» o-Alir.od In roforoooo E, wnn usod. The eoluisQ «»8
iivldod into oS^t —isinm%% »id boo^uso o^' tbo ooooiiwd synsiotry for this o«>ao
only four «4^sB»KQts, or oao fe^lf of the eolu»&, were worked with. To w)rk th®
f«r ^-o >v»T
mft l^
*o4c« •! . - -to--^'^
fog aaoft aetloAc ••• -^^— * «i' r** s^r'ixl tcrt tarf^
15.
Boa»nt »84 <Hiti/«et«d d'?fl«fltloa » n«w diofl*s«tloc sht^p* we» t»orfer«4 out, Tf th«
dttrlT«<i (i<»fl9etio« aiata|i>« ^4 nod «4pr«« with ths f''»9iJBsm^ ^fl«oU^» shnp« the
alUAS ^0r« jter'»t<»d, «s«lnM«iaila^ tfe« •rs?! ieofs«nt flx«d st t! © eho9«n Vfclue. !?b-
ually. If th® vela® of th« ea«aa«d <l«fl«et1r»rj jr^ •»«» 9T«n dose to tft« oorraot
T«lu« for tbe ehosetn «aiS :»-^.T*i«rit, tb« d«iriv«^2 p1u« o-^ th« <50fl«otiea y^ »?ndl H
from oaiy oa« Her««tli»ft ^oui(J .11« oa or v^ry dLoae to the reaJKtnac© eurra.
It ©«o b« e«ea fra-s fl^^ure X? th»^t for l^r??** vi!»tli*#s ?*f y/y«i *^^' value of
H/F^ 1« r«i>»tivc>ly l&diip«A4«at a; th« veXua al" ^/y^ "f"''-** Mgbly d»p4iffcfi«at «a 7
;
fts th» tfarves for T = 'X ' n? T^Zt lipproeah •«eb otbw ee y/y^ b««©a»e lerj;«.
ftiin c»B also b« s««o by l-joklnj? at tfe« vA^^ rs, S/©^ dlagrestt, figto*« 15. For
tfc«! l^rg^r v&lu#8 of "4/%?^ tbe valiui of ©/©^ la iad«p«iid««it of the vaXu«» of <r ;
l>ttt for fi lot?«r rengft, joat eboT» tb» yield poltit, tfe® vslu© of ©/^^ depeoda on
th» T^lu« of T. For tfc$ l^^c•^ dwflftetlnns the ia«l©stic «<otloa mill approsrti
tb« o*nt«r of th« o-slu^jaa r<i«1 r.uch nf t^=« Qoluma V.A^^ will b® «bore tte rtta§tt af
a«tNir=di!»ac« on T» Coaaoquently, ft»r lfri?«r 3«fliWjtiO!5S tfee H/SI^j tb. y/y^ will
nrl'sarili' ^e'pmn^ oa'y, not T^f . !Tow»'r«r, for «««ll«r vp.luas of y/y^^ th© tsIu©
of T will «x«rclae f gr«ftt influ^jr^o* oa th* »hap« of th» K/R^ «. y/y^ Cttr»»« fts
alao shown by figure 17.
4*8 Sloixj of leftist '^•'^.qg Curvoa
Tt is -otffd in fl.i-ur© 17 th^r.t fear tY.^ Tflue of Yr oL t.h*» ^-'/^^ v». y/f^
ewf* eppro^«b«« e fcortaoftt?*! awy^ptotft for tb«* i<^rg;« vf>lu«H of y/y,,. Tliis das
»l(r? b» nhcffftt by th» foUoviuft To fial th«i('y^OL) ralntlon for the r«slet5(nc«
curve tc »ppr<Miet « bori7oat«'l (eayaptot© consider yiaiair.g .pro?6f?rt«t! ap to «©n-
t«p, T):l.*' l?c only »ppro»J3j«t«, n» st Q«&t«r J- •« G; but yjeldiaf, ^«8 epproeob
r«ry «loi<t« tr? o«nt«r for thla swark b^rdaatnij nfeterlal sa «6a foundi in working out
aij^n-jfr i «»1 ©x-'-.-r-lea. Tb«a:
.(U
d« •««
«0 WJ.IiV M .* _>.
;r
•.'JCC '
«. ym b»r
• xV
iB«i9
r»I»«c.- f
rf^Ri* fk!' ' *r«fl««»'
X»i=T ^ V :i T
TO! .T
fno lorrt
f ^r
i*T e»» *>i
•^I • «Wft :f
Y#M9 •<!#
ftXilsT a..j' A^
• *• f J •«
.
•tf o«i«
tt u»
(ob*)''sV = 0^2
mni S% St way |>olat le: 5 Nl = ~ SMq + P ( 5 m)
fro* «4&iob:
/J;( /P Vfjflc Ver<jc -'
frori liiildh B • 0, Tber«ror» to satlufy thf» slopf* st x • L, siN^fcp, L 'suat
tb«n «l8o be »o««l to 0, pnt> for «o«^tlbllity of d«fl©ctJott; Cos \1 ^ La = Q
TbiW >\^ L -^ast •lUftl TT , ••«:'i WtT^ = TT . 1t> fain 11 thl»
M iQ«lastlft netlftti wi «s«i«s«d to p»a«1tf«%« to tl^ o^atsr th<* ^r^luii of
•r - oL ^111 b« flll^tly la «Tror. T^owe^piMr, th« vcluft T = oL ts «ttffici*>ntly eloso
fCT us* «» « eolla|>»» «rSt«»rif'« Tb«n for T^rdL the '1/H^ vs. y/y^ carv« will
«T»ry«*i#r« hb^n 9 alop« ^ 0, '^or'T^cL , •; - '^i"'';re criteria 3th»r th«a th«
point «t «t.ioh tfe« dfmd lo«d[ vlll c* :.;-"? th« structure to oontimi® to dftfl-'St,
will biK'TO to l3« ©BtKbliabei,
for relf^^tivsiy Iwrge d«fl«etioa«, «»»i«ninip inolRstio action «pproisc^«9 tls»
center, th« elopn of tte« ll/Ti' ». y/y^ curT^, for t other tfcen o's oL »*y fe« fo«aatf
in n slrdl^r irmaaer.
^M= -5^fo +iHx +(5y) P^ (4.2.3)
fro!Q ^ihich
:
and «^^^'4)^ ^^ A-f X^O ^L .
Tsklft?? th'* r»l\xi» of'^^^'^J^X » '^^^ «ItsiJa>'tlnc 8v5^ bf tt»« of thn rttlfttlon
bMo:= ^^^/, "*^^^io)/ »«^ subntituti >fi of "^^/^Xd. ~ NpVbT ^ • fliv«s en ex-
pr«3al3K fcr t^#» 81ot>» for r(»l^itiv«ly Iffrg* •lu'^s of y/y 'i:
j. = >. ;
J IjI^V*^*^ • ^ ' ^tni^rfr (i»IAv
.to Y
•••I* ti*r Jo r r « r «<« /Ih»- JO r T
%tp' 9^0 ••Off* i oi ^1'«^ #9 ^r^toc
•lit ••4ir><yiffs ao >: terf
^S {^l)± xHi+ 0^"^- -M"^
-A* n
f„j _ '^'[-'"W^Wh"^'']'{%) -X -'(t-coi-^ir)i.siH^v
1. rr
|jjS,«^Trj.^j,,^^.j]
17.
H."?.4)
*h« co'-^'erlson af tfels «"?.«?*tlon wltk fi,?ur«i IT i« food.
?ABLf I
0,10.20.3
-0.063
O.IOS
17 for y/f^ - 9
-0.070.01c.ic
Wirup* 17 in the dl?j«n«lonl*?6fli for«5 wl?wl f^ls^^ boll for pijm»<!-bB«« fr«««.
[l-,>ojs-[-"l^-'*llj
[a-x:^m.][^][^^.03-l]^^ - ^
m.mii « > > .IC ff--'f'^ r*^^ •->• c-'
tJ •»
18.
^?:^ c( or TI- p
§•1 FOnTi of '¥>3i«t'-P.C8
T** eA«« of fttructar* r-BSfat^nce or *:-*» ft>jn ibom by flgtJB** 18 Is treai«€
In this s«ctioa. ;« beforo, ti.® yJ<il<S poSat is d«fln9<J toy:
?h© second spring forc«, kgy, le tfik«ri to beca-i© arf«atlve pt y , Aft«r
yl«ld th« fftOffftjat of r aittltJplJod by feSilltloael <lefl««tioa amat fe« »obtr*et<»d fro»
t^» '/• r«9lst*'ae^ cuty©. ^or T«rtlcal Iop4, SMp^; '''""^ EI I^mY/s L^ ;
feioti au-t ^« :ubtrrct«d froo 5 Mo= ^'-'^iLCStjWg to deter^in* th« oh^nr:* in
H.
= [ot - yTT^/lZJ K.L(Sij) (?!nned-bs««)
Than tb® •ff«etiT« spring constr^nt eft«r yield for both fixed and '!?Saned-be»« fta4l
frtw»« will be: fot - '771^12] K, . odtr. \f^ -ari ^ .
TJje eollep8« d«fl«ctJoti tslll thft-r: b© !?>?'.' ;^::: by:
?h« rtioffe cf «lidlty of this ^uprftisej 3n 1r for vslu«»s of o^le'^r* tht«a 'i'n'^/ \Z •
for ?^lu«Mi of oc ^'7^7tJ/j2 » « fs^Uur* d«n motion «*» defiawd la the latrci;!uotlon
will QOt b«» rf»fi<^^9d.
Th« form «f tb« r9da««d reslstetnoo ourrts is ths s^am rs in aeotlon IT, Th«
oaly dlff«r»ae« from 8«etloc n li» th« «lop« aft«r y!«ld «a«! col.lo"s« dflfi«»ctlon.
A typlttsl carv* Is ahowa la fifura 19#
.nr
,\c
•I 81 vv
i.a
AllI
(T-0 - X..H , tL^ll
m?"09 •iff
^JSy(;3;ia'TTT =^M6 V f-?
,.«'•
«i •-a ;« •<# •0.1 >'-»: c# sVf^^-'i'^-^ =0^^ ***** ^ •** '**"*" *** "^
19.
&•! Dleouisioa of 'i^tho^a
For kaore VftltMs of ^oj ^f^ ^^^ij'N applied to « otruoturo tb» laMwrio^
«i1feo4 dosorlbi^ 1b r»for«ioo 3 ^«y bo usod to oeaputo tlw reaponoo. Pof^roaoo 4
•Mtolna oamjmTntlr9 lalbrsaetioa on uthor Airtorleal mottioAo* ^mk tbo vortical
tiato dopotuloEit lond ocouro as a ottp fUnetioat osolytiool oquotioM swjr bo qoo4
to ooapato tfco ro^ooiw, Bowovw*, if tho vortlenl Xood is oontlauously rerylnt;,
or If it is dooirod to ooaplfrt^oly treoo oat tho notion tho AtKaorlOttl (i^thod is
BQOli aoro o4T0&t«4POOii8. Iho rofiaoriost bore to tho lUHoorlosl ;«othod of roforoooo
3 So ia tteo aiotfcod of obtaiaioK tb« horizontfil reoiotiaoo oarvoo. 'inits uBod la
tiM aaaM»rieaI .t»thod sro;
QHostity 9m«o of .:«iiiinttty 'Tmtm of
Tbo flrot stop towards oolvln? tho •m^hm i« to ootabliol^ c oortoo of T
r«ilat«Me ourros, ooob m tlgmrms 7 mii S, oovoHae t)Mi rnec^ ^^ fcnrtioal loud*
to bo ooooofitdrod. ?or ^o flot top oootloa WMseBt-oarTat>jro rol^ttioa of eootios
IT «ad for tho oooo oboro tfeo straeture raeistottoo oftor yiold in a atrei^^t
lino, aootioA T, bot not horiaontol, tho ocmstruotioa of thoao ourvea la wyooay. Put for tho isolino^ no««{|-o«irToture ruction oftor ylold, aootloa TV,
tho ooaatruotioa of a aot of reaietanoo oorToa So sot ao ooay. k typiosil aot of
th4Wo «aa abcarn la fSi^uro X?. Tbo aol^bod of obtaialaf tltoao wem outlinod ia aoo*
tlon 4*I* ''ft«ir tho rooiate^o curvoo ere oonatraotod tho aoaorioal atop by atop
proooaa ooa bo atortod. At aa^ ti.^ inton^ t^o Toiooa of y and ^t^t^^x *^
tbat iaatmit doter-^ino H. Tho isalfeod of traaaitloa botvoao tho vorioua rosiat-
aaaa outtoo -mui tkom la aootion 2.5 nnd flf^ro IX. Thla atop by atof prooodoro
tvmt— out tibo doflootiOB, ToXooity eod aooitloratioa at oa«b tl«o iatwrmX. Tbo
• \^""
^9 aotmmi.
(V^i^.^^.i^^)
hmium jit •nit
^i9 el t
,e<i^i CO
o^
DOOO* •if O*
<!•• trf ,!•> acl*
•lUr^O J 7? MPtV Ti
eofiurecy d9p4KMft ^n tfec «ho«»»a tijs# intftrTrsl «R«1 W%« Tela© used (r*?f«r«nc« 3*.
fftr rapid Kpproxl.-ifitlnns t«t& enual to may hm as«»d. Also iris nitsjerlci'l m«tho4
will ^ive tr^' nexisiuffi 4»fleotiaa en5 motion t!i«r«pft«r >*n<l wUl di«t»ct If struc-
ture ii ttaffteble.
Sl«l tfce rertlosl tl'^is d^p^na&ttt load ooenra »» step function tli« feaelytiosl
mi}Hti%i(^ne cejh b« uea'5 to glT« the r«"Bpon8«i« '??v>r »or« th«!R en iaolpted point of
the P08|>oae» tj5« ati»«rle«l mftth<>4i still hns *4v«sit«g««; ««!p(<icl!felly if the 5i»fl»c-
tloa St th» «n^ of th» lo»d ^uls€ i«? fr«!?s.t»r t*-?^r? tJ2« ylal<! Sefi^otion. ?h»
»n»lytic»l •qaetionw sr© ua««i in section ft.F wh«r« the smm •qu^tion ooDatants
•ftfi b<t u«©d anvftrtti tir-dps. Tit® rroo«2ur« «a»d la section S.S is slL^btly aso^li-
^'ted tc. thftt 1st the «in4 ---^ ''•?• lo«d pulse th® kin«%tlc aat^r^y of th* «yat«iE is
a««d to !3e;toh wltfc th« r»p1-t-?sQ# sorlc of th« strvi©tur<? to ^etstsrilnQ v^it^t tfc.* ad-
$\%iotii%l d#flflCtloa will b« Hft«3» th« eat! of th« lofid pula® ^nd to dat^imlBe if
th» «tructur0 1» stf-ble under the &?»plleKj lotde. fJsine tft« mn^r^ in t.hJ« sneanwr
4o«t not d[«t«rmir5S' the tt.'ne to repoh tr-.e 'nrxtr.ufm d«9fl«ct!on»
Ilhr
Ir
»
t» nott
I •i^r t »cltfo i^<^oi $ at ft?
«r- >:»»« f? n. fi n>f (.'•>> .
IL*
Ppof. ^'Z'. >:««anrfe, !Mv€rsny of iXllrjola, has d^vlipi « iwihod fs^- a«ter-
«ln,lng an '»rr7roxi.'np.t#« eqalvslemt si.ni5tvrrr* r»®l»t*^nce t.3 .oro4i«Mi 9 nftXlAtisa deflec-
tion ttquel to Bo.'s*! nr«€«t«5«?3i.i9i.1 vnlui^ for tb© ORa» .^:!;j»rB feg af th«* »tj»'Jictur© }^
bofc of ooaataat T?elu«, In this »«ctlon r-ToT» ?l««3»!rk»« wothad i® Us'-^ejs aad
«4sr>t«!5 to tb.^ c«ae .of tJif» frfe«« frit?- o-:>r'.st«n% T4».rtlCf.l loa<Sa; i.». , »i»t up ia
tonsw of tb® atr•iet^i^^?I i?rop«rtie«f vthf^u c- « effect of tY.a vertical l0e4 on th®
aprSftff ennatdOtB »sd yield point is oorssi^^rsd, ..ftar oo-'jputetlon of «©rtal.a
f?u«ntltl«» pertBfnljag t-s thiti ©tinseiur©, r«;f*»re'ice 5 e«sa te® u«ad to flaJ t;h«-? lostf-
tijn* Tftliitioai f*3-r thft «tr5»5tv{3»«* fyr hoiriKoatEd laf»ds oecurrliSin' tss step fuactioa,
Inltl**! ?®ak, t«rr?;ia«l p^»k and iBi®rffiMl»*te s«s«k:. Also tlw m-ror of this »p-
?roxjRsr't« unifbrs? er^ulveleat -nffthod , for t*n« esse of jaexlriuis feflsotiofi emial to
o«5ll9pa'? d«fl«}Otloa, 1® «bflwa «a4 *>« r««f>rifa«ridetiaii for v. z^f::zi.ntief^l mn'^ction
feetcr l^? given. Tte«* correotlon fiiefcoir oavers the raaiR* of tj_/T^ b«tvw»»a 0.1 <md
1.0. Thl» swtfeod is ft>r « i*t'i=^ T^ppro35.i'¥>: tion of the rJ9«p'»r;s«t wfcaa -Sess^ lo^d in
to b« t«rk«rs Into «icoou?5t ^.M "f^rtittt'l %i%>e d^poad^nt loe>'3 nat Jriolud3<5.
7 • 2 .>ca I » '^'>^%lr.u?y: Dgflootloiat •^o.aal to C^^Uspgo :)Qfl<?ctjoa (rly.ga and riaaa^i"
Th» r^'Bnln'? of ei,n «?mlTsl<ni,t «s>lfo,r« rm»i»t«-n«« »sad relatlos of tfc« '«»uatt-
t1ti«« iRT^lTftd J 9 tllu«!trst»-'i 4c. flj'sure S'O. A^ b)*for«:
0-. T) r / . ( 1
NpTrr 2.
It A«9liSJie;tlon is <.*V ^J
rw»r
acl$o(^ ramuiZ I9V
\ «
at ^ i*-
ftCS
n»*MWr
*jr»|vn-,
Fift'? Tf/»*. '-^(A. U}St£f
p r>«>^f-» or,
•1 b«7i ta«l IMliir
a,?
*oh>
!»n
TTYlr
•tttf
(T-l
fO V 30 'cnear. r")lsi\^^+^-
\
) (V'\)1
(^'\) ^J TT^wiA+ ^ ^^
to
I Ar IS.in'^tmAh i
frir 2 -Xn%/"X
(7.2.1)
(7.^.8)
>>3l3:nstloa)
Th» ratio tj^/T ia a««<l ?»« a p^rarnster in r«^f€»reBO« 5. t. Is t:«* slapsti^ri of tfe«
load pulao ia seKJonda. T in this cfix^e i?? i 277/ - 2T7/_
Tfc» proo«dur« for a»e of refaronc® 5 in t"? co?3put» t'ie r«tl(5 Xj^/.C^,, 7b«a if
th» ti?!e tj_ is knowD the retlo tj_/? o^n Ijo Gom^.?!it«d. Frors th© «pt>rcrrip.t» ofeert
of r«f«r«noo 5 for tt9 typo lond puls« th-i yalue of ^ tn?a vrlll ju«t chus« ths
^RXimxi^ (!}£5fl«etion to teo o^uftl to th^ ooIIrps^ dis^fl^ctioa csn be tsicaa.
1to feetop ^iB ©lursl to H^^/c^j^, If tfc? ?!agaitud« of th© e'S^r^^o of th«
l^'--*! ir?»t«nfl ^t' tj ia Vrov^t, the »!.«» of tj_/T th»:'t -will just oasLUSs ©ollepso oftu
b« found,
^•"^ -^-^gg n .^?teqr<i tb<i D-^alPftd. i:sxi?;i^ r^rieetlagi is Leso Th&ja the Ci^lls'jiae
D«fl^e» jon J,?lX3^^ /^n<1 •"tnnc»a^aa«T'**
Tho ':jua3tltif»<i in'^'^lired f»r« illustrt-tfK! la fivture ?1.
•8 b«for«: iy«.«/. = 5a*«« HeR>g^=: Sam* X = K2R
A-
"/«•«""
"(V,..-')
Then:
/ye /yI- clA
H «" .
O-nO
u X / {aofQr«tf>o« ]
«^«re t'fTfsin tb« rotSo of \/''<y *ftll bo eoaputai'?. ifcon tho «.pproprlat« otferts
Ja rcf«r«no« 5 esn bo 9rter«ct to find the t,/T nttlo for n 9pooifted ^ ; or if
•wv)
^ '\9\
••^liO* *•»•• ^^^
,s»xi-
al r#
('-,.,\>'')"'-''^"
•IS Cr
{' -..w^) ^ - '
'Si
.-lA+sf^.3-
R9
U'-^^^*")
-I i TTrV
inii =
• ^ I
Ax - {
{
[molU^i J^ "=" J^^f»v;^
tol»i fl2
tbft ti/r r^tio ta ^aoncsi th« Vftloe of ^ 4«t9];%la«d tbst will 9«IM« Mui OMlclmMi d«»
fl«otifta to b« •c«u«>l t?> ygj. ^ « H^vj/Aj^.
7.4 corrupt iou^ Factor
la ••ctlon VTT? plots cr© rnjvdn of t:h» arwrei?® Jfeorisontsl nnd vor-tloal loe.4
T». the tj^/^Q retio th«t will 9i^dtae« th« »tsixlma?x> dsflactlon ©qu&l to tbe col-
l«ir»« d«fl«otioR. Twj dilT«r«ftt st3nj<stur«s *ire eafclyaiM. On® is vfith t- K'*^ >
or Pg ttfjeo Pgj,, ?*n<l -^ith d««« load of 0,1 r^y «n<l 0,08 r-^, Ta® othsr oa* is
T»1tJi Ts2'r , !>r "'-^g oa« fc«if :yf P^, nft« vrtth r, (Jwci lo-:-d of 0.2 F^y sua C.4. F^.
•fh^fw two ami«8 oov^r » wld« rsaB«s ^f »» elastic C'3liir:5B *'ith v. r^ry ll3:ht 4»r<S[
load to BR in«l»9tlc »ol<a3ai witli « fll«<*i h«H»Tlor dleaft lo&d. The«3 structures «3N»
aiibj«et«d to lo«4 fuactlons Sn th« form of 3tep» initial and tsKSinal p«ak.
?ttltt«» of ^i/h^ tor 4lffer«nt veiu«» of tj^/T^ h«ve iMMt •a»imt«« ualag th»
npproxls'stft equlTn^l^st uniform ri9i?is?tf5,aa« a<i4 &<mp»T^i&. ^iit t':,« vifsltses from b(*Q'-
tlOB 71 TI fcv o<j^ «qaipl to 0. I'hft ratio t^/T^ Is a««d lui « bas© of coisp«ri»ea ia
•eetion VTII sad thus th« retlo tj^/? aii«d in th© epproxlje)S>%9 ssetbod "a^e to b«
oonvorted to tj^/T.. ?. plot cjf tt« -^ ©rror fcund for t^«* jreng© of tj^/T^^ b«tt«w>R
f.l nnd 1.0 Is ahoMB In figure 2B, Tt is showSi tfcjst tks error for all ommi at a
flT«o t|^/T0 rat Jo is a^t a eoa»t«nt asiouat, l^ut tfe«t » gEBer&l tp«ad Is follownd
without 8 l«r^« s?r«ad. 'Ti^i* spread is the ;! of «rror oould posolbly \m fross both,
•rror is r»f»dlni; th» cfcvrta of r<»f9r»ne« 8 end in th« structure rooponse lts«If«
Th» arror Is flvan ir 7«reentbg« of oorrect ^^^/Hg, <st fi ?5^rticulRr tj^/T^ retio.
A wlua of the error in th.u rmme of tj^/Tg b«t«»»n C.l nnd 1»0 th^t fit© tiift
rr»r««» of tho <>•«•• work«d out 1«: t error « 12 (1 - iti/T^)^) - 4 (pftr»bolio).
,Mthough this «rror 1« boa«d on tfca correct tisIu© of ^j^/H^, to cc*nstruot o^i/^'"©
vs. tj^/?Q ouTTf* for tfc« eutxinws deflection aquftl to t>«< oollaps* d«fl«ctlon,
liai&ft tb,« ••qttiTGlftat ualform r«>«i9t^.a©«» th« rooi»ti(5no« '©y b« flijur^d for tho
•qalrsl^at uaifora r9aiat»ac« aal th«a th-s «rror far ths?t t^/To 8ubtroot«d« B«-
"mo i{9 9a-J 71 :i;-it» ,J ^r-•lor
L »» li '
'3 5»r
>• ,i» in -
4JJJ
94 of m4
^ A^ tt**'^'* rfit mr'i Tf' -rat /^ '
I if
••OOJJWI
,sit •THi^ "'.T ^*^ f.*^
'
'©"'^i*.
.^•^-'^
©iVX'
•»N^
rt •«
..4* V. n > «''i'
/'(Wlllp»
9mtum of th» aEcjall nsxiunt of t-he i,'©TC9Bte(!?« of tfee error th» oarraaiioa mQf %•
fc«se4 in the ' error nviltiplied tyf t*^« «a®p'ut«d<Aj^/B
36.
TEII. .' Tf CF ^FJ-^OT <vF »m.3F :a?'P.K, HSflt « OF 'TRiJCmW
8»1 ^^•<^ T Sectlott
In this s«ctloB a attidy ia nsde of th/? <9ff«ot of tho dead load ©ad tii^e d«-
p»n4tBt T»rt1cel loada on the horiaoatal r«spon8« of a sisipla trmm «ii«a »l»o sub*
j»et«d to R horizontal tine dspenflflnt load. 3^«n in th<? d1v:©pwion.l<»s9 for® the
r^spOBS* of the flxad sad plnnad-baae fr-amea ir, ti.e smse. Pointc to h9 Investi-
gcl*d era: (a) o«n tb« •rtleel dead lood under nuy eii-oursstf-ne^s b« iMiisIected;
(fe) TA«t effect does tba deed Xo»d h^va oa tfe© reapoaae; [&) «hat effeot does the
eb&pe of the forcia*5 fuaetlon hsv© on the response ; (d) when m.fty the vertioal
tlr.« dependent load be ae^leoted and (©) c«n a relcitlon be found to r^li^t© the
response of tt* atructars sjI^iqs subjected to th© diffsrent typs forcing fanetions*
A« & f!r«t step towrsx^s studying tbla nj'Obler, tsro differrtiat fixed -base
•traotures were t«keii« Tlifl two were chosen to bsj very different la chPirncterls-
tlca to deter^loe If botb followed th© S6«*» ^^-sRerwi trend ia respoase. Hb^ firat,
frem K, was with elejitlc coluraaa wttb very ll«:ht d&fid load. For this frenA
T = Jj^'T' '>nd T^ ?;ni T w«re aeeiscned the vfcluea of O.I end 0.05 respeetlvely. The
other fr«r0, frame B, wea irlth iiiMXefitlc colussas, wltb T= ZT «ad 7 «ad t efeo««a
•a 0.£ «nd C,4 reapectlYely to x^pr«9<mt •' eolu:/ic ^:i%h t: nuch hl.J^ber ftsaount of
4esd load.
Tbea to obtain a picture of tfee respoaee of theee tuo fr»rje» they were Miili
flret ep,ub1<&cted to hori7.oat&l time d«pendciit load fuaotlons in the for.'s of step
pulse, initial aod ter?TslnBl peek pulses, fith these type lond fuaetlona It tms
then foujnd for ilffererit sieja^cltudes of th«9 evers/!?e horizontel loed.Aj^, what dur-
atioB of tl ^» In terasB of t^/To tVie lond could be Rppllsd sad her© the mfixlnme
deflootlon eQa»l the ooUeps* deflection. TheD <Cij la plotted ts. tj/To for the
ti»o fr«nes. Th» oollspsfis daflwotlon for frp-;© k la ifllt^ " 1.1.55 nxA for trtmm B
Jf/y^ " «3.65. T%jeee plots «a*e tlkomt ia fleSureQ J'O to :5£.
sffjrrri'n^ .r:- Jo :;
ffrr ci
.dSfioq«.e'i al !
.k.-i'
*h I«i
ftl^ii fli
»i(^ as ate*! I»i»r^«v t«ift«K
~'!i ;:.•' .^"
•rt* !!«• («) '^^T-* be^fgr.-
*!ttf O* fl;
d»»9 •<nv« t*^^
£!--•,-
-^U* t»A* .^A ,1
a4« TOl ^\^3 .ST £r'
Wt §• W.!! - ^t\^x
^ amtti' .actios.^
8?. a»!-B'
isaietaJsi) gj so: 7
^ turn ,8 MM^tl ,fts«Kl %miiti
•il^wU #.0 tea S.O tm
•toci b«*f
' *r ttctfn
LOO 9n^ ijuipa nc t
u»
7f%t, v«rtioal loads in od^itiojj to th« horl-sont**.! ^mvn nii-pli^a to eech of
th*? fr«-.--:«s, %« v«rtie«l loads ^::«r« of the seriia t>p« fuaotior. fes th« r:ariz'.-rii;«?l
lo»d BG'* ^f ttiif 5«aj9 ti«?9 durst Ion (flgrire &3U Tt w&a th?jin fa\ui<i »*»ct c
»»roi1ue© th» a«»l,«a88 4«fl#etiea eqasl to t)m coiIe?a» 4aneotio/u -"'h© vp.lu^e of A,
»• tx/T{3 T»«r» then plotted, for oarin»« wf ooaatwnt ^«?..sou?jt of Vi* for fto^ 9tsrQ9«>
tar«, for «»«ch typ« of lo»,d f^iaction. 11j#«® ourros «r* afeo^s in figures SK to ^-2,
8»8 4Vftttlyti6f;l C'-^r.putetl-as for .?t«p raXae
flt« •rtioel Ic-ftdl, other th«ii tb® 4««i<5 load, oeot^rs «<* s atflip funetioc 'fig;*
ur« 33»}. Tfeft solute on of thn s*qwistlon of :?t^;>ttoa» i??? « Aj_ - H, ia tha «»XR«tlo
raaar* iAtil»^ i^ets will b« of thj? for®:
' -C0S>/(,.y^^^^>^eu,t
(B.8.1)
*»*•*• ^ota -^-^wt ^^1^-
•.liil*/^. ©nd -y^ f;r*» still sotln^ or.d f^mm elftstlo;
/Hs^'^W^^^) SIN coJ((-r,,,,J i
{8,2.2)
lfZij_ is still ?;.etJa3g wh»n y^^ {y^^ bssa«s! o.a rtotgsl^ *^ r®etsh®d, t^, tb« tlat
to r««flJj y^y, "Pill be; .
I -
(S.2.3)
Velocity st ylfi.ldl,^3_ «tlll aatlais, wl.ll b»:
If Aj^ still ctst,-; 'iftflir y^^ t-.« fflK^^irition of .TOtlon will be:
[??•§. 4)
«A-^^«^. -«^e-:!^. 11.
ly« :^e /t^(for vrhldi t « At t^)
•ad «b«r«:
' ^^f im JkfV^kik. n ^^maf fff&lA^ ,fzMr
..mft tftmn^ ^^«4V *>|l|JC
Al'- ?•
rV^.
a^waciiv
fjtff
wit to Ad II
f
'^ bmol
^ oC* ^,,^.v-0\reO
'^s'r^ t^r)- ^^^,r
^ U^.^'^^K^ ^^^ 6.v,v^-0^«^
r :^ x^^'
C.^c^^-»)J=.^^-. ^
C^^c^V-O^oCo
£^,f X^t»oX»*
p..T:;^.u u;7^,.][^^^j= ^^yl^
: 94 11 hi N V nutT
i^ im 4aiJm -mn]
/nt» ^jA-t
e :o V
I
St.
i^ , cOqWU-t^
/^^ ^e
(/-•r}
;action, t-f* tein«tl« s^nerssy oi* th» rtruct'jre *>t t, j:^l ..:-:.
; ..k --.v.i :- ^-.lus --..
txt
23?
yl
(^.i?.6)
oth'-^E' \AiT^.& -t i^ay <;s'l-f^a».j.vrv y&rjt' cUo*
• : .-%> jt.j3 2'».i lo Ufii-ry strantti-r'^ iuat to
.*''«;iuce
JTf with oniy 7T,t '^tSis;:. T:;i« -^lU b? in %» i^-m -:^f n«ur«s 54.
-.i'i!i;:5*^- T?** '!^^"i*'^' .i rt'.- «. ! :^-
i'ltj^/r®.S/ye GJ<
.v''^^'^^tx^^ - A - >T *
J.ti^'.^"'^
.^'" I- \l!f
fV9
C^^-OVpOa = ^^%V^
X
H 1>- ..]/y^;(;s
A ,|Y*
r -r^
"aca
,'00 <^\J^* tAi.'
, tev'
!,tO A..l^
^*f* It
5. If 'iJK*/MeCi ^^^^^ ***** ^ ^* ^*®* *^** i"iilu« n% y, frO!n 8t«p £ •qiistlon 8.2,8
is xwA tot %im cBOtioa «ft«ir 7i»14.
6. vOa«ti its ..2, Bg, J ?nd T era oomrutwd,
?. ~'ic!< b trlfil Ysltte of t (la tarras -^f I/cOq) ftor a«« In equKtloQ P,£,5, Thl«
will b« tli» efter yi«l<! (tj - t * t,). COMpaite jL /y^ end ^|t/ ^"^^ •*•
jwr* with **!«« trov st«p 2.
or tj ^«fB obtalA«d will ^ Id tsraui of I^q, Tim rstlo tj^/T^ or tj/T ti« 49«lr«d
l?«saU8 of this ere p>Xott«d is ft|;«r«a SS» j[!«, it, 31 «m4 ^ for f^i^am« A uii
B« R«»ylt» «r« plotted 2q twras of Aj^ vs. tj^/T^ for lines -^f ooti«t«ttt MBOOQt of
*'^tot«l» '*l/'^o *• •^'^••a Ja l^*» o*" H/*^* ^*^C5'^«^®'^ plots tnk«a fro?8 roftrenoe 5
for <xmpmri»oti. c»e tfe« ti:^ rmtio tx/To«
8.3 *?«a»r!ct!l r^np^teXion for Uiitif^l egd. Tgri^ilttal 'r-e^k F-ulsas
Hi* d*t«rffiln»tion of th# proper oosaMnstlofl of Aj^, tj^ wnd 1i, (in foxra of
fl0ftTOS tS b losd «^ ilons not refidlly l^^»fi its«lf to «»%th«sstlcr-l e-^untlori*. ?5or
tluiae tfes auoMTieel st«p by ster jTocsilaro is usod. %« tiator-iaivtisa of the
proper •ookiatttioa of 'A^^IT*^!^ *® !«>r© t«?dlou« tis^n thu proo©d«r« for th« !;t€p
funetioa.
H«r« »lu€»s of /^- , ^1 1 "i-^ era eho««n i7^ b»l»«j giTanj. ; t> ea
WMflMd oat by n 8t«p ^ ntmp ntmnri&nl ^^rooodure.
Tt» tbIuas &f ^\t AA ^ 5^|ti/y« 'f'^ checked for iii>t«n«>t Mtb «vell«
•bla ttntrgy #»» in st«? S .i .n for th« etep fimotion. ^f ognuKSftnt Is aot ob-
tftln«d, t]^ Is AeB|t*d e»d « nww colutlorv obtftiiMid. fhla procmes Is oontinued un*
%M mf^—mnt le obt«la«d.
"^•sttlta for this pyoo«dur« mr* ahmm In ti^xwma 27, 36, 30, 51 r.nd 3B, H«i«
ngnim p«Niu.lta «r« ulott*^- ^lA'^ «• lAo *"^''* !!»•• of ooastcnt i^totisX*
/^ »t.Vit
^
I'^IKJICM
..ikj^
,t
Y^* A
:? ^CtT":
Ttm rtwrpoae© eurv*> for tmr^in'}! or <n1tr.ij».l p«e5c oula*?^ tn t«iir« Into nooouat
:^t«p ftinctlo!! la icnovG, by i«?i!i/>T th« r^letlan d «• u [^/h). ^h« qursntStiesJ of t.-la
.•.riat:-.«r r?->l't.?:i on t!©t«f*>®n ths> 3t«p srsjid initial ; ©sk p'-slsee* tV.r^t scid*? vary
elo»« 1st ""'"."» rffP^'^mse of » s»trwetm*« 'rA^os ?sufcj««ted to e v^rtiof^*! tijR* dopend-
m?.t lofrd in the f-^rf" of ?«!» lnlt!?-,I ?»«ix pul©*? t¥{t& .^ssxiriur* of 27^ «?sd Jsv^rpa®© nf
y, , 1.3 «-:*ulTHl*nt to tho rsspf^w.p of tea strxioli'.Jre wi'en ^^b jcetsv. to n iit®p t?uIs«
.•,7f -^A?i?tiita^ of •95Yv .C^; tt.ig ^c^xinyuti <SofS.«!Otlon beDsj? equs-,! to %hf> eollfjpso
aefl<»Cition for the d^nd lo«d of tie 3ti",>ot«re. This ia sfeo^m JiriaphJot-Uy io fi-^-
poftX "in?'. b<5#n ^^^'I'ffeaed ia ter^ss cf thfl -^.e.^^fatwde of tha «llov/tsble A^ for
f-.^r at©r '^ul^-.^ .<: . ^.:; l^s. fxaure 35, ?h© g^«»r»-'l t^'">vi of ^hese Is; ssho^s in
ns-^n,? the "p&rtimt^i^ r^lf-tias of ff.-:?ur«P 36 i% ia :oo©3ll*Ie tx:' ocins^truet ap-
for lb« 3t«»;7 pula® is? ^Tjo^a. For th#j e«sa#> of *Y,^ » .it .1;* i>0£4v]bie t& 0:;rjet2'Uv?t
a rsa'TNiawa 11 r# f'.^r ?4^e st4>rf fimetJon tsat a^' ^^s« procelura o?m?n©:; 3n s*st1.n {>.l.
TO oofi«truot r. cirv^ of 6^^ v«. t^j/T^^ for ^^ > ^^-r- t: c. ..t^^-. .ksI n.^ thf^ r^^^l^nioui
.-'?t'-;od of y«i-::tioa SmZ Im befit ysoit, ?5> s3xir9.!X>l.;r te fro,:. %u& &tQp t-i.u%^,ti.oii cjurva
t:5 t>)5 lrat!r;l sRd tar^la^l /•^•©.k <>»«•!!? «3is?r tb« ^•'ppTOX.itmte relwtioft of fl^.ur* :36,
It '
- il^r:^t; i3«c»e«fry to tas^nstraot th« ^t<^- fiitMjil^n t'«<"^-.on'-;i^ evxm Tor- tte eiKr;;*
Tj^ Rud -y^ «9 t*i« on© daslrad for t>m Ir.ltinl *^a3. tsr^tiafcl .^-sr-.k cr..'^C4. for t'c©
this »«etion o<^r> )».l»o ba .is«<l.
rt it is d«3dr«4 to iet<»p2?i.n» «*it^t rdial^uw A^^, in th» farw of h atep .-jlnQ,
7 - 0, a«B o»«8» aolUpfs? wStb t^ -^s long m n®oo»ii«.-ry ti.^ following? expref»eJoii
•^w
jHv:wi7V o^r I
• flwtt •dt Tc^ •riirr
v»
--T., V
'fl 1.*
»t:»**
.^V
TWitB.' OOXlSl
t)
,0 • jT
»/•
r\«y be usfwSl
/He /"aI +
'^»' V.yeR
{S.3.1)
i«fe»r« H^y, yf '-rvd y^y. ere bsj:?sf on T.^-.
9,4 ur-.':-^? oif i^r^'-^es - »n.;:l i; t^'r y^. « ^•
Tt is 9'><5n thfct ^.'^ eff^at ^f th«! Amn^ lo^'l xny not b^ a«gl««^«>te<S for aoy
obtGla*»d for tha '3^J!.f.f- .)f 'y^^ - rf^ » 0, i^ iaarsi^i^'^i. .:a.;.4J rtj.pr&'SK.a-AtlvrT v'-rtiuea
for tte ' .laarc".??* are 0v«n la t-'bl© TT
.
3t»p
'ula«
*;t«p Initial TerasiJiRlr.'.ti
^./j- %,f % -^ "r.1;!A:U--" ^-^f A^ fr%:- A.
*«,15 0.1 ^a,£ 2U.4V 0,1 D m.5 i;9.s
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1 ^(; 0.1 0.5 '51.7'" 0.15
1 «> J s - 0.151 o.c 0.15
27.6
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31.
t/pe pnla^s*
•-. 9'i'AirfilnT.C^ ^,T- n-r-t'^ bet«©«*ri th^ i^apoas© of step usd XftSti?:! ^'^e-t^k yulsi©
.fe
jyS
'^r.le
Tt ' J'JOT
Ai
r^r^ s.T/'W T-r ,>T I" .»l*
dueaa t>9 bo.ri7ont«l yield r»^l?t^nOie of © 8«ruetrur«, i'his yi^ltl (Jofli^ction ajsd
r<»sl'! t^ri.e>^, ^^^«-.•• .y^^*?*^.^ lo-^^, i^ 1 r^cl ^i1 f>rt , i.i ^* funetlort of botb </ wad the ratio
t«fore •'.nd '^ftwr yield* the r^si^tnnc^, yrhen ^srMcsl Im.^ in Inclu^^i, pIso eajs-
•:v;vMv ^jor-r-^ijtn .i . ::. ^iZ^Vfs 6 fi '.lot *?<^?;^ - -^ d«9 'VK:- ^ii^b th® elj^stic --ortio"? of
tb** r«s''f*t:-::o. <* 'XM'VP <jftfl b» tsiksn for il ffer^snt vr.Iu^fj of y ftr. ' t/^ .
'-•hwr: t^;t^ oC jf tjrft •• 'va» ^ rsl>tin.-;i I3 >0 the r®aiet5*ii<?a beyond tb« yield
-oj'ii t^ ', csrtrsi;.! ;->-,th, 1. .-i-^ fjon^ " or tt «(ttj?:.l t;. or I?,--':* th^rs o£ of t^^sv 'ry,
.'-(ti^:»n t'^r* 2-i^yuctur-.5 did t»t heve fi aollftp^e d<»f1.®otiort as prsirlDUisly a©firi«d.
>! . .:3tui r»^l\i« of tlr-9 »iop« of h/H^ vs* j**/^© "'"^^ other vnlugfo of 7 le
d^firr-sJ f;;r If^Ttf^i di--5fl<^cti f>a«,
Ta tfar^p-^'Plng tb® ttv^r^-^^n l-.orlxoRtf-l tl-Mse 'dssi-isaderft loed of th« epproxSwtjte
acuiY^l^tit u5l.fSi.r.i.'. r^'sl* t^ncft to >rif>r«? «i^.*i«t ^tlajgs f&T t*tt> «i-a»ly dlffer»tit r'trito-
b«ti*©an 0.1 *'nd 1.0 m^ y^^ wqu^l to y^,. Thi« € ©rrcr ij? l^ U - i^i/'^^'f'" ^
In wjrJtln^c '--Jt- it^a r<ss5;:-one«» of "^ct t««o 2?t4e»l^ Slffareat i"3r!aT'as, for y_ « y^.
It i»a« fcunj thi^'t tr«5 A./?' ^ 3f th» InitJ;-. .-. , "sarjala^l p««it pi^lsoo see'' '"'>-' ^>voa1. ©
r '/^rj»r>-l srenfl in > of ti«t TRluf? af At/--; for th<» atep r'Ulss?. Frojji tht.«t» -rsnmx4. Q
^ , ^ 0, Ri»prcai }(Bto r<»s'~o!>«<<* i-nir^mn f:>r the inlti&l paiSl tor.'slnp.l poaic pulses Cfi.a
:
-> ''«Tt ^bli'-jHe-: frr>"f: V;?i*f ^..^i-j. @if*5.1 l,v ottnlstibiis rsapome curv?* of *•'- •*.'•'' 'ol*?*.
Tt V/-.3 fa«n<1 t.kf't tm ir-ltiftl p<.»ak pules t>f asxiraus r,iR^!:nitu-a<5 of 8Vj^^ -.wss
''-liTttlast. to * 3t»p "ulgi© >? 'stjenl^-'-'" '^.^>f;r^ . ».... ::'-*-^rK:>su>< --,
•..•-^'^ fv-r-\/r^:%.
j'or the tiwtm tyn« r-ulass fcfi'J ths t-«<:3 iVc*(i«a pro 3^•*y«3i la figUTua J'- -.
Th*Rio (TJrToc af- « t>"i*» rel«:;t.t.oo of th© rets]x:s"n«« for tb« thr»« tf^9 puljias.
» V- T
A^ V
. tNJ
rl t T' . '
tint • v-t* A
Ol A
< V
1» t^ »ft
>«0:!,
39.
BltBLIOaFAPHT
!•
I*
iffc, 1?«M.t
lilt* S«M««
3, ^ewwsrk, ^T.^.:
4. HMHlflC, V.II,
8dad Ch«B» *^,F.J
9* VWMMrilt W«1l*
"^a '^^ngtMMlat Approach To Bleat n«»sl«t«lfe
*inB»arl«iel FrooedsiTtt for OoAputiag D«fLectior«,
tfomnts «id BuokXiag Lob4«*, &ac» Tranisction, 7ol.
IM. 1949, P 1161.
'-,^9t; - . r-.-lvfii- for atroDtizret-
1980.
'sotea to jy-
"A Ooi^piil'ieoA of Kinwrlc^ 'et^ods fbr ^jaalyxiag the
Dyns^alo R»»poa»« of ^tructur©*". Structural ^tmn&rcAi
&mi99 !«B* 35, Oaiv«roity of Illinois.
•"?fe« T?««pcm»«» of sifT^le :truetttpo« to QgnMRle ir *•!•",
Str«!tur?a i?©«««r^ 3«rl»8 rio. 61, University of
1 11idols.
•V
'.rP5;5TDIK T. IlilTT.MI^Y -JHOOKKD SU5TTC OOUHH
1,1 3»oy end '::onptttatlon
A study l8 read* ta»r« of the affsst of initlisl crook«da«fi0 on th«r reapoooo of
Ml «l.tt0tle aolwai idBM •ebjeet«d to sn axial loe^d, equel or grater in nj^griltud*
^«a the or It leal buokliag Xe«4, ^ad «hloh lo«(i roduooa to for t < 0, 1%« Imtq*
a«flootlon theory Is oaofi.
Tbo ooltjsan ^dll b« 8l@ulat«ia by the fsoAA lUontrctted in fi^uro 37,
Wsturblag «oawit • Py
«li«ro ft » l/S«
Lot: u.^(// a=>yo/ ^-^^a: u- = PjJ^ - KZ /'s/a/'u - 5/M""'ao]
«iid let: ^ciZ ; Q = ^2 . p * IC/ ^ K/./^^ ^'^^ or! tloal Io«4.
?fcoB:i;.
3<3 f/^A^i^ - (si/v^'m- -5in~'u.^)j p - q for oritioal load.
At t • 0, tt - Uj^ Sf, u •
Far «p.ah «>?r,b5 nati on of u^ «aa P, P Isolag with in dyaaptie c«pacity of tlio
ooIoMi, tbore exists » uQiqiJW poiat, 'AJJ^^^ of static oqullibrlos at ^d'.loh the ao-
•oleretion will be zero.
Thuei p/q a ^^^^ « sin-lta^^^ - ela'^ (1.1.3)
A plot Of Ojj^^M* «•«^o
^^ thrse caaee, p/q * 0,99, 1.00 and l.OS, is shown
in flgare 38. Tt la aated th s^t for the oaM of p/g • 2.. 00 the plot of u^^q^ vb.
«^, lAwa plot%»4 on lojiflog p^j^op, la « atr©l#t line, ^^b^n -.i loed of the atop
Tttlee fOrw, ibr ^.i<di P • for t < 0, is epplled to tb® yiodel «ith initial
ovoolMdBeee *^ eolassa will dleplao© to so-m r^nxlnosi deflection '^nd then return
to Q_ 8bA reboot the oyele o*«r »n4 oyer es long %a P ROte and there lo no deap-
ing.
Pheee plene <Ua«r«ie of u ». u for u^^ « .'^''''1
, O.OCO, en4 O.Cl end p/q -
i.OC end 1.02 ere hoea la ftsore 3^9, "rtieae dlafr- >-«j obtftinod using step by
^ MUinrpipt «il ISC a6»wb«3iOOTO id t{f# 10 •'Mil •»•« ftl ^IbtfiM, A
i Sjo M ^A JO "^N *\^
• ff « ^fi • ur ,0 • t t;
-M Mit lf»14» Jll fl/Mtft: OltB4» ID <«0t«" •«BiO« M^iOB • t^tlte* •^•M ,«!€•
ml ,aO.X tM 00.x *M*0 « p\q ,«:mm ••iLd* ^e«) ^ .ar j,,jM %> #oif A
to loXq ad^ ^. /^ • p\9 lo Mao adt lol ^ d# ft^foe -» ^'' •« omitt Al
fit* out lo teoX • amt ^1 tM%tm%9m » »l ««»fKr 1»X*D^ ao ibottoXq ao* ,^
XfMtr* !f*Hr Xo6«»ei ofS* o# t. tl ,Q > # irrt • « itel*f -tfA ,mt^ Mijrf
-^Bft an •! onotft hoB stoo t •• |0Oi ao mo *•• wo oXoto <Ui iwmwt tat ^P •#
•ael
• p\i D*; it(. iOO.O . .'o^
'*
ftf qoto tcXaif taBX«#Jo om «E»q>i>lli •«« ' .M ow^n al Aiodi •«• ao.i Mi«i 00*1
•t«p BQB«rl«sl lnt<w;ration. Aleo «te tUe rsqulnid to vm^ i^^ cad u^gy^ fi>r
p/q - 1.00 mi l.OS i« plotted «. u^ ia flgare 40.
TO d«t«r«ias th« voloolty at my lsol6te<J polct on the ptiaau plftao diecTM
without >7orfeiap th« ooaiplat© nwaaricaX aolutlon up to tfeat point tho on«rsr of
tho «78teai 1M7 b« iwod to «let«ir~dae thlo Telocity. Th« forc« st u:
Foro.- Ma^[f5^^ -Cs.r.-'a. s.«-*«.,^] .^^^^^j
1%« »«rk p«rfcm&«d «(p to mmm «u |«i
'^la'k - ^<^<^\i^^ ->u.5iw"'a 'fTu^ + (s»h''«,^)u.1 (1.1.5)
TMs work gpmn ioto ohans* of l£la«tt« mmfnw ^tli u - at t « thl» ylelda:
u.2=>
If ttjj i!s oho»«a |^«ftt«r tii«B i^^^^ for the psrtie^sr u^^ this M^^rAiraloa for
1.2 Dlaeasgloo
Hi* prlmry object here Is «> tfiseBss tl» efTect Initial crookodneos ha« «a
the «RSimM reenoaee »na tfc« ttsfw rm^TM to reeeh this naxloiMi reepo^.
For eueli a ease Urn «signitua« of Uj^^yj; «ad i^^, d«*er«iwe» ae u© (sseesure of
Initial orookftdiiose ) deoroceea. plot of u », a^ for thf ease of p • a.
oritleal loftd, glTos e straight llae ir$m2i plotted on loe-log pfi^er, figure 40,
for tbe ren«« of v^ betimes 0.00001 md O.Oi. Thea as u^ approaebes so doee
the velae of both Ug^^^ mid ^^^A* ''-^^ ^^ fissure 40 th« ttja© required to reaeb
Swj; '^^ **Beut ^' *>o*** ^°^ *^ *^ reaee of Uq betveen CC^OOOl aud 0,01. The
tine to r—tk these v>>auee, i^Mm plotted on loc-lo« paper, tor tfee ease of the
loed equfcl to the eritioal load glvee etrsl^t llnee. fhe tlrve to reaoh these
oluea iaerepseB as tfee laltt^l <rrookedoesa deoretvees. The ti^ in this plot le
expressed la tersaa of l//q; Aa the vrlue of u^ app ro?^<*!«a the tcIus of the tliae
ftpprociehtta <v> ; altoirliMS ^i^ftt ftor a p«rf«K»tIy 8trai^.t oolttMl «itb no disturbiof
eoIoffiD to d«fle<jt under tfe« lo«d uay be «.n iQp.ortsnt fsotor. For Uq • 0,01 the
tl.» to r««<^ *^]3»Y* ^ 8,7{l/\^5 enidi tii«» to rea^ i^u. Is 13«4(l//<f), (p • q),
i«li«rr»«« for Uq • D. 00001 tte tl ?« to ttam,^ is ^ICl/Vq") i^M the. tlf?* to a^,,^ in 68
(1/)/^). la studying a 8p«oiric oase th«se ticss tPDuId bo 00xT,p.'r^jd to th« load
pttlaa <ferfitioii, if thm load zjuIbo 'oration is lotsgar tbaa tfca tl«aa raqulrt^d to
raa^ %UJix ^^^"^ col^MBtt nould Tlbr»t© b«tw««a 0,^5. md u^ wltb th© Tolooity-dia-
pla««snt xwlatlon st^mL by tha i^eea pIma dlegrt^e, ftgura 3^.
ttuai tba p/q ri^io is Inaraaaa^ slsoiro 1 ths T^luas of «%^^ aad Oq^q^ ara ifi*-
oraaaad al^ova th« voluaa f:ir t^a lo?^d squ'^l to tha erttlosl lo6(S« A plot ctf
thaaa, ss wall k« t.*-;* ti^aa to raacfa t&sa© aiuos, la efcowtt la figure 40 tor p/q •
1.02, Vh»9» plots »Tm ao longar llnaar a^rea «ha& plotted on lofflog; papar. Thm
mrTt>8 toe %«« **'^ ^naat» *• ^ appraa^as 0, appro«kCh a horljBontpl aayssptoto of
p/q 11 • atn''ni. vo«ravar» at u^ aqiml to 0, this vaia® of%i^-g^
aj»i %aut ^^ fis«MW»-
lim^aaa, sa as^^a arookadnasa or aaoantrlclty Is ra.^r^a to trlggar off tha
eoluflHi. For tbff l*srgtft v»lw»a of Uq, reaga of 0,<X51 to 0.01, ttj^ tisaa rotjuirad
to raa<fc ^^itit ** «*•«*«* *^*^ ^o** ^* <»«*• ^^ *^'® erltleal losfi rod tho tici® ya-
qulrad to raaab mmrimm dafl^etion is laaa ^an f&r tha eaaa of tha load aqual to
tha erltiOQl lomd. A« u^ daoriHsoaa th© «!«»« cf the tl3«® daeraasee frosi thoaa
for tea aaaa of tfea erltlcsl lo«d, "ys slhnwo In fl^tmre 40; but tbo plots do slop*
upTwrd to tba laft Mtfe iKorassiai? tS;s>.« for daoraaalag u^.
«.5»;-i.?f» ISfrfiv; fifrt/lf-fl SffJ x-id* i^S"*^
^,- ^(W(c#C'/'J
-- ,t- t.
ot fi^fr h»£iT «wrt ^ *«•* Ail' vcft ofutf sctti dl c»Mir«Avi^ rnmitx/m tinman of >«Yl«r
•«ol» •* •toiq •^t «»« ;0* •«•!% «l sw>« •• •>««< i f-^ '
•f.
AV Tl. ^rm
'. atufly 18 ra«de her« of tb« ma* of an Inltielly «trp»l?sJrit ooIasK anlbJAOted to
aa axiel load •quel to or gyaftter thsa th© oi4tical bncklir.e lofi^. Aft«r the load
Th* eff'?ct of th« amtfblta^^ of the displtc«3»r*t oa th« laexlanijss dlsplj»e4m«3t fn<S
tr:« to rcftoi". ti^ls tMsSmuBi dl«p.l»aeci@&t is in b« Irrre8tij|!:««t4»d. A» a spaeiflo ax-
umplA %})• <NUi» of p/q K 1*02 in teJcen i^nd th» r9«p0a«e Invaeti^s^tedi.
(iT.ia)
"•or th» static eqtilXlbrlU;n -o; it Ian; p/q n » atn'^'^u.
Far p/q • 1, ain'-^-u ^ a for fill valuas O'f u > 0; asd Uj^^^j.^ Ibr p/q « I is C. For
all talnaa of p/q ^ 1, ttjj^m^j It graftt«r tlaatu 0. Tfes «ork doaa on t^;« siaas n in
COlDg from Uj^ to mam otli«r tt^ la:
Vv/o n K = Mo."
%(na.e)
Ti!l» lorte ^oaa ioto cfeasgliii ^laetic ssi«?jjgy of th« 'S&,'50, Tb«ttJ
V^a- ^ ^] " -NfT LL' (tl.1.3)
For th» «e««^ baing otudlad &• haa baas tafcea r>» 0» ..nd for any velu© of
""^ > "Wax ^%«x <S**«fralE** ^ ^1/ «xpt'as»lon TI.1.3 for yaloolty baaoeaa lsm(|ia->
;h< sa plRiie dla^rww h&ym h^wa <»oa«tr<iOt«d for Uj o^juisl to 0,001, 0.01, O.l,
0,30, C.r^ n?i!5 0,50, Tc- obtain tte® notion of tb» aoXtixaa « ao^arieal solution of
Ci. 3 \\T^ei. " S/rTtJ wsa aisaa uaiits tixo bate snsthoxj of rnf«rfflic« 3; end bata
taltaa »» C with r ?? t1»e intwTal. 'l»o, th© eawaor of tha aye tan «aa ba uaad
to datar^nina tba Talooity &t daairad points of dl aplaeamant • for botb aatboda
of ^^t^*'
^^W^-
90B .TfJ^'WC
.
-«•
.ftf a«^»t I;
l-- JO
w J
! f ? .f'
f^?*-«JtS .!.-** '^
"^^ #»•«" fcae tO < » It
(lU.n
x-^r
lOJ
"lu-iv - JJi'vwiJi- N^^|x>^4)M = «noWf
'jTmV - Jj'' v̂\ke ja - ^^.^ITTl^V
•->0 9 'J icH
•TV*
\o a»2toIoa lavlrwMw • mbIo* •;f# lo aoitow t^ bam 6e»0 «C .
MtoMm Atr>6 txf
38.
•xtr«ai* ••«i»'««r 9mm% b« u«9d. If i^aor* th^si Hit fn&ooity »t two or tfaro^ points
i« dieelred tte m«Mrl««i. 8i»tti9d l» tttratacwas over th« ua* #f ttM «ft«rtS7 of the
87Bt«t»i.
TI.2 Dlscuiscloa
Hi* i^«o plim» di iier«aMi iQ«1f«l#4 '^r^ ^^.'-ma ns figure 41. ^at* Hm plot i«
tt int. u« A pLst of u^,ix rs* U]^ is «lio«m is flfar* 43. A plot of tbm tin« to
rMMfei <^ax ^^^ ^«at ^^ ^'^owi la ft^rs 43.
A« i^owa bjr botb flgttres 41 sad 48 ItM iai«m&t of tbo iaitlal dlctitrlixmee,
'•#.*« l«a« t?isa ni]^ iiqual C-.l, h*s very llttl© eff««t oa tbs flM&xlmm dS»pl«^oesi«Kftt.
7hl» l9 born9 otit ^i^a th9 sssfell itagiOunt of wMriQr rwaoTsd frofr tb« iqretoai by tlM«
aaall«r 1 itiftl ais:>lKO<»t»«it« is «on«ld«r«d«
AS Oji^ (tt]^ < %wut^ ^^ l!3or#flui«d tbo Taluft of u^^p^ will d«cr»8ao, but lAll b«
sr«Bt«r tbea tij^^ut* ^-n^ut ^® lad»iMMid«Bt of t^. If n^ is |Er«et«r tb&n OQeaf
bat ls«8 th«n %||]( for »x-»-0» tbo 4<iHn«otlnn will not isier««c« wfbmx r«l««is»d, but
«111 dosrMA** enS tlM» gslaia^a^ «d«Mi vill b« flM una* »« a^ nooli !»-?• b«d to b«
to p2*o<Sa0o • ^uat ^^^ ^° tas^nittido to this Isduood m-j^*
If a^ is «x'«et«r thi^n ^wax ^'^ vti-m-0, tbs assss ivlll hsYs klnntio es^rcf
9duMi tiM ssXiinii 7«siiM>s tht «bpo dofle^otlan j»08iti«dn. Tbs ooltssfi «t 11 ti;<sa ooft-
tlnuo oa pe«t th« o'T»iHlbylt» position m& will repset tb« loop on the oo-.-osite
sltlo. \ osso of this is sbOMB la fl«:uro 41 f<a- uj^ sqaal 0,60 fend %a3i for «i
is 0.4S«. For oj. > Ug^^^ for Uj^ -* 0, ttj^ is M^^^.
f»r t&o osso of p/q sqvtftl 1.08 thf> tls« required to re»ofc ths at^itic equl-
IJbriu:^ posit loa and imi&mwst dofleetion is afcotm in fl^wro 43. Tblsi Is a plot of
ti>« Ts. I'littel deflftctlon aad Is jrirao on loss^-tos? iwpsr. For p/q •> 1.02 tho
tl^i» to rss^ tbe amiimm. dsflsotlon plots -sr- : .-,i;r*:ight lins ifesa the laitisl
dlstorbtsctes i« Isss Ifesa tlis stetio eqnlXlbriws '>^08itlon. /.t the r^lm of a;j^
•qosl to or inmattr th?»a ths atstlo o-itsjltbriun fOsltlon tte® tiffi;« r©d?ics» to 0,
es uj_ Is then '-Vfisx* ''^^ s*^*^ *^^ *^-* ^"^ ^- "^-'^ r^TC ..;e ,-t.:ti-- oquillbrlu^ pofii-
>r- yt- knm jif. UC-fftLBtt^fWta %i b<
^r<f :-J
, .u Mtani
«Xft ^* ^^ ""^ ^
»'•"
i-(R'*rt P' ?'a >-t I? E r « J'lifl^Mk^'
•
*w«wii aMia «i#a*v
-,-*
^ «^ft
••Kit «» upo
tloa la «l«o abom In ft^r* i» for p/q « 1. 02. This «»•« aot ploft m • ttrnii^t
ItB* on ttM Xog-Ioc ps!F«r. Howwirwp, fbr wilsiofi of uj^ ^ **aaut» ^^ ^•^ ***• •
•CMBStent <5!ff«•«&«• of ti^an fawn th» tSrt# r^nuirisd to f«»<^. oiaxiistKt deflection.
In thi« e«ai» 1S>ti tui« })m9R ^i^mi :«t to n^.cm the Icfliienoe of the Initial
41«turbtta9o -^u th* otximin dofl<9etlo;i an<i ti^tis to dl^IftO^ tli9 «sA<mAt of tha ftxitiL
load has been tatmn en only X.r^2 of the ariti^^l load mvl tM lsr®e defX«e%i&m
th«or:(> u»9d. ^«c thifi ontftll e»oaiSEt of loai! abovo tte oritjosl load miritmi •
l«rs« dlff«mnQ* In th* «ii9<iflft of tb« a«flactlon.
1.0
fi.
1.0Ji
Fig. 1 - Stress - Strain Curve forIdeal Plastic Material
Idealized Curve
1.0
\/[/Vi
1.0 9/e^
Fig. 2 - M - Q Curve for Ideal PlasticMaterial and H Type Section
1.0
}]/lI
1.0Y/Y
Fig. 3 - H/H vs. Y/Y for P and a Equal to
^1
I =M
H
H
Fig. 4- - Pinned- Base Frane
I = tt«
77T7T
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27.
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nTA
nmry t«««rd ntcpMu «i« tern on lo^mAmt 13, 1921. in feirburn, r^ttorgie.
".1« high aohool ©duofitioa at C«Bpb«U Hl^ Pohool, Fairburn, C.eopgla,
in X9S8 «r. red 0«orgl« S^ool of TM^aoXogy In July of th«t y««r, «h«r« h*
ytudi^d %t«>v mnAtxMrliif QRtll June, 1941. To ^^aguat, IHl, hft raeoived an
-ilat??!»nt to th© ''., n«val ^ORda-^y end reooivod th* d«ifr«« of riaofcelor of
oi«QO«, tulth honors, fro«iai«b laatitiAlon la 7uim», 1944. K« hfis atrwd oontlsa-
uoualy la ttm TT.T'. )7fivy fsoa Atuwt, 1941, to tho pr»sent , nti& ««rT«d fxoa July,
1944, to 0«tt>b«r, 1945, 1» tti« i aciflo 1fe«it«r. no «Tit«!r#4 a*iiui»ole«r T'olyt»«fe-
nic Instituto Ic Jenur^ry, 1945, us « ??aTy in!*nduH(i stad««2t m.& rtoolTsd th« tegrot
of Bn^f»lor of Sol«ad« la CItII 'SA0ina9Vlmsi fror- th;^t Institution la SmptmUbmr,
1S'47« Tb ^^ptaeaber, 1947, ha treosforred fzon ti»» KttTy Lina to th« ^?i<iTy ^iTll
laglBMir C^srps «ad oontinaad hi« |Erc).da«t« ^xK^rlt st 6«DSMiI«9r Folyteehnie Institute
«!h«7<i ho rooMved th« 4^gr«« of ^jfestor of Seionea la Civil i^^laewpli^ la Jua*,
15^ • Ttot.. ,'an<f», 194S, to Jua», 1988, b« sorvad st T«y «at. Florid©, end Yopk-
tows, Vlrglnlo. At ^©y fast hA hed «ip«ri«es« la daslga, eonstruetlon, aalntea*
naom tad tPRnff-vortctlon, Tfe «wi h»ed of 1fe# Fublio forka D©p«Ptis»nt , !f«T«l
Schools, L!> urfRrw, Torktown, TlrgSoie, froa Jsnu'^ry, 1960, to Jttoa, 1952, In
jxtm, 1952, h« fi»t«rftd th« Onivorstty of llllBola ss a cra4«at» atndflet ia tha
^truetuTftl fyasMSilca oorrioulm*
Ha ••• alaatad e K«snb»r of T«u B«ts ?1 ^*a Chi :pallon la ©pterobar, 1947,
ad an es^ocli^ta mmAm of SlKo* Xi la Juoa, 194S.
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22999
j'hosis Stephens
S692 A study of the effect of ver-
tical loads on the dynamic re-
sponse of structures.
. ibrary
U. S. Naval Po<5tcTadxiate School
Mi»iiterey, Caliluriiia
thesS692
A study of the effect of vertical loads