STRESS ANALYSIS OF ~ABABOLIO ARO~ AND !HEIR DYNAMIC BBHAVIOR

Shou-n1en Hou

1'h•aie aubaitiei to the Graduate :raoultf of tile Virginia PolT'eobnio Iutitute in oand1dao:r tor the degree ~

JUSTER OP SCIENCE in

SfRUC1'UBAL ENGINEERIUG

2

Title Page • • • • • • • • • • • • • • • • • • • • • • • • • • • • 1

table ot eantent. • • • • • • • • • • • • • • • • • • • • • • • • 2

~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ ~ • • • • • • • • • • • • • • • • • • • • • • • • • • • 6 ReTiewot Li~ ••••••••••••••••••••••• 8

x. T'NI» and ~ ot Parabol1o Ardl99 ard the

'••Sana ot l..Uq Ccn:S1Um • • • • • • • • • • • • • lD

n. statical AnlL1.,.u ot strew and DaH.ng licueute

ot ~c ~ • • • • • • • • • • • • • • • • • • • • 22 m. 1'ha Nama1. .m ~ Detlect.ica18 ot

PaNboli.o ANle9 1D1elt std.14 IOldSng e • • • e • • e • e • 16

xv. AJipltcat.t.cn to st&.to Solut1ona • • • • • • • • • • • • • • l3 (A) For fwo-td.nged PaZ'abol.1c Aft.bee

(B) Fort ~ Paabolio ~ (Both with

.... and tunaUcnal ic.ttna m t.lngd,S&1.

amnorSJ.~)

v. o,n.d.c Dahav:1or ot ~ ~ • • • • • • • • • • • 32 VI. RelatSaw bet;uaari !lorcal ~ ale! T1DI

~ ot Pa:aboU.c Archce md !heir~

1n Free V1bratJ.tlla • • • • • • • • • • • • • • • • • • • • 38 (A) hr ~ Parabolic .A.rcbee

(B) rw ~ Pa:rabo.U.c ArctlB9

3

VII. ~ ~ ~ Datlec:Uana and

'r.lm PuncU.cna et ~ A:rdM9 a'Kl 1htd.r

~ 1n Prete ~ •••••••••••• • .•• 49 (A) For~~ Archea

(D) Fw ~ Puabo11c A&-ctl08

vm. ~ ~- ot ~ Panbo1t.o Azidle9 ••••• 57 IX. Nmar1cal fi\rarc>1 .. at Find end PaftboUc lrchee • • • • • ~

CGnclllld.on • • • • • • • • • • • • • • • • • • • • • • • • •••• 73

~ • • • • • • • • • • • • • • • • • • • • • • • • • • ?S ntb110£lllli:\Y' • • • • • • • • • • • • • • • • • • • • • • • • • •• 76 Vita • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 71

4

x - 'l'he abed-. oc ~ ~ 7 - '1'he ~ ot ~ COOi~

11 - ·~ ot C'd1

L - SS* JAnetb ot ard1

1 - TJUe cxmet;c1' or -'* B - 8t.reea nonn1 to the °"* ~ v -

e - Slope ot a ll8CU.ao

• - An ~ Ill 8l'Cll ~ ts- ai.gtn

p n - Caipt-' ot ]Olding !n -.Ha.1. W IA-1 direcU.m to the Vaoa ot ard1

,, - Collicndt ot lotldtng !ft ~ d1zieotJ.en to tho

'--flt arch

n - Rad1&1 ca~ ot det1sotlc:m

' - ~ OCl:;G .. at detlAoUm

- - BGMUGll ot a 8"t1cln G - MDdu1ua ot ~~ tor abear st.1'989

A - Crom 89ct.k'n .._

E - Mo61lus ot el aat.i.citq tor normal et.reaa

I - M•11L ot !nertS.a ot a 9Gct.i«l

y - Det1eeUaa Sn wrt.1oal C01CMllm4'

x - Det.1.ecU.cn 1n borbmlta1 oaapnmt,

s

• - i-. pr unit ~ ot ardl

o- - Unit. at.re.a in a aeotJ.cm

• - D.L..._ boa tJie ~&'Ida ot a ~ u - ~ flt a ee~ ot &1"dl

G - st-ra1.n ~ a '8'd.t langth ..,.,.... ot ardl

" - ~ ot hoe ~ ill -'Sal direot.tm

p - ~ ot t1'9G ~Sn rd&l. ~

1.c:.1 -~~

f.; - ~ ot tree vtbNtlon 1n ~ 4buaUm

q - ~ ~ heo ~Sn~ dS.rect.1an

6

~lTlpOOCTION

'l'h1a thea1.a 1a ca>.oo1"ll8d ld.tb. both the etatio and t\?MmlO ~

- ot paraboU.c arcbe9 with conatant. crw llllCt1on8. In th• d1'fcd.c ~. 8P9Ci al at,tct.icn 18 given to the tree v.lbraUana ot 9Uch U'Cbe8.

'l'he tol 1 Old.ng procedure 19 tollouad. The tlW ot the paraboU.c

..m Md th• loading aonditJJ:lns are a•ned and an Winitem&l ..,g-

mat ot the veil 1a takal - .. to develop the govem1ng ~

equatiarw tw the atruoture. The l.Unp mxl .tre ..... eJ;ll••-4

in tc. ot the Nd1&1 and tclgentJ.a1. 004aoneut. tor the U'Ct1 u1I.

Theee Jield the llt&Uo eol\&'lona ot st.ree.a and bead1ng mamt.e in

cl~·tOl"lle Then, the ditt"1'8ntJ.al. equaU.a:w "l atSnc datlec:U.one

Ind elope abangea cm 'bot.h cwld8 ot the W1nlt,es,al ~ are dewl·

oped. BoUi -. ot ditterctJ.al eqnmtima rel&Ung the ...._. and

~ &N t1nallT ccri>ined eo u to obt.a1n a general equation

tor elutio parabo1.S.c U'Cbee.

In ..-. •• the ~s .. or the general ourwd ~ are tunat.1Gna ot u., de.f'1ecU.Gn m..-...ta, m! ~•ae and ue de-

'"1oped th1"0adl ~ ot t\?MmlO equ1U.br11D. A •lddln re-

mon.1. ot lMdfng 1a ...... to -- the ~ '° v1brlto ~.

A method ot ~ Y&r1able8 tw puUal dltterwit.1&1. ~

1e 1l98d in order to pt, the ~ ot the det'lect.t-. .......

t'he dittermtt md ccnd:!.UcN glw d!tteront bomdary ooadlU.. and

dittermt ..ta ot ~. Since all the ~ Sn each 81\

_.. bol&ob._..., non-t.ridal aobzUcm a19' on1;r 1t the d8t.er--

nd.tmlt. ot ooattic1Gnt.ll 11 ~ Eactl epeda1 ~o

• ._CA qo.19 ~

·~ ~ .... pU9 .-i18Cl..,.. PU9 ~ ~ ...._ ~

nP' rpl8tft lltl\ edoq I ~ • 'uqo.ra Pl .uodeu ~ •tn uo

11tqt UMCl nq Jl.llCllll ou JO enin ~ eq mo • a; ay ~._l"[d&!9 twn

-au! MR~ O'i UM"S' MW ••t .... ~ wwi t..cn-zu ~ tpwe ~ pe.uppuoo ...

~ 'AlA .Z81ft0 •'ft pa ~ ~ tn'P' .. •tteq0.n

al"(OClWl&f JO 8Ptpt Cll'4 'ucrJ:_.[dxa ~ • M,S °' ap.10 UI

-suon -o8tJ8P •:J ~ IUOJ..-pell • 18' 9' puw ~ 011'9.L(p -n. JO

91U91DtJJMO -n ~ oi &*ft\ ss ~ .zermoa eiu. ewn ~ ..eq 1"911eq180 ~ • """'08 fiu:A*'t O'J1919 .lapJI\ llUOl\'JPtl'O

MA • -91ft an 8UD'J1tl.l<nA -.iJ a1 ~ Tl'nlUi 9\U.

~ ee.t,J JO 9apoll 10 tieqam

pe\piS[tal UI 9849 ~m.1 8ltn .ZOJ etto5'41t(OI JO .teqlill ~ Mf.t

.... -n JO 9UDl\lPUCIO pm JO '88 te'PtJd8 qowo JOJ peA'f-hiP S'S uonoUl\1

L

8

Moat ~ or &?'Chea or curved ~haw bee in-

teracsted Sn the cue or atat.io loading ccn11Uma. !he tunoUane or loading,~. and do.t'l.ection ,__.,,.,.~are e:ip:aa•l3id Sn~ at M•pumta parallel. to the ~ dXlt\Unate --. In tact.1

lllGot .... ot load!ng ld.tb 1'd.ob - deal in pria.ct1ctJ. probl.al9 .. act.-a1q dJnaud.o 1-l:1ne;a. Ho1•ver, a tar aa the author can a.temtne, there 1a Vfll!T little l.U.er&ture Cl1 the lllbject ot the reepmee et &1'dlll9

to ~o d1sturbancoh

Dr. H. ~iarcua (eeo no. 1 ~ OOiU~) bae tllde a bu1o 9tat1o

~and a~~ ot the bee~ ot qUMr.lcal.

lll'dtaa 1d.tb oamt.ant. oroaa eecUGn8. In that Wl4c, be oanaldered. tbe

pt'Obl.m b.J' 6411 1 H~lll \be co:pananta ot loac!1ng8, toNeo1 at dlltleo-

Uans u i-a11e1 am taneenUaJ. to the tiue ot arch.

1he ~o bebador ot bem!8 and DW.U.·et.;r buUd!ng benta bu

belsa d:18Q'1111&1 br o.r. a. i.. I?ogers c .. iio. 2 or ll\bHoCfttlil1')• The

~ .tbod or aol~ tor t.lnd1ng the ~ ot a at.ruo-t.ure 1n tl'9e T.tbre.tJm WllBd in tbllt, wen hM been fdapt,ed u an ln8t.N-

-" in w. thesS.a.

!he det1ecUm ~ ar pmiabolio arohee uatna the dUterm-

tial eqm.tJ.ane «iqnmed !n t.. ot ~ campanmita .........

M'ltod b;y Mr. E. P• Popcw (919e Ho. .3 ot Blbl!ogzep\f ). '1'he tw.te ditt8NDOI ll8tbod ptopoecd b;r l'da cm an1T gift 814D'Od.mte IObzt.Sai ..

The BNet Fq.vsticrut (eee IIG. 4 ot lllbl 1~) gS.w an ~

lnatrtDant tor ~ the l'9latJ.ane ~ the reJ.at1ye die-

.. ;o -nOUl\1 • • l*l•aaf.xa eq mo 'IO'N'L ~n-t 10 Pl'Pl Ila

~ •tqw'4P 08tw SJ U . ~ opnwl&lp a O'J11J111 .19p1m ~ .lM

-~ UJ ~ pM1.tm 10 P1'PI aino .Am ~ o-. •cqs-;ad ts .,_ pm ~ JO .IUCf\i1U Ml\ l80ftpD.t ~ ~ eqoa o~

• JO ~ 1).19 ~ '• JO ~ .. awrnn>tat pm ...,,~ luR40[

rot __.,,,. ~ i.eq lllQ 8'p8l(\ SM US 1*11' """°19 JO POIQ18ll -U.

~~10~•1119~01.l'\10~~

• tnort 0'4 89'fo& .--p edota ~ q ~ ~a .--.q

91f\ aJ -n t»'flOS..._ e.DI SJ ~ 9tl\ •OOQ.l\ ~ ~ uonnae . °' mno-llOZJ ~ 8l ~ atR OOUTS •Q(faqa D~ JO

-.... 8'A uo ~ meet Nt l»j!98CWIAJ Mt\ ~ 9l'ft UI

•.AOCfod JO ~ ..uu.tOJ ecn. >PCMP °' .. 08 ...

-mloo ~ ~ x ~ us sm~ ~ Mt\ ~ oi pell!\ maq

"" 'fD'90llfifa SJtU. •a.m~ pu.rnn • ui auona• c::wq JO ~td

6

10

~ A.'W.l'SIS OF PARAOOLIC ARCHI+3

Arm THEIR D?MAMIC BmAVIOR

I. TnACE AND nPPfWt'JS Of PNWplJC AR£f\R tJ!P U MSY!fTIPNS OF

UW?p1G CONprttONS.

!td.11 thee.la ~ a d19CUll8im abaut, arcbee haY!na puabolio

lh8pte. A-- the trace is u 11¥Nn 1n the F1guN 1.1.

Figure I' I

(101)

"'-9 H JllfJatUI the glwn ~ ot arch, and L the llaD l9lgth ot arch.

SSnce both H and L ue oomtant., thq can be replaced b;r mother aan-

111:.ant '• .. - 4H A--;, (laa)

.. JO UDl10UV -.. ~

-~ 'd PU9 a, (t) pie •qon ~ ~ ~ ~ susi-c vuonun (t) . l-'1 MW epe'q\ wpn U'J ~ JI01 PJR 8C\ 01 tnenlfilUDO IUJPWt q

·~ \'PS' ad ei-t ~ ••• ,'M ~ et(\ .. ecfot'8 q,. awnouw •'11 ui !uttJIOL ·~ lu'pzelm •i.1 pt.'W ·~ \'P' .-1 'vPNt 10

. ....._ \WWW Gift a qoa 10 ... ,,..._ Ml\ 0\ A(hO~ 1'o&JPIO[

Otl1 ~ ... cl ~ICJC·-Oil\ Al 1IDlft .,... -'•P'Ot Mt\ a; ..

"1'P"nft't'9 "'lnlftPCID MR P1W .ta~ JO..,._ etn \Win.,._ 8ll8tQ.

·~ Bl lpltf' ~ -Mt1 JO .. 8l qoa ei.oqa aq\ ...

~ ei qoa ~ JO uonoee eeuo *" ,. ,,.. 9ll\ _..., Oll't9 ltA

(cat)

n

12

:a. RA'We&L w.LlSiti Of n'Rlmi!l Ai!I> mw10 mmm OF PAIWpUC ARCJG• I.- • take a ..ii eeoUcm ot arci1 ae a true boctr, &1¥1 .a a .t.re.. ~ en it u *-t 1n the t1guN 2.1.

0.--~~~~~~~~~~~~--

!:I

Figure 2: I

Where the .,mbola ~· (a) N -~ normal et.re.

(b) v - red:l&1. ahear' at."98

(c) M • bmd1n{! llGltadi

(d) • - al.as- ot ~ A.

/VI+ ~els as Nr 2.t:lc1.s iJ.S

(e) d9 • chango ot al.ope chi& to ahange ot 89c:Uan f1'QI A to B

vith uo d18t:ance u. Far th• lltaUo ~ Vie total t0l"'Ce9 1n the nGll'mll di.NoUon . n l1lOUld be in h&lenoe1 orZP0 : O. a.noe the toll.ordng eq,uat.1Cll ma be wittma

( av ) d9 di.(• t!H) d8 V + - da Coe - - V COIJ - + I. • -·de Sin -a. 2 2 a. 2

+ N Sin 1- + Pn&t : 0 (201.)

Since tho lmgth ot dll 18 eo ..U, w can ocmsidel- it. ae a pflJ't, ot a

c1rcle. the di la alao a "lfJ1!'r ..U 9*lt1V • bmce an apprad.:ate

wsaptlon aan be ued eo aa to car.t.derl .

Sin a : 01 and eoe a : 1 2 2

SU'beU.tut!nc !n the esplUcn (%>1), w &Wt•

!- • +'rid&: 0

a_ ---·P da l\

(3:¥!)

(2>3)

t ebould &180 be in balance, or~r,: o. 11mce 1t loads to the toll.ow-

ing equatiml

(U + !i da) Coe 1!, - N Coe £ - (V + £ ds) Sin !! aa a 2 a. 2

• V Sin~• Pt.de: 0

Subetitutinc (D) in the equa\km (D.) • get.a

!da + Ptde: 0

(ft ... ; : - pt

(204)

(a:>s)

For the IJtatJ.c q.d.librilSlf th• tot.al. bending nm•& or all tcrcetf

about MT po1nt 8t.cW.d be in balance. !fare• let ue take the beading

1D111ltr ot all tho toroee about the cent.ml pc:dJ$ c a.a dlOWn 1n the

t1gure 2.1, or ,D10 : o. Thie )"!.elda

• (N + ~ da) Sin ~ (~ Coll ~ + N Sin ~ (9j. ~

• (N • Ea> eoe Si1f C1'f ·sm 41> .. n Coe 1 <1ft am d!> • (V + f. da) S1n ~ (~ S1n • • V Sin ~ <5' Sin •

• {V + ! da) Coe ~ (~ Coe r> • V Coll Sf CS'f Coe 1> + (M + E, da) • M : 0 a.

Subllt.it.uting (D) !n the oquat1an (2>6) 1 w pt

•Vde·~~ +~da:O

(206)

(3>7)

Xlhdnate t.he ~ order tcmm ot the above eqwstJ.c:n lnl JWl'ite 1t

1n a "ftll7 .,.,.,.. ., aa

! •V•O

F:. (21))' (2os) end (200)' - get,

V : -fosP uda

N : •.fosPt,,da

K: fo~ . . It ,. dittermt.1.ate ~'le eqtat!al (:m). and eubet!.tuto the ! ot

oquat1an (203) 1n 1t., the real11.t. 1l1lJ. be

(D)

(3>9)

(210)

(21.1)

$ + Pn a 0 (212)

Tho~ ot (»J), (21.0), (211) and (212) IN qld.te accurate 1lblll

th• ardl S. llha1 ,., aad bat.Ii Pn and Pt are not, equal to ISGl'O.

Since - have UMd t.he ~ &1'C length • to crxprea the ~

~ V • 1, and ~ 80 w rust, \r,r' w omp.it,e the value ot • in av- glwn . -1.\11 ot X.· ~ to the re.l.aua. et ooord:lnatee, w know

c!a2 11 m.2 + ~. [ l • <i>2J a2

•.tcpa•....,. MR

'[\'It w.p tmO .. '(nz) p.19 (at%) (6<2) ...... Mn UJ • ~ mq:.

(m)

• Bot li • (t!'r,rl • 1r+ • X) lot+ • ~ • tr i' a s

z!otlf-:

(Z Dot • t Sot) + • : <lf > Sot + • : O

0 • ( ~ 0) :Jot + . 0 : 0

80\»q •o • x • o : • Mml •

O•(rfi~•tf ~ ••)30t+• rftW•tfi~ Jq) ~ • '[ r"J= • 0

xp ~ + tfxJ: • ~ ~ *'.II • t : t'P

zp X',,f! : '1' 901.9q

rx = .1. st ~ eitn ui lp.l'rl O'J [OCl&.a--ud '° 809il'\ aaA\2 eq;r,

16

m. pm HlAL A?lD TAjlG1~:mL mlHSIQ! OP PAP.looyp N?£Wj! m1tm

sttm IOOWp

F•1 let, U9 di.NI .. the ~ probl-.

O~-----------------X

F/gure 3 =I

. . In t1gure ,.1, let, ... denote

(a) ~n • rad1&1. ~

(b) ~. ~ d1apl&eamant

< c> I - rotation

et a ll9ct.1an.

The relat1.on betMees\ st.reaa and st.rain &JS to ll1GA1" 18

~:a

17

(~n + aSn ds) .. bn +ht Sin ~ + , de : Vda a• 2 GI

~ v .. -•f:-da GA

Ae we lcnew, the atreat ad stra1n rel at.ton ot nonal. etl"G98 1e

stl"'lUI - .. S'Git-1 - 4ll

~ B 1a the •iulue ot elutJ.citq tor normal streaa. Hmoe

~he Nlat.S.ve ~ displaoemnt; or d8 : 11 We oe en vp \he eqmt.t.m tor ~ cU.apl.&oemmt .. ton.,.,

'9 tor taa re1at1an of l'Otat1oa am bending ·mr:nent.1 wa knoll

(3ol)

(~)

the 1'91.atiVG rot.at.im ot aoct.iclus at the fMo ends Of da : !L de EI

1'he191 I 1a the m:.rL ot 1ncrtJ.& ot the eectS.cn.

Hmoe, w mm ..t up tile tolladng equation ot rotat.:S.cna

~ • M •> ... , = ti da

- .5111• +M -- ds - ?!!' (lo))

•mndao •UOO a~ 'l\l" (90() puw (~) '(~) StOl~ 4J.111A Im 191 'MN

(OU:)

(SOC)

(LO()

(90()

('<>')

(fJO£)

D -~ -.. -...-i H u'?'f!

~ -(LOC) O\Ul (90C) 911\~~

o:~ • ..t! fl' u~i'

\al oi (410C) ~ eqi ~·1na

D sp w•= ii

---ep

o=-· -U~p •• eq _, (COC) pue (ZOC) (to() ~ eqi

'eomtf ~ 1SU .l.IM .:I.OJ ~ eq wo .llKn •w ,..... IU'AJUICl Ml\ p.11 N 8hl1S t1llMOU "[9TIJU •Q\ .lq ~ 88CMA 0\ pa.n&.mo ..

t-r-e ..ZW A ao.IOJ ~ tw"A'M Mt\ ~ 8ftP ~ •tn ~

19

A A

Figure .3 :2 F/9ure 3 :3

In t1gure 3.21 ~ I 1a the rat&1cn ot eect1Gn A about, the neutral

ala ot the •aUon, the ~ ot .. ct.ion B v1U be (- + ~ d.8). . da

SSW da le Yn7 wl 1 1 w w oanaidel- the det1eet1an ot B due to l'Otao-

U. enq wUl be ' eta. Th19 • oan .,. 1a al.,. Sn t.i. noaw.1 direo-

Um ot a mt ..utat.es \be .sn part, ot no,.i det'l.ecUm. '1'be

~ caaeed \tr I or V 1n t.he --1 dbffUan ls ..U ~

'ld.t-h thia ettect- and hmce ce be neCJ.eated. ao that, w can write

... df>g: - , da

~·,:o da

1'hia 1e the - equat1Cln .. (,30l..). a.., - OINd.der the oounter-cloak.s.• IC.U... and IAll&lte at. the hont tw ot ll8ct.icm (eectlan B)

are poe!.Uft•

Nw, let U8 look at t1gu.re 3.3. 'l'M detlec:Uon ot B due to nora1. at,1'988

N anlT vill oanst.1\ute the -1n ~ ot tangent1&1 de1'leet.1on ot B1 and

d" - lJ cb ot.·-AE

dht, N -= ... d8 AE

Thia la the - equat1an .. (305).

Since the detleoUon due to the llhear1ng etN98 1e '¥91'7 ..u u aca-

pand to the dotl.ecUarl c&l..S b7 nona1. etre• and -.ni, - neglect

l\ 1n prut.lcal applicaH,ons.

ot ocuree, 4118 to the pr~ ot ela8Ua uatedal.a,,.. knalf

~ • M ..a .. ·-·-"W EI

d!J M --·-da • EI

!he aboN ecpatSon ls ju8t the - M (,306).

How, u - Wilt '° 'tll"i\.41 thm in the d1Ncst1ana ot x and 71 ....tb1ng

'tdl.1 happen .. fol, ....

Dm to rot.at.ion I ~

.6 h7: • <--.> Coe. : - ((Ida) ~

A ~X: • (-ib) Sin 8 ! + (flda) !?. da

(,:U)

(312)

A b 1' :: + ( ~ da ) 81n 8 : + ( • ) & (313)

A £, X : + ( N da ) Coe 8 : + ( Nd9 ) dK (314) It tr 3i

2l

!he tot&1. ett'ect wUl be

d ~ 7 = - " c1ae • < h da > f; :-"dx+!... ..... ,, M "Y

d 6 x : • Cl de) ~ • ( !.... da ) !!. da AB da

b to the _ _, oanvmt.1on ot stsna, - know

Ml M :C.: ·-da EI

.S , 18 a1.llO .qual. to ~ : , (x). It. la a ~ ot x onlT•

Henoe ~: a.t ~ • ~ ~ da ax cl& 07 els

: <!.! ec.. ax

Dltt•~ the equatJ..an (.'J.18) to get.

h • N dJ.r -=-·--dill dx ~ ctl!-

(315)

(3].6)

(317)

(318)

(319)

8ublU.tute (317) into (319) to ..

4S 2 M N dJ., _,_: ·-Seol + - -~ EI AB~

(320)

hoe the equaUcn (316)

d~ d.1 N -=·--·-- dx AK

. - (321)

'l'he equaU.orw (:s:r;>) Ind (321) .. af.mtl.., to the wl1 knollD bedc arch

detl.ectJat eq1at1om iv r~. Ee •· ropw.

Now. let ... app1.T the lltat,ic equatiala that - haft tound 1n eect.1ana

n and m to ~ parabol.io arahee and tbad-seod parabol.1c

(A) For two-h1nged Puabo.t1c baheet Jt~o V=O 'P:;: 0

~-l = 0 ~" = 0 M =o

.!:I

F/9ure 4:/

(a) cw&• With Ccmtant, ~and TangmUa1. Imde

(Or Pn lrd Pt, UV oanBtant) •

Froa the eqqaU.arJS (20.3) (3>5) and (ml). - a1."'8d7 knowl \

fi-: - pn e: -,, and ~-v:o

Henoa1 for OC'INJta'lt P 8 and Pt,1 w awt. v •. Pn ••cl Ma• Pt s + ~

M : - Pn .;. + Ci • + c'2 2 ~

(401)

(402)

(403)

Then, trm '1'8 eqnat.iaw (30l.) {)OS) and (306) we~

knoWt

"' M -- . -da.. ~

d4, u and - -~ da ... Al~

Hmoe1 aDlt1.tute \he M and V ·: that wa - 1n the equatSAlw ;he .

(/em) am (403) int.o,.above equa\icnl,. tbe7 beoata

l P. ., ,jl , : ~ ( • + • Ci, 2 + C, a • C4 J (liOI.)

l r.a4 ,,, ~ ~n : if l + T • Ci 6 • C, 2 • c,. I + C,) (I.OS)

l Pt,~ ~t, : m ( -a. C:i •• 06 J (406)

Now, w ._. tho boundm7 oami.Uana or \be t1IO hinaed a&'Cbea

as lhown 1n the fieuN 4.1 to llOlw tor all the ~ in

the ... equat.1ana.

For V: 0 ate: 0

FarMsOata:!._

~<St, : 0 at • : 0

For~, a 0 at a a •e

For~, : 0 at, • = ....

For': 0 at, ~ a 0 C4 :.O

For~ D • 0 at, 8 : ! 8e C5 : i P n8e 4

(tor O<a<+ aJ

(tor ... <.< 0)

2S

I: t- ("e • 2a) tor O<a<+ •e N: ~ C-., + 28) tor• a8 <e<O

M : !!!, ( .. 2 • a2) 2

~ n : ~ ( rf* • 6 •e 2 ,/l + S •e 4 J

~t.: 2,,~ (8e .. a)• fol' O<e<+ •e

~,: ~~ C•e • •J • tor - •.<a<o

(b) C.., 2t Wlth PtanctJ.cnl Hol9l. an! ~ I ....

(""1)

(400)

(409)

(UO)

CW.)

(412)

(413)

1w, let. ,. dieom a Id.rd ot loed!ng wtd.dl la Sn ~

ot .. Suppoee thC'9 ia • ~ 'td.th oamst.111' m&Bnitude p par ll'd.t l.algtJi alanlJ ~

Figure 4:2

. P0 : P Coe 8 Pt. : P Sin 8

r.oa e = }~ e =A .\;;;z,, =}1.\'fJ}x>2 Shill: f!i cae'-" : J1+1e«.2e : )1.\ ~>2

""~ At•: O, I: O,,..eet. Pn: P ad Pt, a 0 . lJf At•• .. x:;. !lint:~ a r~ l!it'

PL 4 HP

t'he tUl:wt.f.cnl ot Pn Slid Pt within • : 0 m:I I : '• will be

the oont.btQoue f'wcUaw aniS Vat7 "'9tb t.he lltvtpe ot. arch.

1At\ the t'l.mot.1aw 'be

Pn s P (l • J.i,P) atlt Pt : ~

tlhel'9 Ii am ~ aw OClnlltanta.

At\ep aubet.itut:lng the ~ ot 'n am 't at • : 0 anS • : ..

into the .,.,. tw ~ - get L l

K1 a ( l • j L2 • l.6li2 ) ~ (414)

~: ;up J-. (L2 + lQ~)

Proa the equat1aw (2>3) (20)) and (2»), - vUl get,

! : P ( K1.ft • 1 )

(415)

Gr

.

sn- - -~ da--~

sn-v:o da

7T

v : P [ t1ri3 • • • cl J

' N:·J12.3/2+°2

H: p (Kl,/+ - !. +Ci_•+ C,J 12 2

(416)

(417)

(418)

'lben, •abititute into tlae equaUA:m (3oi.). (los) and ()06),

to get,a p 11-5 ,;J .a

- : - ( - • - + Ci - + C,S + c4J (419) EI (IJ 6 2

P K.a6 J. ,,3 ~ < • -... [ • ..a;;_. + - • t"- - • C3 -0 n - ·~ )l<> 24 ~ 6 2

• c4a + C; J (~)

~t : ~ [- ~~fl • ~ + c6) (421)

The bamSarY ~ .. ~ 1n t.llO ttcure i..i. . ,_ v : 0 • • : o, got, ~ : 0

ra.. , • o at. • : o, Git c6 = o ror-:o a e:o, ~c4 :o

< 2 4 Fw M : 0 at, • : •-• get C.. : i • E~al v- --;, 2 - -

.... ~, : 0 at • : lie~ «"" C:l : fs "2 .. 3/2 Fer an:: 0 et,.:~ aetr c,: -lirKi .. 6. h •• 4

Subet.ituti~!nk ~ oquat1an1I (416), (417), (418), (419), . (42>) mid (421.), it boccw'el '

y : ?f ( '1,;. - ' )

B : ~ ( 2 .. 3/2 • ' ~/2)

M : ~ (J+ • • 4) • ?. (_a ... 2) · 12 9 2

• 21+~'3. ( ,/+ - 6 •• 2 a2 • s •• 4 )

a,: !a_• C•.'/2 - s'/2) ~

(J.22)

(la:J)

(424)

(425)

(426)

Tb• abcne ~ ai. ettecU.w tflt' O < •<• ... S1noe

the l-SS ml the ettape ot \be vm &N ~cal. tw we

- 8e < •< o, "ldll baa the ... tom ot equat.io'ia.

(B) Fol- Fbrad-d P8NboU.a ~

!he ~ oancHMo18 1d.ll. be • lhcAe'l in the ~ 4.3. ¢=0

'!' =o ~,, =O

ci-t =o

4=o V=o

Figure 4:..3

¢=0 ci,, = 0 ~-t = 0

29

(a) gpgg J.s W!.tb ConatArit. !ic.-1 Sid ~ I48dat .. * ~ •

ham t.ta• equatiam <401>, ci.m>. cw>, ci.ot.>. <405> am . (llJf>), - ~ knaws

Va• Pn a+ Ci

H: •Pt•+ Cz M : • '\ ,;l + Ci 8 + c3

• l p ., a2 I - ~C-T •Ci r • c, • • C4)

l . Pn rJ. iJ ,il ~n : ?f' (+ £ - ~ °6 - c3 r • c4a + CS )

l . ,, .a at,. ili(- ~ • ~. + c6)

Forv:o at aso ,_.,.o at aao

For ~t. • 0 at. a : 0

For - a 0 at. a : 8e

,.. a, = o ... = •• For ~n : 0 at • : S.

11moe, tho rewlte 1d.ll be•

V: • Pn I

?J: ~ ( •• - 28)

c1: o C4 • 0

c6: o ,_._2 c,:. -E:'

c - !t.!e. 2- -r p 4 c,: i ••

M : !t ( 8e2 - ,,,. )

(427)

(428)

(429)

Ulaml~­

'(lZ'I) tm9 (~) '(61"1) '(st'r) 1(1.1') '(ntf) ~ MA 1IDl4 . . . -

dH'I

i •• ( rJ11t • 11r -t > = ti T.. '1 ..-.f ':x•'•--

( 'l!li • t)• a "• M Piii ~ft ~--8tA em • 18illff

(tt'I)

(°"1).

. ... 1 '[W~ pa "[9'IGR ~ 1RTA It iii5 (Cl)

.. m .. •(• -) T: ·~

(9"1)

(4K"J)

(zt"r)

• (1iff • rl' .. ) ~ ~ : '~ fJZ oot

( -r. + '• 'tJ - ., 9 ~ Z't ..,, (1)( :er

-e9 • t9 --• -• --) ...,.. • u~ ., •• ti c·• ti' vat1 d -

.)9 c• )oo CtJC•z• 1 •.,•-.,as tij:H a "'1

(7J~ S • £/C SC:) °!i' : N

(C • t'~) ..i : A

llftO([O.J ft-~ ...,,.,.Iba aA0qV 8lft IGOUDJI

t/,··; *1: ~ -"-• M --~o .,.. 9 .. tJ -

09 9 •.'L --;; : Co ., -x c

0 : 9{')

o = 'o 0: to

9a:a19 0:,.-1,

o:• 1Bo:1~aa o:• .. oa-a4 0 : 8 111 0 : A .ZO.:J:

V. p!NAHIC B!fAy:toRS Of CURVED ~·

Now, 18' u look at ngure a.i. As• 1cnow, the l'98li1.tant. toroe alcna 81\Y d:S.reotJ.an ot a moving boc\r J11J8t, equal. to the product ot _. ml

ac:oel.aNtian ot the boctr• Let. ua lltJIT'I the torcea 1n the .-.-1. d1rect1Cln

.t11'9t. It a.wt be

Pn da + ( V + ~ da ) Coe ~ • V Coe ~

+ ( N + ~ d9) Sin ~ • N Sin ~

: II da 0~

where • le th• .... pep 1Slit length ot th• arch.

Let. ue ue \he - awz-cb .. wore, and ..... ,

eo.!!! :1 ... Sin. :o 2 2

~ the equat1on (502), - pt,1

P + il : • a~! n ae at

(SOl)

(sa2)

(503)

'l'hen, let. u eum the tGl'C89 Sn th• tangenU&l dS.rectJdn. It. aUlt, bet

(504)

Pt. da + ( N • ~ da) Coe ~ • N Coe ~ • V Sin ~ ll ~ ~ - ( v + aa da) S1n ~ : a da ~ (505)

il ~!_~ - 'i • aa : 11 ::z at

''

. Then1 let ua t.alce 1DW1t11 abot& the, cmtnl point Ca I:. M0 : 0

or veo.~ da. M -(M+CU!. da):O "' a• s~ the ~ (So7)1 w pt.a

u - ' as -

-Hiis SubeU.tute,.!nto the~ (503)1 -t:o ge+

p + ~ • II a2~n n ~ - at.2

~ a26 ~ : • pn+•~ or

!:J

oM Mr ai"ds

(S06)

(SO?)

(SOS)

(509)

(SlO)

New, l.t ua 1nnstJ.g&to tho propert.1ea ot the 8t.rengt.b et nater.Sal.

In \be figure s.i, the 1nt.en1S1.t7 ot normi 11t.ree11 produoed b7 tm blnd-

s.na ... it. M in the eectJ.an A wJ.th a d1stance • trora the neut.ral aid.a

1a

a- - !il .+ I (SU)

-.re w ban ._IDDd the bmdinc -1\ le poe:l.Uw. S1nOI a poeitive

cbl1limd s 1dll oreate a tcsll• or paet.tiw nolml etftee1 benae w . pat. the poaiUw sip cm the r1ah\ hand aide ot the equat1Gn (SU),

. "'8N I la the mmm1t ot 1Pet1;1a ot \he •ct.1.m 1.

'lhe d1spl ~ at that. plaot la u. md tdll be ( u + eta ) at, the

corsellPODding p1aol ot the eecU.al B *ioh baa a lb1tt. of da. Henoe

the =- at.r&tn ~ da 1s

1'1is

{., : ' u + ~ .. 11 : f:· a~D u: - • aa

""'9 a2cS - :.a- ... ~ - u - aa2 St.re• rr • -l-

11: st.ail> = Z ~ • aa)

& M: - EI a-~

NMIU.tute ;mq t.h• equat1Gn (SlO)io got

a2 ~ 2 -a,1- ( - EI a a) ) : - Pn • m a a !i

(.512)

(Sl3)

(514)

(51S)

'' -tMf-

Since w have U8'aed" t.he arch la made ot the - mt.er1al. t.luOu,£h

the ¥b01.e lcmgth and has tJ-.e ... C1"0U eect:lal area, t'heee 71eldt a~ ~;

EI--!. + a ~ : pn ar/+ at-2

or

now, i.t.. • .,.,.. th1a '4th \he etatio oandit.t.orus.

Fl'tll tbe equatS. (3.>3) ot pl.89 3•

MW haft baa the equaticln (200) ot pig9 4, IO that.I

p :- ~ .n a,/t . ·we

then traa i:h• ~ (300) ot page u, get. "

M: •EI a2h; a-2

SubeU.tut-e !n the ~ (Sl.7)~ ~

a~n • !a -o~ EI

(516)

(Sl.7)

(Sl.8)

I.-• CGlllplre_ th• etatio equation (Sl.S) to th• ~ale equat1Gn (Sl.6).

It w let the aocelerat.1cln 1n the equat1an (Sl6) equal to 181'0, ..

a2~i : o~ 1t 1d.ll be cbang9d t.. cb'-.ia OCllllltian to at.uo oaadl.U... at:

an4 theee - eqat.iana vUl be !dentJ.cal.q equal.

Then, let ua !mestJ.gate the properUes ot the ~ ot -.nal

aga1n. The lhorten1nts ~ da 1a equal to the Nlatiw tangct1&1. d:S.a-

pl~ or two er1.1 eeetJ.ona or de, •

N : AE a~t aa (Sl.9)

An41 thtl clittermoe ot ~ DIDll9l ~ flt t. -1 eact..:lcN

ot da :la equal. to the ~ 1-1 alq da VS.th ~ .s.,,i, or ( fl + dH ) • H : • Pt d8

(SZ)

(S21)

~ 1n the ecpat.1a\ (506), t-o pt

't • .AJ~ a2h : • a2~) at

a~ aa~t- Pt - - __._. .. --a.a A.~ at.2 AE (522)

Naw1 19' • OCJil81'8 ld.th ~ statia -.!S.U... h9a the ecpat.1a\ (*>S)

et Jllge '· - know

P - CUL t - - aa (5a3) '

- aa the apt•-.lon Sn tlJe equatSai (520).

06 N : ll ::1.. aa

1'd.ah 1a .- we pin equat.1ca (Sl.9). 4?hen1 dltt~ the

equat1an (S24) and ..W\ute 1\ 1n t.he aquat1m (S:O) tie get,

't. : - u ~

(S24)

(S2.5)

IA • ~ the stat1o equat1an (52S) v1th the d;vnand.o equat1an

(522). We 1d.l1 t!n4 out that. u ... let. the ~ ~ in the . ~ equat1an (522) equal. t.o llGl'O, 1' lf1ll be ahanged ha the 4'•*40 a.lditim to et&to ocntt.U. t.a-uateq, and t.t... tw equaU&m tdll

be !a the - form. 1'bu81 the eqt.U .. ot (Sl.6) ml ('22) are the but.o equat..iaus tor the

~o tlam&1. oedl l&ticm-. S1ilOe all the dim"'ratt·ves ua ~ on the

nature ot 8IV' curve, and. • parU.Cl11arq buad m the l*'9bOUo pro-

~ tAt e•1n '516) & cm> an stMble ror •11 qgrnd

~ '1'b.e ~thing~ tor~ tll8le ~to

l*'9bOlio arm. la to CIXPI•• all. t.be a 1n terml et tbe propnUee

or the paNbo1a, 1'here • 1a expreeeed u 1n eq1atim (224).

VI. ItU'llQNS ~: l~!IU. VZL,mONS ANp 'mm rot1rnqp 2[ lft.!J!JRLIQ &'!Cffit! ARD nm f1Ul?lllJIC£¢3 ~l lR.'i vmwwis~ 1"'98 ~ to1l.w a audde ~ ot exkm&1. 1oacsa. At 8U.ch

U..1 bGtll Pn an4 Pt oqual to nero. fkmoe1 the ~ (516) ~ti

a 4~ m a2~n - o ( 601) 71 •-;a at2 -tw, va vUl \17 te llOlw ~not the~ (@). 3uppo9e t.t.

llOlut.jan laa

~n : S (a) T (t.) (tal)

m... s (a) 18 a pin tmlotJ.cin ot • at T (t) 1a a PJl'9 tanatton ot

U..\.

Let, - .-Utute t.h• ~ .... icln (002) In \ha ~ (~).

It 1dll beows 4" 2

T ~+LS~: 0 rl+ Kt ,a

.5 1 a4 s : - 1 g : rll II S --;, f at

... P2 19 ..... rDSo '° be •--4 tor ~ tMo ~ ~

((OJ)

Fi. t.be r!ght hm1 aS4e ot t.bo equatJm ( !03), 1t am be wi\\en Ml

T (t) : c• Sin pt, + c- Coe pt

T(t,) :C Co8(pt-ol)

(fa.)

(!OS)

39

..._. et, en, c al4 are utd.trarr ~. Aa r.. th• left harxl Id.de or the aquaUon ( (:03). it can be WJ"!tten ..

toll ...

Let • 1ntroduce a rw ~ )'.. , ..s let,

A.4: ~ EI

-#lis:

Wltute,. !n tt. equation ( OOS). I\ beoameat

a4 s - )\ 4 s : o aJ.

S (1) : Ci Sin A e + , r. A a

{(i)6)

(lD7)

. °' Sinh~. • c,. Coeh )\. (lat)

Artei- the ml.Uplimdilon ot tho ~-T (t.) ~ ~ (•), the 811 ma eS.x arb1tzw17 --.U. The;r *"' <;_, C7 c3, c4, P1 ..So< ! ~ce /-t. Gin bet~ VMlt P 18 lnMl ant C bu been abeorbed in °t• Ca' c3 w C4• '1'hen, wa CID uaa the~ ornU.t!arJ8 ot the ardl an4 the

bMhSal7 ocnllt!lnl et Um to 801.w tor th ... oonatld.a. Th• ttmcU.an ot 8 (a) vUl. bave the follOld.ng ~ tor ditteren\ ~ oan-

ditSonaa

(l) Falt~ «1.:t to~· an: o. M an: ST. For CT

u., '1' u nat, alV!V8 aero, '° that s : 0.

(2) Far al.ope~ to woa It, .... a!2 a o. a. a:: : t ! and T s 0 . '. £ : 0 or S' ; $>•

40

~2cS (3) Por, ..tt equal to lllm>I Froa tho tOftlel' equat1Gn M s • BI . 0

: .. m 0:; • S1noe T 18 not. sero :. a:,.s : o, .. ;r; o.

(4) Poi- llllClle.r eqtal t;o eoroa hen ti. equation (SOO), V : ff : -m ~· Stnoe T • o~ •that, lt. .- be a:;$ : o or S"f ; g,

(A) For 2-td.nged ParaboU.o Archeal

'1'htt boundu7 ocnU.U... oE tba eblpe u. tho ~ .t the

orS":Oa'ldS:O.

:>" --=:::::: I :::::_:...... <::: ~ x

!:i

F19vre 6: /

Ve ~ S a C1 Sin--'• + ~ Coe A • + c3 S1nhA8 • c4 ColhA 8

hmoe s• = cy. CoeJ\8 - ~A S1n A •• ~A Collh )"'\.. ~S!nb A.

Stt • • CJ.A2 Sin>-. I - ~A2 C. -"• + Cy>'2 ~ a+

C4)\ 2 Coah .A e

Subet.itute the ~ conditla1a to ~ (609) +Ci, S!n.A .. + ~ Coe.>..t1o + C, SinbA•e + c4 CtMb>-.8e: 0

(610) • C1 SinA. .. + c2 CooA .. • C, 51nhAa8 + C4 ~ .. : 0

(6U) • CJ.A.2 S1nA8e • ~)\2 Coe.>--18 + Cy'2 M.nh.?\8• + c4/\2 GolhA.Se

:O

+ Ci_A2 SinA .. • ~;-.2 Caal\ .. • C3A2 SinhAe0 + C4A.~• 0

(612)

h tftv&l. 90l1Jt.1cn ot the above erpit.1arl8 are &Uthe ocmtaita . . Cit C:z• c3 am C4 equal to..,,_ It donntt main~. Now,

w \17 to t1nd out, tbo1r non-\r'1.val. 80luU.m u tollawa (..t. \l'lll

dete1'ld.nlr1' equal. to :mio)

• SSnJ-.•·

I A I = - Sjn)\ ..

-"2 81n1'• 0

+/\2 Si.al\ ..

How

• Coe"•e • Coal\ ..

.... /\2 Coe1'1fe

•A2 CoaJ\8e

+ Coe"-•. - "2 Coel\8 • •Al Ceo" ..

+ CoeJ-.•o

-1'2 Coe"• • - "2 CoeA.e • +Coe" ..

+ Coe,x.a0

- J\.2 Coe/'. ..

• Coe" ..

- Sinbl\.8 • • "2 Sinb" ..

• "-2 S1nh/\9e

- SinhA. ..

+A2 S1nbAe0

-A2 SinhAa • • Sinb>-. ..

+1'2 SSnb)l.89

•1'2 S!nbA8 • • Sinb1'.•.

- Sinb1' ..

- )\.2 Sinb1'• e • Sim>-. ..

+ COlbA ..

• Com" ..

• "" C.aeh )\ .. +1'.2~ ..

• Coab!'-..

.. "2 Caab"-•.

+ A.2 Colb" ••

• Coah ""'•

• "2 Coeh "·· • )\.2 C'ioefl )\.

8

• eo.tll'JJe + Ccal)'. ..

+1'2 ~"·· + Coatv ....

:o

•A.2 BmA... + Coe J\8e - Sinh)'..89 • Colh".. = 0 - "2 Coe)\.. • "'2 Sinh.A...,

: + SinA .. t + Coo>-.lo (1'4 !limA.8e Coah"'"e

• "2 Callh ,.,.. • + ,...4 mm" .. ColhAte}

+A2 CoeAa9 ( •)\2 SinhA89 Coeh/\ae +l'-2 Sinh)\89 CoehA•8 )

-1'2 Coel'-8e ( -A2 Sinh>-.e9 Coeh.A841 •J\.2 Sinh"-•e CoehAa9)

+ S1n.A•e { + CoeA .. (A4 S1nhl\le eo.bA .. +A4 Sinhl\89 CollhAae)

+)\2 ColS,>\88 (A2 SinhA.18 Coeh)\8• +)\2 SinhA•e Ceah·A•e}

• 1' 2 Cos >-.IJe (.>-. 2 Sinh A.•e Coeb >-. a8 •A. 2 81nb A•e Coeb >-. 89)}

•/\2 SinAae f + Coe>-.ae ( _,,,_2 Sbah"-ae Coll't>-.8e +A2 Sinh/\88 ~·.>

- C./\ .. (1'2 SinhA8e Coeb.Al8 +A2 Sf.nhA8e ~A89) -;.,.2 Coe/\Be (Sinh)\ .. CoehA•e + SinhA8e Coabl\ .. )}

•)\2 81nAa0 { + Co9AS8 (•A2 S1nhAl8 CoabASe •J-...2 atnli)\80 Coetva9 )

- CoeA•e {+A2 SJn?a)\ .. CollhA •• - " 2 SinhA•e CoahA•e>

• )'..2 Coe;-...llo (+ Sinb;._ .. Coeh>-.88 • 81nhA•8 CoohA8e)}

• + 16 A 4 Sin)\ .. Coe A .. Sinb /\•e eo.hl'-i•e : 0

W 81'4 81n 2Alle SinhA88 CoabA.89 : 0 (613)

.law knw1 A.41a not. sero. SinbA.•e la anq..,. at. A: O, it ia

lltaU.o oandltiane ~l"-•e bu a ftlu. ai..,. s;reatea- thm\ cine and

le nner .ro. llence1 the ClllT pcw1hle conU.t.ion 1a

Sin 2-"•e : 0

2~•e : Alf" (mere n a O, i. 2- :J • • •)

an a 0 111 the RaUo candit!Gn, becml89 it Jielda : o. Frm ( 61.,). - get,

)\ -n -

: ~ 1n (to6) ., that EI

(614)

(61.S)

..... °"' ~ Oq\ U'I ~ tslOU Oq\ .. JIUlJ -111'[ '"°" (819) ---s ., 't : u

(u'/J• \"d) 900 • ~ llO'J uo ~ : u~ amooeq ...,_ arrnoeuep 1W-lOU oin 10 ucrnmht ,,.... .. eq;r.

(u>" .. ,t\I) tlO'J • ~ ao:> u0 : "cu~)

., 0\ (l.19) pa (~) '(~) amnvnba MR ~

•u JO 9Gln"[VA iiat&JnP ~ ldtaqo ~ ~ .l.tu\ltlift ~

(Lt9) ( • --' '£ 't 'o : u) • ~eoo~:us

·---·~ •c 't oi t1ftlbo eq ptmql u p19 •cwa11 oi

twnbe 'o pm 'o •"to ~. ••~ , • • "' w 10'.I ea • ~ llllOO ~·

pm • -n 'lL1W '• :-g uis ·~ uoqw «n 10 8P'PI puwq

~IMft.&O.i ~=· -iw o:spm .. ,:• 19o:s Mmr8A

(9l9)

(a) Wlth c..tattt. ?lanal. and~ !Qadaa

For tJi1a oaee1 we have a1."*'1' 1'canS \he ncmia1. cWlectJ.an 1n

.t.at.lc oandi'1ml (o.r at t : 0) to be

~ n Co, o) : ~ (,;. - 6 -.a a2 + , 894)

: LCn Coe~ a Colo(n

Jn at&! ce, the wlocitq ..t be ll81'01 hmoe

~a~ C • ) L:. DJL. a\ •• 0 : • n Ca p n Coll 28e • S!no<.n : 0

Thia gt.wa g : O •

(619)

(620)

'l'hc, lA • aptnrl the equaU.an (619) 1J1'o ·a ~ c-5ne

sedeea . ~n (a, 0) : 'ii Ar. Coe ~ a (621)

: 'n J+se (rl+ _ 6 •• a,/-• 5 .. 4) eo. a..n7r • 8 24Eia " -ae

: ( -1 )D:f}.

4S

tl ,\ 511 (e,t): 1i (-1) ~-4

~ Pn•1c , Coe DJL • eo. Pnt &I n?n-S 2ae

n-1 6r. Pn& 4 or ~D (•,t) : 5° L (-1fT ) Coe~ a Coe Pt.

r.:t 7T n 2a D • . ., n : 1, '' S --- - (622)

(b) With Funct1m&1 Norml. and ~ 1-dal

Fw the cue of' Pn • P ( l .. Ki a2) and Pt, : '2 J the DOl9l.

de1'l.ocU.on ot ~ arobetl 1n st.atic candit.i.m 1a

an< .. o>: !:...(-1.6+!....J.+(r.~4-··2 ~ 3W 24 4

) .a .. ( k •• 4 - ~ •• 6 ) J (62'J)

and it al.80 '*l be v.r:Ltt.m in the -font u .

~n ( .. O) : Z Cn COii :.:: • C:O.o<n (624)

. alt, ~n Ce, 0) : z. Cn PD Coe r . S1no<n: 0

• 31no<n = 0 °'n : 0

~

Then, w can tap&nd tho eqmtJm (6a3) 1n Four1ar cotdna .series

..s ts.rd out. the OOllf'tidentt °" u tollowel

cn=t/: !il-~ •6 .~J++(~-¥->; • ( k So 4 - t .. 6 >J Coe s; • da

n-1 ( 6 6 : (-1)T r£ (C· !lt~ ) ~ + 61.ee4 (1-11># • ~1.!'I )

n7rr7

(625)

n: 1- 3., S • ... •

!he tean:lat7 OClnd1t1one 1ldll bes

, ( ... ): 0 , (19 ) • O

~n ( ... ): O ~n C-ao) : 0 .. •< ... > : 0 (ta&)

•<--.>: 0 (627)

••( .... ): 0 (628)

··<-.. > : 0 (f;Q9)

0 x

Figure 6:2

Ci SlnA8a

- C:i. Sin)\ ..

<;_J'. Coel l\89

Ci_AColll\ ..

•ea. eoe~-. • c, S1nhA•. • c4 Coab"'••: o (630)

+ Cz CoisA88 • C3 S1nhA.S9 + C4 "-hA .. : 0 (631)

.. t2J'.Sinl\•e + cycoanA... + C4>-Sinh"-ae : 0 (6,32)

+ Ca A51n >-•e + C3 A c.oeh >-. •e • CJ/' tlilm Aa9 : 0 ( 633)

or

• S1n1'So

- Sin;-. ..

+1'Coe)\.•.

•1'Coe "-•.

+ Co91'• e

+ Cos"-•o

-J\SinA9 e +f.,Sin1'8&

47

+ Sinh"•· - Sinh1'8e

+ ACoah..1'&8

• l\CoahJ\•.

• Coah"•· • Coehl'-JJe

+1'S1nhAS8

-J\Sinhl\a8

+ t.oe.;..tlu • SinbA.. + eo.i)\.ae

:o

• 1'$1n1'88 • /\ Colfi)\. •e + !'-Si.'lhA•e

+A Sin )'...98 +/\Cash-"' •e • 1' S1nb" 8 e

- J\.$!n ""e + J... Coai'lA•e + 1' Sinh>-. 99

• "sm "'e +" Coah "8a -l'- S1nb "'••

(634)

• )\.Coll,..... + Coe "lie

• Cea"'•· + S1nb;.,,ee

- Sinb-"••

+ Coa A"o • SSnh 1'.. + Coeh Aa8

• )\. S:L"l.A Ile + 1'Coeh7'8e + A.5inh)\.G9

+ Co9)'9e ( -~ Coeh1'•e Sinh1',8ll • ,..._2 ColhA•e

S1nh"s0> +Asm"•• (•;-...ru.m2"•• -)\.w,,... •• > +/"51nA,S0 ( •)'S1nb2"ae •J-..~Aae) }

+ Co8.AS0 (•1' 2 SinhJ\.89 Colb>-.80 -Al Siml\Se

~J-..08 ) •A.SinM9 (•!'Sinb2-"8e --"W"a•) +}\$iJ1ABo (+,1'Sinh2)'9e -)'...Cotil2r..'o)}

+Mo&J\•e{ +Coe.Ase (+)\Sinll2.A10 •ACoeh~8) - Coe J\ ~ (-As:lnh2)... •e -"'Coib2M8 )

+ J\S1n ,\ a0 ( + Sinh /\ a9 Coeh1'.•e + Sinl\A •e CoshA •e) }

•)\CoeA .. { • Co&Mo {-J\S1nh1A•. -ACoah2A•e> - C.O.J\s• (+A.5~~ •e -ACoeh2>.e0 )

. -/\SinA&e c+ SJ.nhA•e Coeh.A•e + S:lnb>-.. .. ColbA8.)}

: •. 4~2 Sin8>-.•o eo.ti2A.•• + 4~2 ee.2Aa8 sim2"•• • o or stn2 )-,. a. ~ ~-. : ~"' •e s!n!z2 A 8e

(63S)

!he ..:LuU.clna ot the abow oquaU.Gll «*1 give ua the_,._ etA•

Thell troa the"-'•• we can t1nd .-the~ ot tree~ \ta Sn the IW1. ~ ~ - aplatn it in the ~

49

ARClP AND T!!§IR P'Rml!!fC:W IN FRF.E yIBRmOmJ

irm the equ.t.ian (~) 1 we abeld7 lmow

a2~t. _ !.. a2bt : _ !.\. a-2 AE at2 AZ

For tNe Yib~ Pt, equP to uro, hence

a2~t. • o2Jt -~-- • 0 a.fl AE ai2

(101)

Now, lAlt. U t17 to -1w ~t, in the abcm equat.icn. 5uppoee the IOl.u-

t!An iet

(702)

-..,_ a (a) la a ~ .tuncU.an ot • and T (t) la a pare tuncUon ot

tilllt t. Tben, .w.t.i\ute ('102) 1nt.o (?01). It lMtcmaaa

T~-Ls~-o ~ Al at:I-

AB 1 a2s • l a2' • -2 -;; 8 ;a - i" atr .. - ... (703)

Mlere rf la a eanatMlt, rat4.e te be UIU'D9d tor th ... t.w aepaNted

ftriab1.ea.

rra the right ·hand aide ot the llq\at.t.an (703), :lt. CID be written u1

~+-2T•O at,2 q- -

T (t) : Ct Sin qt. • C" Coe qt.

T (t) : C Coe (qt,•"'-)

(?Cl.)

(?OS)

~.Lcf-s:o aa2 MI

I..t. U8 introduce • DfN ~ (J cld let.

2 - • ~ f3 -uq-Hmoe, th• equaUon (706) beCCIDle9•

Sf' • i52 s : 0

8 (e) : Ci Sin {!> a + ~ Coe ~ I

(?06)

(707)

(?t»)

(709)

wtm"8 c•, C", c, o< , c1 and Ca are ubitrvT camrt.fmt.a. ~ tlle

ml.tJ.pU.caUcll ot th• l\mcU.ana 1' (t) ands (a), the ~t bu tour arid.•

t.zv,' omiatante. Th97 &N Ci,1 c2, q and o<, a1noe t> cen be found tllben

q la knoNn and c hu been abeort>ed 1n Ci, Md c~ Then, ,.. om ue

the boundary oandiUana ot the arch 8Dd the u. to eolft tor \heM

cantMnta.

S1noe t.he rib ~ ~' 1a -.1n1T ""8med by the noiml llt"88 N

and th• ettecte trom ta. other tacta ue nagltgible, ~the~

bamd&l7 cond1tJ.cn to be 1288d 1a f; t : o, or s (1) : o. Aa tor the

al.ope, danec:UGn and bmd1na .-.it, - DIV' oana1de1' that Uiq p-

em the nor.l. det1ect1m ~ n «L°IT• (A) For "9-binged Paabolic Archee.

h boundary eandit.Sala are u 8hCMl 1n the t1gure 7 el on ~

tol.lad.ng pigee

Now, let, - .. the equatJJwl ( 709)

For ~ t, : 0 &t • : 0 C2 : 0 (710)

For ~t, : 0 at, a : .. Cl Sin t' •e : 0 (7ll.)

The \rival eolut.Sm ot (710) .S (711) are both Ci and ~ equal. to

./

§. -::;O t 0

Sl

=-----=-=- I :z:==.. 2<:: • x

!i

Figure 7: I

aero. Th• nan-tr.lv&l eolut.ian 1d.l1 be

I.Al: I 0 1 I : 0

81nt6 .. 0

or 81n,8 .. : o Thia giw.

~·· :· nrr n : 11 21 3 - - •

Fram (714), - get ~ nJT . -n 1te

art. - haft ••1118d ~2 : ii' rf 1n (707), hence

fEca2:~ 8e

«lat : D!I rM .. J•

(712)

(713)

(714)

Cru>

(716)

Thm1 let ua llUblUtute the f> ot the equat1m (71.S) into the equation

(709) to get.

... •

n : l, 3, S, - • •

For 8 : 0 at a a 0 a"l4 S : 0 at, a • •et we get C2 : O.

Sn (a) : 0n Sin !!.! I (717) . . ..

n a 11 .,, s, • • • 'lhe artd.tJVT ormet.a1I; Cra Clbange8 tor dU'termt ftluee ot n. Com!ne the equatim (705) and (717). w pt

(~t)n : Cn Sin l1ZZ I COii (41nt. • 0(.) .. Hance, the gtDllll'al -1ut1cln ot the tanpnt.1a]. det1ec:Uan -- be•

~t : ~ ~ Sin WL._rr a Coe CC\, t -o<.) (718)

n : i. 31 s, • - • . Now, let.,. 801.w the........, equat.1en 1n the tol'Gldng tMo _...,

(a) VS.th Cawtant. !--.1. an4 Tangmt1&1. U.S.1

FNll tho t"1'91" OO'pltat!an1 W9 haft alrwt;r ptm the tlnpn-

Ual det.1.actJe in atat.1c oandit!.cm .... 1n tJJ.9 equatim

(412). 16here

~ t (a~ 0) : i ( •. - • ) • 2A.I

a& bQa the GqUat.1m (718). ,,. alllo know . ~t (e, O) : "i en Sin~ •Coe°" , n: 1, 31 5 • • -

.. (719) P. . . . !1.. ( e • e J a : L c Sin a.!! s Coeot- n : 11 31 J • • • 2AE • n .,. •e

• a~., <-.o>

- :. ~ ~ClriS1n~ e Sincx :o oi -. Tbua w get Sin ol. • O or o.. : o. ~ the equation (719)

'beocnoel

(Coe nrr - 1) J . For n : 01 21 4 • • • 1

Fw n a 11 31 S • • • ,

n : 11 3, S • • -

Q: 1, '·' -('r.ill)

• sin azr • da -J._ Jl aSn !! • u] .. .. 0

(?Zl)

(b) With ~ ?*'-1.. anS ~ Loade•

Let WI ealect. tho _. tunct1m U W haw uaecl in ~.

,._. Pn : p ( l - Ki •2) c'4 Pt : '2 J.. \fe haw a].rgacty got.ten \he t.angentJ&1. dotleot.ion !n the -..Uo

oandlt.t.cm aa Gbawn !n the equaUon (426), ~

at. (e~ o) : ~ • ( •83/2. ;/2)

. M \ (a, O) : ·On Mn L8 a Coe n : 1, 31 5 - • •

n • \be ftloal\y ( Qp -t ) at. t, : 0 equa1' .....

and '. .

. . = o,hll10t

i:: .. 3/a • • .S/J : n C. Sin t' •

Cu

n s 11 31 S • • • (?22)

='- .. •• 0

:~

SIA!!... .. - !':a -~

-~ - '

1See2 -~ 2

~ .. ~.-~ S1DL acte .. "2 .. .. J/2 90 aS!nL 1de· ..

0 0 ... 7/2 ..

- ., ~ n - J/2 l:ln L • di ll 0 ..

- ~7/2 C. D - ( .... 7/2 ll n

.. J s.tn IL. ... .. 0

• • l L . •SSA .. 0

• da

Cal II

• 3/2 -1 _ • ec.n • ~ ;.c.L .. aa ll 2n

- a,"2 D

.-3/..1 Sln I- ... •

(£ti.) -.. -~ ., 't -u -

nutr :iv~ nwrc -t : uo ... --. • "''~

St t

DJPJ.trfS. SY r,"-r;WJ ~~ • uo ---· ti,~ -ti~ • r/~~

mo • '1'lft • •• -~ •c 't : u ID=! • ·~ l'.zm 11• .

...,_ et{\ °' pwnroo .-..re ercnn ._ ttPt pm """ ~

... • ·~ t"O'noazd u;c •tram J1JaA a.re ""°' ~ 8lR ave 8GtttM 81.(1 '~116 UriM. ~ ~ lt'l\ 8CK&'tS

~· ~ eaAtoW'J ~ MOU3I -·~ UGq'8 9tR Wd

:t~~ . . J~)i:

~ .u.a ]'rt~ ~di -.11 u mo 'e/C 9a -'Bii' :

( trp • ;u 1111() ti~-. f '~ . iltl 900 ~ -lLU tlO'J J.LUI -J ?!ttt!:

"' ttc• ~ '~

(724)

n • 11 31 S .. - -(9) ,.. P1xin4 ald Pnrlrhe.llc ~

Baae on the totuw- dert.VGt:lcn, the tansmt1&l ~ 81"8

~ _. tbfl nomal ~.. Mnce the bolmdar7 <*ldlUona

et~, Sn 11-d·end Pueollo ~ ue ~the..., thing• ~

in the ~ 7-1, (\.\40" are at : 0 at • = 0 and ~ t : 0 ••• t .. ) .

• that.1 tor tho S'lmt lOGd.ing oandiU.C.. ..S the - ---1 at.re ... ,

,.. can ..,. the.y v1ll ba¥e the .., ~ ot ~ det1eotJAal

• 'Mh& w haw der.!.V9d far the t.uo-b1npct parabolie arct. 1n the

tor..-~

S7

VIII. b11MmtCAL EXAMPLES Of tf.?:IJI?&Q? PA.YJPL1C ARCJ!Q,

A parabol1o arch ld.th au it.a ctt.wlarw 1a giYm u tbOm in th•

t1guN 8.1.

0 =--- -:::::::: r ::::::::- --= • x

A .!f

1. L ~ 12 ' _______ ---.1

Crou Sect.1Cln

DI ~9~

Fi9ure 8 :/

A : 12 x 9 : 108 Ill• 1n. l : !. x 9 x 'JiJ : 1296 in. 4

12

E:: 3x106 pe1

HL _L

u • omo.rte all ftl.Des or x and 7 Sn S.nabea, the cout.aa1' 1 tdll be

lit 4x3x12 l ' = L2 = <12 x 12>2 = m (801)

8l1d 7 It !ha2 w. ban alNld;r got.ten tb• relat1cn betwen arch lencth • Ind the

~ 4Utanoe x ...

• • f J 1 • 4 xQ • d log (x • k J l • 4 KY> + k log a

S8

1.'he oorraepand.ing • wlue to each x hu been ~ and 11 stet u tollowa1

Table S.l

!}i+4KQ h-1.os(x • tz}l~) x ..! lOG 2K • 2 4K

0 0 • 154.08 - 154.ai 0

l' : 12" 6.00 ·~ - 154.CS 12..al" 2' : 24n l2.6S • l.6S.96 - 154.08 24.53" ,. : 36" 20.12 • 171.36 -154.~ 71.JiJ" 4' : 48" 28.84 • 176.40 - 154.08 51.16" S' : f:Dt 39.05 • lSJ..44 - 154.f» 66.41" 6t = ?:!!' S0.91 • 18S.l.e0 • 154.08 ~

lJow, let th• loada be Pn • :Jlal l/tt and Pt a 2400 11/n. (or Pn a 300 I/in ml Pt,: a:X> I/in) Ve ccm UM tlle ~ (llJ'l) to (413) and pt the

.tnaea and detlecUona Sn stat.ic oandit.t.l u tol 'Olla&

Table 8.2 M 1'n ~t,

x V (I) tl ("') (in - I) (1n) (in)

0 0 S2Z3 1.014,.300 0.7.350 0

12" 3,660 5783 m.m 0.7157 2.6'7'/ x 10-4

24" 7.3S9 3317 924,<XX> 0.6571 4.368 x 10-4

36" ll,220 743 b.,4SO o.s588 S.174x10-4 I#' lS,349 • 2009 t'al.t750 0.4157 4.905 x 10·4 ID' 19,923 - SOS9 352,800 0.2223 3.242x10·4 7J!" 24,669 - tJ2a"J 0 0 0

~·

/

. ,9

0

(.a)

1014800 in-#

+

-822.:3#

~-------~ . ~ / '

(C)

Dat"lect1an ~

!ncl·~n& ncll9l and ~ CCl~lte 4=a73S" + -~,

~----- ~ ~>---- ------<,~ A-</ (dJ ' .... ,,.y

For a ...sden 1w..i ot ateftal loade, (Pn mt Pt,) a tl"lflO 'V'ibNt1on have

1dll. toll.ow u w l\deri'ftld 1n the eect1taw VI ..S VU.

1.'be ~ (Q.5) gl'l'Olt th• ~

)-. n : ( : a·~ : 0.0191 ft (802)

and t.he equaUm (616) e,1.we th• ~ ot tree ~ ot tw-

h!nced arabM 1n nGPl:la1. ~ .... theee load1na oancH.U.cme.

p D : ~2 J ~ (803)

v... • •W the .... or tho a:rdl per ml\ arc l.mgt.h. Here w ,..

the lnGb u the mtt or ~ 1n OCl!p.lt.at1Gn, • tJC

!!. ~ i 1-~ • • g : (i2)J x lSO x S£2 x n : o.02la3

.._ n•l

61

: Cl.649 x :ur4) ril J 16.0000 x lfilD : (3.(49 JC io-4) r/l ( 4 x uP )

: 14$.96 Bl '1: w.96 (1)2: u.s.96 radian/.second

n: 3 Pl a 145.96 (3)2: 1313.64 rod.On/second

D : 5 P' a 145.96 (S)I : '649.00 radian /-'4!cond I I I I

n: n Pn a 42.1? ,,. radiay.sacond

(804)

,.,..., .. .-U.td.e t.1w mto ta. ....... (622). We .w. "' the

~ 1n ....i d1NOtJ.m .. ton .. , . 61. P. 4 n-1

~ n (•, t.) : D j' L. ( •l) T 1. Coe !!.. e Coe ~nt. - a n rt' 2ao • {;;x300x ta;m)4 · L. ~1.. - J x JJJ6x1296 x (3.W,6)5 n (•1) ~ Coe(0.019lna)Co.Pn'

n-1 : 0.'1719 ~ (·l>_T" ~ Cot (o.aL9lnl) C. Pnt, (805)

. hr_. u. t. at. mv ..u..i ., w a1n ~ tile .._. ..,at-!m w ge.

• the an Sn a eer.l.ee ten. wb.-o n s 1, ,, ' - - - • ssnoa the '-'-

'

.. eblasbw ta t.b9 -- _n.tth palm' ot n, - tll&t, tld.a .n... OGD-

,,. ..... ~ &f\elt a tw ....._ •

Fw u el•, the .s.n.ou.on .In IWl. cSSnoU.cn a the *""'-' -.. • • at ( .. a•> _dudng ' • ' -a. v1ll be

~D • 0.7379 f Co. (0.4581.) Coe CUT.as> - .. Coll (l.3752) Coe (3940.92)

ca:c-c = 4' ' : u tt'l'I : tr,, t : u a.i

(9<.li) u tt'1'1 :

am. x a;~ u ts(o-O :

~x~ ~ {~ I r) i Mi f9i'ft't) 'i U :

w :; :t;.

twpmt"*.1 Mr+ "4'lO tilt$ ~ (9tl.) ~

MR em-=» M ~ ~ vtt\ UJ ~ 8MJ Mn "°1 tty

•,JOtil8 ~ ~'P' ~GO Gq, Ull'O ~ .-no Tl"'

Pit ~ ~ .... ..V ell\ .ftUI 19tt\ .,... comaa.erJOO pptu t(OaS

... ----• ~O+ a:'trXlO-o-~ :

~ ----• --• (0)60-0-) (a7)!1-0-) T •

c~nc-o) (®6-rG) ~ .. (6~) (~) 6/IJ..,-0 :

• • • • • + (.9"19 -0-) (89-e'I 900-) ~ +

~ ·"·u -., .u-eL _, T .

( eaJ-W eo:>-).~! 80() 61.£.L *() :

----------------·

n • ' n • 7

n : n

Now - llUbeUtute the99 ...iue. into the equst1an (721.) and get the de-

t1act.iona 1n ~ dU9et.1en .at. arv 88ct.1cln am C\1 t.1.- t with

u. tree ~ ot oanat.ant nrma1 and t.angmt1&1. i-..

"'91'9 n : 11 31 s, ? • • - - -For ftMJ'l •1 the det'lecst.iGn 1n a t.angent1al d1rocUon at. the 88cUan

--.. • • 2' (ar 24") during t : 3 eec. will be

~ t, a '•381.B a 10·4 { Sin (0.9169) Coe (J3a33)

• ~ Sin (L7'°7) Coe (39699)

• ;, Qin (4.SS4S) Coe (6616S)

• ! Sin (6.418J) Coe (92631)+ .. - - - -73

: s.3848 a 10·4 { s111 sa.s3• eo. 0.0924 + ! Sin lS?.60- Coe 0.%773

Zl • .!.. S1n 2fJJ..f11• Coe o.4622 l2S

+ ;b- Sin 367.74• Coe 0.£471 + • • • - }

: s.384S x 10·4 {Sin 52.sJ• eo. '.294'

• 1.. stn 22.40" eoe is.89• • .l.... S1n 82.6reo. 26.Jelr Zl 1Z

+ ;t;- Sin 7.74• Coe YleW + • • • - }

: s.3848 x u:r4 { co.7934) co.9"7) • ;; (O.)al.l) (0.9617) • ~ (-0.9918) (0.8949)

+ 3h' (-0.131.8) (0.7976) ... - .. - }

: 4.2539 • 0.0731 - o.0382 - 0.0011 • - - - x 10-4 : lt.2S7l x 'lD-4 1n .

The above oon;utatJ.cn ebol'8 the •rie8 vUl camvga NpSd.\J' atter a

tw terr.. The atarU.ng three ti.- cmi giw a quite eat' etaotor.r re-

.alt. Nw, it the loada AN not. OCIWtant but ftl'J' with & tmct.1cn ot 81

aud\ .. th• ca88 ot unltorml.1' ~ lOlld• alcng the barlamtal

project.1.m, t.he equ.1.1'&1..ant tunct.iana ot 1cMda.,.

Pn : p (l • 11-2) . pt : ~ "191'9 'i md ~ are ODl18tll1ta. 'l'hq are

L 1 Kl : (1 - J L f,j + 161J2 ) •e 2

4 HP '2 : j e0 (L5 + 16112)

Here p ... the tntenalt.7 ol ~ diatr1butec! loadll.

r.. .,..1., let P : 3000 IJ/tt, (or 300 l/m). Then

[ 144 J 1 (808) K1 8 l • J (144)2 + 16 (36)2 x (82.a3)2

-• J ~ ( cw.'2 • 16('2>2J

t..'2 g uJ+ J 3.4UR a ilJ6 (809)

P0 : 300 [l • 4.3319 X w•S ,j'J #ftn (810)

,, : ~.39'Jl .A (811)

For the t... ~ Sn t.ho .,.,_ dS.ftcM.Cll ot IMb ...U.t W9 OM

~in tbe eqaau. (62S), .. &ft the det.1.ecsUnl. toll ...

. ~ l 6 4 ~n ( .. '): k 1i (•tfT (. ~ + 64(1•1i,) ) S

51: £,.J.le 6 J I:. TT

• ? 7 Coe - • Coe •n t. n 11" a.. . -~ n-1 ( . ~\..../ >' • 3 x l1T' ~ (•l) T(• 4.3317 a 10 1&\82.23

- (3 a i06) z (1.296 z iOJ) n 90 z 3.1416 a 17

(82.23)4 • 61. (1 - 4.3317 x 10-5) (3.ll,.16)3 rr5

• 512. (4.331.'1 x 10-S)(~)6] Coll 3.l4l{n Cl.1416)7 n7 2 x ~

(812)

'1'ba ,..._.. )'. lld the trapnc1ll Pn u. llJIMlmld ~ the 41._... n eku IDIS the md Olmdit.ima ~ the arch. The dJlrl89 et 1-ISl\I omdl• . . U.. m1cee no dUte.NnDi abclit "n ll1d P n• eo tut., ..... w haft

.... 111 the eq,uaU.on (tm) .s (SOI.) &N 8'ill ~ to tht8

66

1'mcUonal l.old1ng. Aft.er ~ the equat1cn (812) sw1 eett..tna a : 2' (er 24") BB1 t. : 3 eee., w get,

. ~ 1 ~ n (24. .3) : 7.71.6 x ur8 L. (-1>2 ( - 4.737 x lD4 ii n

n : 1, '• '• - - • (8]3)

(l)a n 1 ' s

(2). -4. TJ7 z u:r ~ -4.771 xu/t -~m x lflt -0.9414 x uJ. (3)• +O.<JS{Q x UJ? ~ .0.9SG2 x 3.07 +3.9.350 x u:J+ •3.0S93 x u}

(4). +2.2710 x w6 ~ •2.2710 x w6 +l.0'.384 x u} +0.2'1J7 z ufl n

(S)a : (2) • (3) • (4) +J...1880 x lD7 .a.use x uJt -6.38S6 x w (6)1 Coe 0.4584 n +<>.8968 .o.1900 -0.6taz

(?)t Coe Pn t, ..0.3(49 +0.215S -o.09(0

n-J. (S)a (-1)T 1.7l.6xl0-a .0.300" -0.cn:m" •3el226z lJJ-S

x (5) x ( 6) x (7)

~ 11 : o • .1' - O.CXXXJBI' - o.<XXX>3" • • - • • Whcm w lock • tho a1>0Ve 1'08Ul.ts, "" Imm the ~ tezm are r .. -11 W thm the first. toft8 in the IJ01"1oe, 9C) that WI CIGrl 18' D : 1

4ft'J. D9g1.eo\ the ot.Mre.

Ae tflr' the tree 'l1brat.1ona in the tangent1&l d:1.rectJ.cm. the ~

l3n and the ~ 'In ldll. rota1z1 the ... ftlueo no matter ma \he lOlld!ng canitit!a1e. We oan cnqend tho 991'1• ot cqJGt~cwa (721.)

term Iv' tem. SSnoe oach tarm oonta!M a h1l#l mime power ot n, it

ld.ll OOl&tel'b• wr.r tut.. It we 'flllDt to check the eer1ea ot def'.leotJ.an, . . we om ccainec$ 1t with tJw 8tat.1o c:md1t.1Gn1 beca1188 at \ : o, the

tniual Jm:S,t!Gn ot the tree v.1br8t1cn ld.11 be the -. poeltJ.an u

in -..U.o load1ng.

For equat1al (005), at \ = O Coe Pn \ : l. We t1n4 . (4 p 4 l'l'-'1 ,

~n (a, O) : El Jt ii (-1)"T" :S Cce ~ •

6t. x 300 x (~)4 lt:1 ~ : I 6 b )S L. (-1) 2 COii (O.Ol.<Jlne)

3 x lD x 1296 x .1416 n

: 0.1Y79 ~ (-1)D'ff- ~ C09 (0.0191 .. ,

._.. n : 11 31 S - • - (8.14)

bcoe, the detl.ecticln ~ n at, the ndddlA eectJ.Gn (a : 0) ot the arch

v1l1 bel . 1S1- 1

~n (o, o) : 0.7379 2ji (-1) ~

: 0.1379 ( 1 - ~ • ~ - - - )

: o. 7379 ( l - o.cr..nas • 0.00137 ... - - )

: o.~ - o~ • o.cxn.o = o.71..58" (815)

'fti8' w oaque the ~ 'f&l~ ot def'loct.1on ~ n at; that eeo-

Ucll m the Tables.a. It. a o.13scr, the erNr •have r..se 1n th•

...s.e. 11 am.it ' Jm'OC!lt• . 1.'hm1 let WI chodc the .m.oa oquatJ.cn ot tan&mtJ.a1. detled.1al. At,

t = o, Coe Clo ' : i. 'rtlC ~ (007) 1d.ll beocae • 4P. 2

~ (a, O) : ~ L. !. Sin !!! • \ ii7TJ n r;J •e

• 4 x 3lO x~t'a.'¥J'J2 L. 6. S1n all- 1 - 108 ~ (l z) x (3.l416)j n OJ 8:2.a3

: S.381.8 x w-4 L: ~ Sin !!.!.... a n ll"' 82.23

,._.. n : 1, '' s, - • -At, the ..ct.:lcn lltd.ch a : 2' (or 24")

~, c21.•: 0) = ,.3Bl.8 x io-4 'i. ~ Sin o.9169 n

: s.331.S x io-4 [Sin (0.9169) • ~ S1n (2.7507)

• ~ s:t.n <i..sm.s> • ~ mn c 6.4183) • - - - J

: 5.3049 x 10-4 ( Sin S2.S3• + !,. Sm 157.009 Zl

• ~ S!n 214.(:ff- • ;h- Sin ,r:n.74• • - - - J

: 5e38lt8 x 10-4 ( 0. 7934 • ~ (0.'81J.)

• ..1.. (-0.991Jl) • ..!... (<>.1348) • - - -J ~ 34.3

(816)

: [ 4.2723 • o.rn(:JJ - o.04ZI • o.ocm. + - - • J x ]J)-4

: 4.'»n z 10-4 in (817) '

the OOi ~ Y&lue ot detleet.1m ' at tll&t aoeUm 1n the '-bl.a

8.2 le 4.368 x 10-4 irle '1be enw w haft .S. ia a1*1' 1.4 )lel'oeut;.

p I

l 1. ~

i f •

~ f

? l

' r.

~ f

~ r I I

f -

I "f

t1I

r i1

• '1

~1

r ~~I'>~

R. >

> r

~ ~

~ t

: i

t t

, .~ ' I..

f f i

f ,

! e.

re n:

; ;

~ i:t

it •

1 I

I I

8 "

.. ~

i if

Ii:

!f:

i f

$i:

B t"

~

-~~ l

ft rt

IJ.

f,

f t

• ~

i: 'U

f

§ i

- ~ -

--

' I I I I

70

Funcfions

Figure 9: I

Ml - A .. ot t.i.. t\llCU.one are

f(+J.. .. ) : • t(-A .. ) (<JOI.)

eo that• the ll01ut1an llh1oh w set at, the oanCUft'Wltl paint .& 1n the

ti.gun 9.1 la 1dant1ca1.q tho ... value u .., - g9' hal th• paint,

''· 1'he f\l\cUm ot tam" 1fe la a «dimlou8 cmw "1th cno tmd ~

1ng to •l an! another ond to -1.

Sinoe the eolutione tram the ccncu:nrent. po1nta A, B1 c - - - ot thw

two tmc:U.cm in the t1Nt qua.rtv ~ the coordt.nate .. the - ..

1n the third quart,er1 (or At 1 Dt 1 Ct - • • equal to A.1 B1 C • • - ~

peet..i.wl.T) - tl1&t the plott1na cinq to the ~ qua.rtv 1a enough to tsnd aU mWt!ma tw" • For the 8*e or accurat'l'T, lot ua ~ the nnt, quarter ot the csunea

J\Se =..3.9Z63 1 I I I ' I I

1.0

~ 09 ~ I -lanA.s .. _ I I / I -i-- -lanh .>..Sft

~ 08 ~ ~ ~ 07 (l

v!J 0.6 ~ -<

§ 'f- 05 ~ ~

~ • <:) -i:

04

IJ o.3 c:: ~

0.2

o. I

o~ I I

I I v I I I A Se • I 7T 2 3

2 .,, .

Fi9ure :J:2

1n the t1gure 9.2 with a larger ecal.e. At the b~ theee two

OUl'ft8 ..,. ~ at p:dnt. o, 81d • pt"• : o or the ~c;r

Po :: o. It. .. ,..._ .. a lltat1o oanc11ticn 'Without. ~

Artez- that• the eurve ot tam"'-.. CIGIJWrgn rap.ldl.7 tClm'd •l and inter-

18cte vith tanJ-. .. at pQ1nt. A. 'fb1a mk8e ~ 9C)]\1UmAl .. : 3.926'e

Sinoe W CM ocm!dep tanhl\S. : 1 tOf" the nsl.n1ng pan et t.h9 cmw

att.r the Plln' '•.that due to th\J ~ ~ ot tin>.. ...

the other llOluU.Gna vU1 be

An8e : n 7T + l (90S)

._... l u a ~ 111.tb a wl.ue 0.7847 (w t1gure 9.21 \be dbtance

l8UUNd ~ ll'Clll ... to 7T ). lkmoe

"n : .!!.. n + .!. (~) 8e ..

n : 11 2,. 31 4 .. • •

Nw w 9Ub9Utute i1Jto the equaU.m (901) am ~ the ~ or tl'M 'Y1bftitJJ:ma ot tbed•d ~ archee.

Pn : ~ (n 1T' + lt)2 ff:! • J-; (907)

n : 11 21 31 4 • • •

73

CONCimIONS

In th11 thesia a eolution ot parabolio arch problem 1a preaentecl

which h&ll eneral a1gnit1cant adYant.agea over tradiUonal methods. Vint,

\be arc length a ot the a.rah 18 ueed u the cnq Tariable 1nstead ot the

rectangul.a.r ooordinates x and 7• This «id mpl 1 ties tho eqaation8 1n ditter-. enUa1 ~ and mlces them easier to aolw. Seoond1 the radial and

tangenUal oanpcnenta are used 1n the anal1'l1a ot detlectiona and

8\reese•• Thie giwa tangmt1al CCIJp"esaicn stresaea 1n the aame direo-

Uan aa the arch rib shortening and the rad1&1 shear 8'resaes 1n the . eame direction ae shear dist.ortim strain. Hence 1 we can write the

equation ot rib ehort.ening as a tuncticn onq ot normal 8\reaaee and

the equation ot radial derlectian as a tuncticn on.11' of sheari.ng streaeea

and bending DICID9llt. These a.rrangaEDt.9 reduce the number ot unknown8

1n the ditterential equation or equUibrium. .. In 9ecticn m, two mt.bode haw been Wied to develop the defieo-.

ticm equatiana, and the J"ellUlta are identical.lT equal. A tranatoma-

tian haa been introduced to aepar&te the va.riable • 1n x and 7 caapo-

nenta. Thie leads to a set ot equaticna which are stm Jar to \hoee . dft9loped 'b7 Mr. Popov. ~' bT using onlT ane Tal"i&ble a the

rad1al or normal CC•'l>anents are given b7 id~er ~ which are

euier to eolve.

The basic d;ynamf.c cquat1ona which n~"e \leen developed 1n eect.!.an

V are the general equations tor mv r.urved structure. Since thq are

twacU.ona ot • and the stresaes, a ditt'1Nr~'- to.a~~ •lf atr"l"J and ditter-

•

~JO~~ Wf ll'pq, •-.q ~

JO 8UOl1WUDO ~ etn. luJWt pm ·~ -.nri °' ~ ~ pie -.. twUOi--n> OA!4 *ft ~ At 8tP'JI onoqued

30 Wpi80DGj:p W 0'4Ul ~ eq C9tl VIO ~ ~ 'Ml>Fhtl

-..uoo .. tuonMtaP lltft ~ ~~ 11\tNIM

MO"[O MU ~RJ'IOL Onw18 .l8SUl 8UOJ1Utoe MR 0'4 ~ .t'rr->11D8Pl

eq nTA ~ SIOJ~-UJ 0 : 1 .. 1tf8 M tlDN" ~ JO

.. n ..... aqi 'UD'J11PUDO ~ ell\ °' 1U9'f.11Ald» an ~ WJ MA JO

9UOJ1llJUDO (9~•4JUl 9'l\ 80a'S9 ~LOI ~ ~ 8tR .1101 ..

• • pil8I\ eq 090 ·~ ~ -n UJ ~MU

~ 5uSP10t ~JO 11~ an .&aq:, ...-nwaoo Pt&

."' ptl9 '""4~ --·~ .. tpW ·~._ ""' JO -~ . . -oeaqo etA .lq .f'l,1IO pea.ta.ad .,. ~ ae.tJ us qo.n -JO ~

~ MA JO 98ftt1tA Mft ~ MQI Ill 1IIA P.l8 IA ~ UI

9'FWP1tl0 eq .ttu~ao ttTA ~pwa1 .-z•Jm puv ·~

75

S1noere 0'8UtAm le ~ to ~. a. L. Rocen, both tflt'

et.1a11aung IV 1nte.reat. 1n ~ d;J'lrad.ca 8ld tor Id.a pS.dmce1

..-de:tcn, ald ha1ptu1. ~Sn prapar1n& tb1a theada. He

.. best 90 Jc1nd to - dur.lng \he put ,.,...,..

~.um la ~'°Dr. n. Morda, Head, ~ ot . . Civil~~~· ImJU.tuto, ... ettorMI rate

pnrfble the authorf 1 ba1n& cnmW a ranma1 ot h18 ~ .-. ..

t.antAlldp clur!ng the l.aot. ~ or ~. I • daapq Sndebt,od to tho p&'Of'eiw'9 Sn tho ~ ot Civil . ~ ~ ot ~ NechanlCI mi~ ot

~ ot tb1a Insti.1.tut.e tor the ~ tbe7 have made to w.. thtJ81e.

~ ia &1ao ... to • oamtrr ad.na, ,,. bM encour-

aced • in •NZ7 ~. ~ or ~. I cmno' eetabl1eb

.,udnc .S.thaut he~.

srosY 9il ~ 1.aodod •t1 ~ pa ~OSY ~i •t ·~1 . . •o ~ .fEI --.. 1110UNOJ.Jta ~ "*' 8DtJ\-lY JO ~. -C

Ul1lou 91 •n -.ia 4tt aa90'p&WQIQ ~ 10 ~ • l810fl -c ~-ii·.ra-lt ~10~aputa;o~ ~

9l

The vita has been removed from the scanned document

.lbatraot

ot STRESS ANALYSIS O? 1ABABOLIO ARCHES

AND THEIR DYHAllIC BBHAVIOR

Shou-n1en Hou

Th••i• eubaitt•• to the Graduate Jaculty of the Virginia PolT'•ohnio Inatitute in oand1dao;r for th• degree ~

14.l.S!ER OP SCIENCE

1n

SfRUCfURAL ENGINEERIUG

Thia thesis ia concerned with both the atatio and dynamic analyeia ot parabolic arohea. In the dynamic

pa.rt:, •pecial attention ie giTen to the tree vibration

ot auoh archea. The following procedUl"e is followed. The loading

conditions are aa8Um.ed and a inf1Jdteamal segment ot the aroh ie taken ao e.a the dUf e:rential equations relating detleotiona and •lope·, change• 011 both ends ot the •esment are developed. 'l'heae obtain a eet of

general equatioD9 tor elaatio parabol~c arohea. In dynamioa, the equations of general CUZ'Ted atru.-

oture are developed through cor...aiderationa ot dynamic

tQ.uilibrium. A euclden :remonU. of loading is aasuaed to oauae the etru.oture to Tib:rate f'l'eely. Then, a method

ot eeparatin& variables for partial ditf erential equa-. . '

tiona ia uaed to get the equations of de!lection com-~ 1' ~. : •

ponent8. Eaoh special ~rlstip flmotioa 1• derived for each epeoial ••t of ~oundary oondit10Jl9 ~o .a.e to

. # ; • . • •

. get an unlimited number of modee .oi. ~ .. :,vibrations. ·

The Jourier aeriea 1• employed to determine the ooeff1-oienta of the d.ynamio equationa, and to get a seriee•

tol'll solution for defleotiona. FinalJ.7, two nUller1cal example• are given to repre-

een~ the practical application. Two kin.de of parabolio

.uehee, one with ho-hinged aupporta and ~•J" with

tixed-enu are ooneider•d in each procedure.