Requires D052‘ 2.? or hlghgrt?
`(mar graphics; and 256K memory
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`phlcs require 384K)
`
`Costco Exhibit 1007, p- 1
`
`Costco Exhibit 1007, p. 1
`
`

`
`SECOND EDITION
`
`= .!'uu 0 '.iJ: &
`Lehigh University
`
`r rig =; j .'rip, ]ri
`of Connecticut
`
`University
`
`With the collaboration
`3 P)e l!1/o/f
`, lol /n
`University of Connecticut
`
`of
`
`P.'Iv ~3.'~.st=i.'ll'll:,
`ttrt ~=:
`St. Louis
`San Francisco
`New York
`Auckland
`Mexico
`Lisbon
`London
`Montreal
`Madnd
`Milan
`San Juan
`Singapore
`Sydney
`Tokyo
`
`Caracas
`Bogota
`Paris
`New Delhi
`Toronto
`
`Costco Exhibit 1007, p. 2
`
`

`
`i&s
`
`IlAI 1)1 Nlc(:I;ttv-l(ill,
`(",o)nvffiht
`I»(
`I(0&0'),
`t.i&~his!.(s(*1»(*(1.
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`st st(*It), (vtfhottt
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`
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`This 1)ool& (v,ts s( t i» (:itic(lo»I;I hI
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`I h(.''(lit&)l's vcvl'c Joh» J. ( ol'I'I'i »I i»«I J»cl» AI»Iscl;
`((HS Ji»1('ll(* ). 11»CCI S.
`thc pl»(I»Clloll S»pcl Vl'ioi
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`I I&!hei'»i(l Jotl» (. I c('»Ilvl(l.
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`
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`
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`
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`
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`((It!tel.
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`Ntcclt'tt»cs 01»'I'lt('I»ils
`thc c»II'II)OI"tt«») ol Joh» I D&'U 011. -'(I v(1
`p.
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`I!tel!I(lcs f»&IC)
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`
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`I(h 'I'itl(*.
`Ioh» T.
`
`(I'Iv«00(1 11»s'icll),
`
`(01-')0&A;)
`
`Costco Exhibit 1007, p. 3
`
`

`
`for stre»g[h,
`In thc prccccling chapter xvc lc'lr»c«1 to design hclu»s
`i» thc next., w('hall hc conccrnc«l xvith a»othe) asp( c[
`In this chapter
`lmcl
`thc dc[el'»in)lltlo»
`of thc clcslgn of hc(nns, nlnnclv,
`i:hc (/cglcc[IOII of
`of
`interest
`is thc «1«-
`Of p;lrticular
`prismatic hcams under given loadings,
`of thc HL(l r»1»1111 (lcjlccfto» of 11 he(un undcl
`11 glvcl'I
`loa(hng
`t(.rlrnnatlon
`i»elude
`a m,lsi-
`of a hcam sviff gener(Illy
`since the «lesign specifications
`value for i[s «lcflection.
`muln allowable
`XVc saw in Scc. 4.5 that a prismatic hcan) suhj«ct(. d to pur(. 1)ending
`svlthl» thc el((sile 1*»lgc, thc C»I'v,'I-
`I» c of cll clc 1»)d tl'lilt,
`Is hcnt. 1»to 'll)
`turc of'he neutral
`surface ml)y hc express(.d as
`
`(4,21)
`
`is thc bending moment, E thc modulus of elasticity,
`where('tI
`thc cl oss scctlon 1140»[ lts neutral
`n)ol'ncnt
`of 1Hcl"tla Of
`«i%Is.
`to a tl"n) svcl sc 10'lcllng,
`lx hcn a. be'un Is suhjcctccl
`I" q, ('1.211 I ('O'Illlns
`fol cnly glvcn transvc) sc sec[lou, pl ovldcd thll[ Sault
`I)cnant
`s pl'n'I
`vllhcl
`110'o'cvc)', ho[1'I thc 1)cnchng nlo»1cnt. 11»cl thc culvllt»lc of
`ciplc appli(.*s.
`surf'acc will vary f'rom section to section.
`13cnoting 1)y x the
`the neutral
`clistancc of the section from thc left cnd of thc hearn,
`xvc shall vvritc
`
`a»d I [hc
`
`Thc knowlcclgc of thc cln vatul'c at vln 10»s ponlts of [hc he(un wlff c»af)fc
`I cglu cling thc clcfor»1atlon
`of thc
`to ch aw sol'nc gcncl 111 conclusions
`Us
`lou«i»lg (Scc, 8.2I.
`hcan'I Unclcr
`the slope and deflection of the hearn
`To determine
`at any given
`f'irst derive the following second-orcler
`linear difIcrcntiaf
`point, we shall
`thc shape of t'hc
`the elastic cl(rue characterizing
`equation, which governs
`
`Costco Exhibit 1007, p. 4
`
`

`
`d(foi'i]]cd hca»i
`
`'S( &
`
`8 ]
`
`INTRODU( T IQN
`
`476
`
`I/j&l
`
`(II } C iltltlf«vcl
`
`l&cinll
`
`[I/, = nj
`[I/,
`! /& } Silnpf) ~npp»I tc(f 1&»,nn
`Fig. 8.1
`
`=- OI
`
`Ffg. 8.2
`
`&/'-I/,5 J(x}
`cix-''/
`foi dl «lu
`In&] h(- I«pi(scn[ccj
`If the he»din&'«»»c»t
`I» thc c&lsc of fhc l)c&»ns an«I 10&ii(lll'Igs
`sj]0'o&B I»
`$ /&.] '- &ts
`I'Ilc fol'N'tjo»
`[hc, s fop( 0 = (li/ (tx an(1 thc cl('fl(!etio» I/ at,'tl]v point &)I [hc h«iun
`(j I
`] hcoh[;]inc(l ll»')"gj'~vo s»eccss&ve 'Ht('g"1["»as
`in thc pl occss xv) jj 1)c «1ctcl'In»le(1 fi 0»a [1'lc
`of jnfeI&I ][jot] jl ill od&i «'ccj
`In [hc flgur(.
`ho»i«1&LB'on(hi«)»s»ichcittccl
`if (Iifj( re»t
`functions
`'H»tllrtical
`ffon&evc r
`I» & i»10&is pol [ious of
`fli(* hct](jjn«H]OIH('Hf.
`tj d c(f afjons y& jlj,ij so hc I c(I L»reel, 1(.*&tcjlng [0 (hffct cnl f»»c[10»s cleflnlltg
`41 lhc case of thc
`lhc 1)cain.
`[h&'ji])f]c ('1»'v(* ii] lhc vi»'10»s pol tlons Of
`itild Ioa(hiig& Of I'g f) -; fol cxa»]pj(*& tv'0 «jiff(.*I V»ll&tl
`(&(ILla[10»s i&i
`h('iln]
`&)Hc foi'li(* poi tlon Of 1)ca»i ~(D i»lcj fhc other 101'h(-'ol t io»
`I/I & a»cj thc sccon(1 thc
`Ihc I» sf ('(i»,'I [«)» ]'Icfcjs th(& funcf lo»!I 0I it»cl
`J)J).
`0& iln(1 I/&. Aj[ogc[j]c I & foul co»stiulfs
`of 1»leg&'&&f10» i]IHs[ 1)c
`finlctloil!I
`I)/'vrltjt]g [ha[ lhc «I(.'fleet&0» is xcro a[
`dc[co]ln]ccj; t)vo v lll I«'ht&H»c(l
`thc por[IO»s of bean] AD
`th& oth(*r hvo I)&& espn sslng fh;it
`A;tn(j fj, an(l
`thc sitn]c «lcflectio» at D.
`slope a»«I
`lh& s,ii»(
`an(1 Dfj hi» «
`in Scc. S.-'1 that
`c&tsc of a hcaln s&tppor[»]g a
`i» th(
`O'L shitlf oj)q(i»(
`distributed 10;«1 « I &}, lhe clastic curve H];ty 1)e ol)taitlc(1 «lire ctjy fn»n t&;(x)
`f&]&ir s&&cc('ssivc i&it( gi'Ll&0»s. Thc const'Hits
`hi[to(luce«1 11] this
`tltrottf&j]
`]& vali»&s Of t, M, 0, I&II«I
`f)r»co&& i»ll h('I('[("t'ti »n(.'(I fronl
`the hout]chit
`1.c.,
`»I(t("/ci )nt»(//('dc&(/»Is,
`lvc sh;ill chsctlss
`f),5,
`s/(itic/il/I/
`In hcc,
`ts»]vojvc*
`'Ioppol'tc(1 Ill s»cll a vv&1 & tj]at thc n acfio»!I al thc suppoi
`hcillns
`Since 01]j)'h I c(& c(ILllllbl'1«llii
`f»in'i
`('(IH,'Itious
`iti (
`n]»l'(& Hi]j&I«)M"Hs,
`fl'I('sc &IIII«110%'ns, [hc c(IL»hhl
`to (I('t'('I-»litic
`IL»H c(fu&L[toBs I»List
`availahlc
`froln thc I)0»»«fary conclitions
`c&fttittioils oj]tainccj
`Irc s&tf]pl& incilfc(l hl
`thc s»pp&)i ts.
`fin[)oscd h]
`of the clastic
`for the clctermination
`Tj]c n]etj]0(j (lcscl il)e(l c;trlier
`I e(I»» c(1 to I cpl'c!Icn l tj]('en(h ng
`i»»etio»s
`iu'c
`seve I'al
`cill}xc ivlici]
`Il mal h«'t»tc'(thorio»s& since it rc(I&tires matching slopes ancj
`nlonlcnt
`]I'&/e shall sce in Sec. [).6 that
`the usc
`&1&*11('ctioi» itt cv(*rl
`tl',tt]sition poitl[,
`in scc.
`f««(/io)ts
`»f sn]/ittfinilt/
`(prcvio»sl)& cjjscussc«I
`/.51
`lhc 1)caB]. At]
`I/ af. (t»']I po»ll Of
`fhc cl('tcl'It»»;&tlOH of 0 it»el
`base(1 (»I ccl [;]ln
`ill[('I'n&lfji('nc[ho(l
`foi. [hc cfc [crinjniltjon
`of 0 an(l
`I/
`Neon]cfnc f)I opc'I t j(*s of fhc elastic c»rve it»cj involving thc cotnputaf
`ion of
`illcas &lnd »ion»*It ls of al('as uncf('I'hc
`c»1 vc.
`h()ncji»g-It]0»]«H[
`'O'Ijj 1)c
`in [;h;tp, 9.
`djsctts&c&j
`Thc I»[ piirt of'hc chapter
`to thc
`If."/ an(l S.SI is cjcvotccj
`(Secs.
`vvh&ch cons]sts of clctc] In)ning s(.p&tratcb, &u]cl
`(/Iio(l of &III)ei I)(»'&/ion,
`cn ilddin'h('h)pc
`an(I (I(".flection. C&tllsccl I)v tj]( val lo»s 10&«js apph«'«I
`This f)i'Oc((l»re inav he f;tcjjjtatecf l)v the us(. Of the table 1»
`- ppendis 0 zvhjcji
`g&jx cs the slopes an«1 cleflcctions off)cams for various
`&jtt]I'& i]]]d [~I)('s of sttpp(
`
`arc re«I&lit ccj to represent
`
`('((It»lcd,
`
`consjcjclal)j&'in'IphfiL"I
`
`Costco Exhibit 1007, p. 5
`
`

`
`DEFLECTION OF BE&-')MS BY INTEGI:)ATION
`
`ill(* l)(*gl»»»lfg of [ills ella))l(.'), ))'c rvcallc«1 I",(j. ( j.'
`) of )cc.
`&'It
`))'Iucj) r(.'I;ll('s ilK'&ll villi»'c of tllc nviltl
`lllc l)el»ling »K)
`ill su) facv Iul(l
`Ii I)«'ll» I» pi» v 1)cn(ill lg.
`j)c)lilt('&I oui
`» lcl'll
`lllls
`M~«"
`tllili.
`c(jil&l[ioil
`ill
`f«r lul)'in(*n [Ia»)sx vl'sc svcti&)ii &)6 'I I)(*'un sul)j«*«tc(I
`rcnrains
`to a
`villi«l
`tll;1[. 8;Ii»t-vvn,u)t
`s principlv;ippfics.
`[I".»)Sv(*)s&* 10 Kli»g, pl 0'f'i&jv«I
`I joqq
`a»«f [I)c cur»;it»re of tllv»ciltrlll
`vrvl, 1)otll [I)c I)cn«I»ig i»on)cut
`s»I I;I& «.
`1)'ilj Y;lr1'ionl
`svction lo scctio». Bv»oting I)y x tllc «list;ulcc of tll«
`)cc )) rite
`sec[i»» fr»i» tllv Ivf't vn(l &)I'li(* l)v;un.
`
`ITI(x}
`
`p
`
`[:0»si«I«r. Ioi. Crul»lpl(',;I c;ultilvr(*r 1)c(un A jj of lv»gill I. s»hjcctc(l
`to
`)vv I);Ivc,TI(x) = —P)
`101«l P;ll its fl cv v»&1 A (I'ig. h.3&&).
`,I concviitratv«l
`;ul«l, s»l)stltiitnlg»lto
`(IS. I),
`
`))'Ill('ll SIIOX)'S I.I)ill tl)C c»1')alt»I'v of tl)C I)Cliir'll Sill'flicv 9,'u 1(.*S 11»C'u'li') itl)
`to —j&l/7'./,lt
`jr»in zv) 0;It A,
`jj, )rj)vrv
`)):I)vrv )01 ltsvlf
`ls inflnlt(*.,
`jP„I = I:'llPL (Vig. f'j.3/)).
`[ 0»si«ICI
`»01('* [I)c 0)'CI'I)a»gi»g 1)v(11» AD »I Vig. 8.'j(I, whicj)
`siip-
`t)1'0 co»ceo lr(ltc(l 101Kls (1s sflo))'».
`Vl 0»1 thc frv( -I)0«jy «Iiagr,u» of
`poI'ts
`ar«* jj I =
`tllc lcllctioils
`tllc I)&.* un (Vig. N.4/)}, xvv I'ln«l
`that
`tllv supports
`lit
`j 1(jsj an«l 8( = 5 1«N, I'&*spvctix el@, 'u)«I «lrllw thc corrcspon«fing
`f)&»«li»«-
`»101»cnt (llllgl *u)l (Vlg. &)L5(l). A&0»otc fl 01» tjlc (ll'lgl"un tlla[ M, 'ul(l
`[11»s
`[11('ul va[»I c of tllc l)c'1»1, iu c zvl 0 a[ l)0[11 cn«ls Of [llc 1)c'un, an(1 Iilso ill
`
`«f)
`
`Costco Exhibit 1007, p. 6
`
`

`
`83 EQUATIOI&] QF THE ELAST]C CURVE
`
`/
`
`C.'
`
`;I j)0»lf. /s Iociifc(l ilf. x = -I In. Bc five(*H A an(l
`thc hc»chug& »10Hicnf
`I.i
`/
`thc I)(;u» is c&»)cave Hpvvitnl: hchvcc» E;In(1 D thc I)en«ling
`j,osjfivc;uicl
`(t']g. Is.'3/)/. 8 c
`ls Iacgat]v('ii(1 thc 1)ca») Is co»ca) 0 cfo)vu)val(l
`H)0»tent
`thc hug('st Yiiltlc of fhc cuin itttil(" fi. ("...fhc sn]illlcst %alt](* of
`i]lqo Hof.c lhilf
`rachus of cu]a';Itt»(*) 0(cuis (if. fhc supporf. 6, xvh(!rc ~Mf Is i»ax»nu]n.
`th(
`t'()nl
`thc H)foi »)at]0» ohtan)c(I on Its ctn v'ltu]'c',
`)vc »lily gcf. it fiHI'lv
`Ijovcxci i thc a»alysis an(I
`&q)0(1 i(lcil of'thc shape of th('Icf()r&n(*(I 'hc'1»i.
`cfcqj&gn of (I h(."I) n tistlilllv rc(ju» c tnor(.'i cc]sc I»for]Hilt]on On thc (fcff(cid:30)lc-
`thc 8/Oj)c of thc hc'i&HI af vitrlou'i
`Of jaai t]ct]Ia] Hnpol-
`pol]'Its.
`cf(cid:30))II an(I
`In thc
`f»]('c Is thc I(»0)VIC(lgr(* of fhc Iii(&xiii)i(iil deaf/(&c/io)i
`of thc hcaiiu
`tisc I" cj, (S. I ) to 01)ta]n a I clatioi'I hctvvccn tlic cfcflcc-
`H( xl. section '&vc shiill
`'lt. a g)vcn point. f) 0» fhc ilxls Of
`thc hca»1 i»]el
`thc
`tjon I/ 1»casu]'c(l
`j)01»t fi 0»1 so»le fixe(1 01'lgi» (frig. 8.6). Thc rcf'Itio»
`cljqtiu]cc x of tl'&ilt
`is thc c(juatjon of fhc ('f((sfic c'(II ( 0, i.c,, lhc ((jt)ation of thc
`Ohf;Ii»ccl
`"i of fhc I)( al'0 Is tl'an.'ifonnccl
`thc give»
`un(lc]
`thc asl
`ctuvc Into vvhlch
`loa(lit)g (f'g. I).tal)). I
`
`0'« fj]st recall
`thc curvill»i c of'
`f'ron] c lcn)c»tary cillculus
`that
`t)(x, I/) of thc cul vc »lily 1)c cxj)i (*ssc(1
`('uin c iit
`(I j)oint
`
`plane
`
`(t-'I/
`
`(IX
`
`di/
`
`p
`
`iu (.- th(.'1 i'!if. 'IH(1 scconcl (lci lvilf ives of thc huic-
`xvh(*I'c (ft/lc(x 'an(l
`(I I//(fx
`I cpi'csc»fc(1 hy thitt curve. B(It, ln thc case of tlic clasflc cui'vc
`t]0]I I/(x)
`thc slope dp/dx
`'ui(1 its s(juarc is ncgligihlc
`is very sin(ill,
`of;& hc'um,
`9'c nlay xv]jtc,
`to»»ity.
`thciicfor(*,
`con)j)'uc«I
`f
`
`(f-j/
`
`St]hsfittltu]g foi
`
`I/p flu»i
`
`C/X
`
`P
`(8.3) nlto (4. f), )vc htlvc
`
`t,
`
`Fig. 8.6
`
`C
`
`Ii
`
`D
`
`d-'I/
`
`M(x)
`
`dx-'J
`I-or I ..Ii c(
`jiff r- t'il ( 1 ition;
`cljffcrcntji)I c(juatjon for thc cl;Istic curve.
`
`ol tii c I i,
`
`itio
`Tl
`j
`thc g&ovcrning
`
`it is
`
`tft sf&on)&i be n&)te«j
`tf)e nest.
`«& (I&is ebapte«n&&f
`that.,
`'»&'nt. «efnje &I seas «se«1 n& f)&&*%&noh ej)«f)to& ~ to ret)r&'se&)t
`r«&s«ase section iron& tb& n«ntraj;&x&s
`tb&t see»on
`
`&)f
`
`a. ) ert«.al «f&sf)jaee-
`&/ represents
`tl&e «list*alee ol a &",I& en f)&)nlt &! 3
`
`i&
`
`Costco Exhibit 1007, p. 7
`
`

`
`DEFLECTION OF BEAMS BY INTEGRATION
`
`0:.
`
`Fig. 8.7
`
`/, — 0
`
`)I I, -- 0
`
`(() ) s)n)plv 4) )pp«)'Ic(t 1)c ))))
`
`—0
`
`)I )
`
`(1)) Ovc)'1)())lg))lg l)i.",))))
`
`- 0
`)/)
`(), =-0
`
`ic) C())1&)lcv(.')')c())))
`Fig. 8.8. Boundary conditions
`beams
`determinate
`
`for statically
`
`is 1&no)v» «s th('l('x(l I (II I igi(lit(/
`if it v'ivies iiloiig
`lrl
`iu)(1,
`Thc pvo(liict
`In thc cil!i('f il hei&»1 of v,u ylng (1('plh, vv(''nil!iL exp&'es!i
`the bc&11»,
`it
`il'i
`;is a fi»&ct&o» of r hcfovc pvoc(( (liiig to in tcg&.,itc I'.(1. ()L4). Ho)v( vev,
`iil
`ih«ciisc co»si&lcr&(l hcv«,
`ii pl&sn)«L&c hc;iin, wf»cl»s
`the cils(. of
`th&
`in( &nh( rs of I.'(I
`»Vc i»ay ll&i&s &»iiltiply bolli
`11(.xi&ral vigi(lity is constant.
`())L-I) hy El an&1 int«gv;itc
`i&) x.
`)»Ic xvrit(
`
`El—=
`
`(IJ(
`
`&Ix
`
`(TI(x) (l.r + C:,
`
`)
`()
`
`(/), 5)
`
`&
`
`&vl&e&'&'
`
`I)(!»Otlllig bv 0(x) thc iulgf(', »1cil-
`il con!itin)i of llltcgl
`ill)011.
`l,i
`to th(.'l(istic curve;&t C) fbvins
`tl&c ta»gc»t
`lliat
`xvitf& th&.
`in r;ulia&is,
`siii'c&l
`this;ulgl('s
`v('rv s&nail, xvc ha)(
`1&ovizont;&I (Fig. (4. /),,u&&1 recalling&g
`thiit
`
`I hilsi,
`
`)v('»ilv vvl ilc I',(1.
`
`(i)L 5) &11 thc 'Ill&'I'l&i&t&ve
`
`fol'ln
`
`I:I 0(x) =
`
`)
`
`()
`
`I)l(x) (h + C:,
`
`I&it('.gi"&ting both &nc»&hc&'s of L(1, (i5. 5) ill x,
`
`)VC l)i&),
`
`&''I
`
`I/ =
`
`)
`
`()
`
`l'.'I
`
`ij
`
`()
`
`.II(x) (h. + C,'i
`
`(h + C.,
`
`lsl
`
`I/ =
`
`(lr
`
`()
`
`()
`
`/II(x) (h+ c,!x + c a
`
`tc) m i i) th(
`vvl &cv( C,', is;i sccon&1 co»st;u it, iul&1 xvl&(. v& th(. firsl
`ri«lit-1)aii(l
`th( 1'iinction of'
`ohl;iin('&1 h)
`i»ember
`ii&tcgv;iting Lvvice in 1
`vepvcs&n)ts
`thc co»stanti
`If it vxx v«not
`for tl&« f;icl thai
`the b&'n&ling n)omcnt hl(r).
`(1&.I'in&'hc &leflection
`C; I &in&1 C;ivc as yel un&1& tcvmi»«&l,
`I'.&I. (l)L 6) avon)&i
`of'h«bea&)& «t an) give» poiilt Q,;u&&f L&1. (l).5) or (l).,5') cavo»1&1 si&nil;irl)
`slop« of thc bc'lln,'it C).
`(lcfi»e th(
`iu'c &lclcl'»1111«&1 fl'o»1 lhc l)()II»(l(I I I/ &'(»I(II-
`rl)e ('0»'it&nit!i C
`ii&1&1 C
`i&nposc(l 0» lh(
`hei&&n hy iii
`fn»n the coil(litions
`tio»s ov, »)o&c precisely,
`in tl&is sc«ti&)n to static»ll(/ &let&.I
`I,i&nitini'iiv
`analysis
`siipports.
`reactions;it
`i e., Lo hc;lms
`in siicli a ivi&y that
`th(
`support«el
`lolly hc obtill»c&1 hy Lh('1(.'tho&1!i of slilllc!i, vvc»otc thill
`0&lh
`sllppol'1!i
`t) p&..s of'b(. uns nce(l to be co»si&lc& c&l 1)er«(I'ig. 1)L))): (») th& si»II)II/
`thrc(
`'u)&1 (c) the c(I»til('cel
`(I)) the ocel II(I»gi»g I)e&I»I,
`l)e»I»
`Ia( d»l,
`s&I ppovte&I
`txvo cas(s,
`consist of a pin ail&1 bvacl&«t at 4
`In thc f'irst
`supports
`th(
`tin rlefl«ctioil h«z& vo «t each of
`an&I of a roller, it II, an&1 rc(fuivc that
`Lett»&g first r = x,l,
`&0 = I/,1 = 0 in k,&I. (b.b), an(l
`th«n r =.1'll
`I/ = I/8 = 0 I» thc siunc c(I»lit&oil, vvc oblilln txvo c(fuiltiolls vvhich 11&ii)
`solve&1 f'r Cl an&1 C'e.
`In th( case of'th(. Cantilever
`he;un (I'ig.
`I).laic),
`Lett»&!-'
`that both thc &lcflection an(1 thc slope at A &»ust h(. z&ro.
`= r,, I/ = I/, = 0 in E&I, (/)L6), an&1 x = x,, 0 = 0 = 0 in Is(I. (d,5'), ""
`Obli&1&1 ilgil&11 LU 0 c(Ill«1.1011!i xvl)&ch lililv hc .iolvc&l for C: I a»cl C
`
`th&'-)&'o»lts.
`
`)V('ote
`
`»&i»(it(')e&I»)s,
`
`tile
`
`Costco Exhibit 1007, p. 8
`
`

`
`.': & I & 1) l (
`'hec»ntil«) «r hca&n )(B is of unifortu
`cross s«ction &»1(l
`c,»aa(s;& lo;&cl 1'lt &Is lrcc «nd A (I'&g. S.c)). 1?«tcrmn)e thc
`( 0»;ttion of thc (.'last ic c«rvc'mcl
`thc d( Il&'ction 'u&cl slope &&t c(
`
`8.9
`
`Fig. 8.10
`
`f'&'ec-hocly &1&&&grun ol'hc
`portion AC of thc
`i&sing thc'.
`l&«&n) ('1'ig 8. 10), )vl&c rc C: is loca t& &I;&I a ebs tan cc r ft cnn encl.'(,
`)vc I'ind
`
`&11 = -P.&.
`(8 ()
`Sill)st&t«tn&g k)l'lI n&to 1.'. (8 -1) 'u&cl »)»it&pl)'1»g both n)«»1-
`thc constant El, vv('vl
`I tc
`h('I s b)
`LI ——', = —I'r
`
`(l&(/
`
`dxa
`
`8 3
`
`EQUATION OF THE ELASTIC CLIR&/E
`
`-'f83
`
`El —'
`
`dt/
`
`dx
`
`— 'Prs + &PL&
`
`(8.9)
`
`lntegr&tt&ng
`
`(iS,&?), wc &v&.itc
`c)f Kq,
`1&oth Inc&nb&.rs
`El p = —,';Pr'
`)PLxx + Cx
`(8. 10)
`But,;tt 13 wc have r = I., I/ = 0. S«1&stit«ting into (8 10), we
`h&n'c
`
`0 = ——,',PL) + &I I '
`C;. = —
`thc value of':. back into Eq. (8.10), we obtain the
`Carrying
`c (f»ation of'he clastic curve:
`I&PX + PI. x ——,',VL
`
`Cs
`
`—,)PL'ig.
`
`Ol'I !/ =
`
`( —x'
`
`3L&r —2L«)
`
`(8.11)
`
`P
`SEI
`Thc. deHcction an&i slope at A arc ol&t;&inc &1 h) letting x = 0
`in I'qs.
`(&8,11) and (8 ',)).
`YVC f»1(1
`PL'
`SEI
`
`(/& =
`
`&1&lcl
`
`f).& =
`
`PI.
`d&g 'I
`dr «2EI
`)
`
`(
`
`I»t(" I"lt&ng In x, &VC ol)t&1&n
`
`cl(/
`
`(h-
`
`El —' —'-Pra + C:,
`
`M'( non obs('rvc tl&a. &tt thc.'irecl &'ncl
`0= d(//dx = 0 (Vig. i8.11). S&tbstituting
`«nd si&lv&ng to&'I, wc
`
`h«v(',',
`
`= 'Pl-
`
`(S.iS)
`
`= L and
`II &vc'ave.'
`thcs«v&&lues &»to (i8.8)
`
`['i = l., &/
`
`-= o]
`
`)vl&ich wc c&&r&y luck it&to (8 8).
`
`Fig. 8.11
`
`()?
`E!&3'' -'&I('
`1 h('t»&ply sui)pc)1't'ccl pl'Isn)atlc
`hct»11 AII c&u I'&('.s a &Inl-
`length (Vig. 8. 12).
`I?Ctcr-
`lo»d tc per unit
`Iorml) &listributecl
`thc'narimum de-
`tl&c & I&&stlc c«rvc &tncl
`&»lac tl&» c&i»ttt&on ol
`flect&o» &&f th('ca&n
`
`I
`I
`
`tT(iR
`
`Fig. 8.12
`
`Fig. 8.13
`
`f&sec-bocly diagram of'hc portion /&D of thE".
`Dr;nving th(
`bc&&») (1'g. 8.13) ancl
`taking mom&»)ts about D, we find that
`IU = 1tcLx
`2
`1&('r
`
`(8. 12)
`
`for III into Eq. (S.q) and multiply&ng
`Sul&stituting
`l&ers of'his eq«ation l&y thc constant LI, sve writ«
`
`d 8 1, 1
`El, = ——tcr- + —&cl.r
`
`dr
`
`2
`
`1&oth mcn)-
`
`Integ&"ltnlg
`
`twlcc «1 x, vvc have
`
`cl(/
`dr
`
`1
`
`s
`
`1
`6
`
`C,
`
`El —' ——»:ra + —(oLx'
`1, 1
`El (/ = ——(ux' —u Lr + C,r + C.
`
`(S.13)
`
`(8.14)
`
`(8,15)
`
`Costco Exhibit 1007, p. 9
`
`

`
`I)
`
`DEFLECTION OF BEAMS BY INTEGRATION
`
`)/ = 0 at both c»ds ol'he b«am (1'ig. 8.14), lv(
`Observing
`that
`let x = 0;md p = 0 in L'(I
`(8. 1,5) and obtaii) C'4 = 0. U(c
`first
`x = L and )/ = (I
`tlie» inak(
`sam« c(luatioii;uid
`lvntc
`in tli(
`0 = ——.,',«I.'+—,',«I.'+D,l.
`( i =
`I)CL
`C(u'13'1)lg thc v(lilies of C ) a»d ( a h(ick l»to L'(f. (8. 15), lvc oh-
`th«c(tiliitlo)1 ol
`tl 1(" el'istic c»)ice;
`t()ill
`
`Ol'l )J =
`
`r,) «'x + I ''LX
`
`.) «.'L
`
`Fig. 8.14
`
`(c
`2-1EI
`
`(- x'
`
`2Lx'
`
`L'x)
`
`(8. 16)
`
`Fig. 8.15
`
`I
`
`II
`
`8»bstituting
`into Lz(f. (8.1-1) the value ohtliinecl
`lor C), wc
`the slope of'the beam is zero f'r x = I.!2;mcl tl&at the
`chccl. that
`the midpoiiit C of'hc hclim
`elastic curve has ii
`at
`ininin&uin
`(1'ig. 8 15). Letting x = L!2 in L(f
`(8.16), wc have
`. I.'i
`L
`8
`2
`
`(c /'.
`
`16
`
`dc(lcctioil ol',
`1 h« ll'iaxllniiln
`;ibsoliit('al»c of'h«d«flection,
`
`lllolc pl'ccisclv,
`is thiis
`
`thc»&ax»nu)))
`
`5)CL'8-fi:
`
`1
`
`5(OL'4EI
`
`884EI
`
`I» e;ich of th«hvo cxa)»pfcs considered
`so f«r, only one fic«.-bo(11
`rc(loire(1 to deteri»inc
`thc bmuling mom(.nt
`cliagram was
`in the lieam
`As;1 result, a si»glc fi&nctio» of x wils used to represent
`lt1 throughout
`thc
`beain.
`This, 1&ovvev«r,
`is not ge»ei.ally thc c;lsc. Con«cut)a&tcd
`lo;&(ls,
`1 cactlolls lit. suppo&'ts
`or chscontlllultl«s
`ll chstlnblltcd
`load w&11 nlllkc lt
`nl
`necessary to clividc the beam into several portions,
`and to represe»t
`tlii
`fi&»ction M(x) in each of'hese po& tions ol
`bending moment by a different
`beain. Each of the functions M(x) vvill
`then lead to a different
`expression
`for the slope H(x) and for the deflcction ()(x). Since each of'the cxp&essio»s
`obtained for the cleflection niust contain two co»stailts of int«gi.;ition,;i
`large nu&nber of co»st«nts xvill have to be determined.
`As vve shlill se('i»
`th«required
`th( next example,
`aclditional
`bounclary
`conclitions m«y
`by observi»g th'lt, while the shear and bending moment
`can
`thc slop('f
`at several poi»ts in a beam,
`the rleflection anti
`the bea&» ('(&»»ot f)e cli,sco»tinno«s
`at any point.
`
`h('iscontinuoiis
`
`h('btained
`
`'). (),)
`1',i;t)1)1) I(
`For th(. prismatic beam and tlie lo(«ling shown (1'g. 8,16),
`determine
`the slope and deflectioi) at poi»t D,
`
`WVC must divide the beam into t)vo portions, AD and DB,
`and determine
`the function plx) vvhich defines the elastic ciirvc
`for each of these portions.
`
`I./4
`
`31./4
`
`n
`
`Fig. 8.16
`
`Costco Exhibit 1007, p. 10
`
`

`
`I'o»&,t
`/o D I r & I /4).
`VVC cfr,uv the f'Iec-bocly (lia-
`«r,ini of a portio» of 1&&.'anl AL( of ln)gl.h x & I./4 (I'Ig. 8.17).
`];&Icing Iilo»ic»is
`ill)out L, wc
`
`hav(.'3('/(*&»Ii»&I/io» 0/
`
`tl & I o»st&»l(s o/
`l&ii& g)a&(i »I The co»-
`ditions which inust hc satisfiecl hy the constai)ts of iiitegration
`I lilvc hccn su»1»l,u i'Icd ln Elg. 8. 1',3 At the support A, xvherc
`
`8.3
`
`EQUAT(ON OF THE ELASTIC CURVE
`
`(0
`
`[
`
`- » ((i
`
`- &)]
`
`] &
`
`=.- /, II, - &) ]
`0
`
`((I —— (I, j
`
`D I
`
`/.
`
`oi
`
`rec;illi»g Ecf. (8.-1),
`
`(8. 17)
`
`L'l, = —Px(lr-'8. 18)
`
`(h/,
`
`3
`
`the clastic curv« fo&.
`ulu i'e pi(x) is the functi&&» &vhich defines
`1»teg)"lt»ic& I» r, svc write
`))&)I I to» AD o3 (/Ie I&&'(»».
`
`Fig. 8.19
`
`LI 01 = El
`
`(IIJ i
`
`(IX
`
`3
`= —,I'x + Ci
`8
`
`LI I/) = —I'X'
`
`1
`
`C JX + Cs
`
`(8 19)
`
`(8.20)
`
`8
`
`x}&
`
`F.
`
`Fig. 8.17
`
`ii
`
`Fig, 8.18
`
`&»&i D /o P i.l. & I./4).
`)Vc»o&v dra(v the f'rce-body
`portion of beam AE of length x & L/4 (V)g, 8 18)
`&ling(,uii of'
`
`an&I
`
`hI«= —x —P r ——
`
`3P
`
`L
`
`&ir, r&.c,illiiig E&f. (8.4) ail«I
`
`rearranging
`
`tenn»,
`
`cl-'y(;
`
`(Ira
`
`1
`4
`
`1
`4
`
`Cx
`
`El —,= ——Pr + —PI.
`thc elastic curve for
`'v)c're (/a'&I
`) is th&. functioii which def'Ines
`(lga 1, 1
`} I»'iiv» Dl3 of the l&e&i&»
`Integrati»g
`in x, xve writ(.
`= ——,Px + —PLx + Ca
`L'I 0« = LI
`1, 1
`4
`8
`El (0, = ——Px' —PLxa + C, x + C
`tf
`
`(8.21)
`
`(8 22)
`
`(8.23)
`
`(8.24)
`
`)vi&tc'PL« 11PI;
`
`tlic clefle«tin» i» dchned 1&y E&1. (8.20), we must have x = 0 «nd
`I/I = 0. At th«support
`/3, xvhnc the defi«ction is clef'ined by
`E&1. (8.24), xvc must have x = L a»cl 0. = 0. Also,
`the filet
`that
`th«1'('l»1 bc»&) s»dd&. i) ch;inge in deflect)on or in slope at point
`(Ji = I/ &ulcl 0) = 0» when x = L/4.
`D i«quires
`WUe h«ve
`th;it
`tlicrcf'orc.
`)x = 0,
`I/, = 0),
`I':&f. (8.20):
`)x = L,
`iya = 0), E&1. (824).
`
`(8.25)
`
`0 = Cz
`0 =—PI.«+ C,l. + C,
`1
`]2
`;3,
`7—,
`)x = I./4, 01 = 0.], E(fs. (8.1(J) luicl (8.2'3):
`PL- + C, =
`,
`128
`128
`I/, = i/,], E&f». (8.') and (8.24).
`+C)—=
`+Ca —+C»
`, I.
`PL'
`11PL
`L
`512
`4
`1,5,'38&
`Solving these ec}uat)oiis simulta»eously,
`
`]x = I./1,
`
`PL- + Ca
`
`(8 2G)
`
`(8 2()
`
`(8, 28)
`
`we lind
`
`128
`
`'
`
`128 ''384
`
`PL
`
`S)lbstltutlng
`fol'
`fol x ~ L/4,
`tlillt
`
`I &ui(1 Cz Into Eels, (8. 19) (ulcl (8,20), vvc wl ite
`
`3,,
`LI 0) = —Pr
`8
`El gi = —Pr'—
`1
`8
`l.etti»g r = L/4 in each of'hese e(fuations, wc find that
`sl&&pc a»d clc.flectioii at point D are,
`respectively,
`
`7PL
`128
`7PL«
`
`12&8
`
`1
`
`the
`
`0(& =
`
`PL«
`
`a»el
`
`(Jo =
`
`'5GE I
`;32El
`IVC note th(it, since 0» 0 0, the deflection at D is»ot the mlixi-
`mum deflection of'he 1&earn.
`
`3PL«
`
`Costco Exhibit 1007, p. 11
`
`

`
`C H A P
`
`T
`
`E
`
`R
`
`N
`
`I
`
`N
`
`E
`
`5ÃPLICN~OM F IRWWS
`WIVHP35
`
`O'9'OMIV='I''i'4Ã%4
`
`In the prcccdillg clliiptcl, wc»scd ii mathcnlatlcal
`lllcthod lrasc(l oii
`thc integration
`of a differcntial
`cquatio» to d«tcrminc
`tlic dcflcctio» i»ii1
`slope of a beam at a given point. The bc»cling moment
`svas expressed as
`a I'iinction M(x) of'he dista»cc x measured;flong
`the beain, and txvo suc-
`cessive intcgrations
`lcd to thc functioris 0(x) a»d (~(x) representing,
`respec-
`thc slope and deflectio at any point of thc beam.
`tively,
`In this chapter, we shall use. certain geomi
`tric properties of the cl;is-
`tic ciirve to determine
`the deflection
`and slope of a beam at a given
`Instead of'esprcssing
`point.
`thc bending naoi»ent
`as a f'unction
`iII(x) an(1
`f'unction
`integrating
`this
`svc shall draw the diagrai»
`analytically,
`repre-
`senting thc v;iriation of ltllEI over thc lcngtli of the beai» [Sec, 9.2].
`iV«
`two mom(nt-are
`then derive
`i theorei»s.
`The fi»st »101»ent-al«a
`shall
`tlieoreiii
`ivill enable us to ciilculiite the iinglc between the tangents
`to the
`beain at
`the secoiiil nioinent-aieii
`two poiiits;
`tlieoisein will be used to
`calculate the vertical distance f'iom a point on the beam to a tangent;it
`a
`second point.
`In Sec. 9.3 the moment-area
`theorems
`xvill be used to dcterini»c the
`at selected points of'antilever
`slope and deflection
`beams
`anti beni»s
`I S C.9.4 cslallf ielth;t
`et-cli&adi gs.
`i
`ths&
`i
`areas and moments ofarcas defined bv thc itIIEI diagram mai be i»or«
`if wc draiv the 1&en&li»g-inoi»ent
`easily determined
`1&ft t&ai ts. As
`elia~~rain
`we study the moment-area method, wc shall observe that
`this i»«tliorl
`effective in the case of l&canis of carial&le cioss sectiaii,
`Beams with unsynanaetric
`beams will
`li«
`and overhanging
`loadings
`considered in Sec. 9.5. Since for an unsymmetric
`loading the masii»iiiii
`learn in Scc
`deflection docs not occur at the center of'
`beans, wc shall
`t"
`9.6 how to locate the point where
`in ordci
`the tangent
`is horizontal
`Section 9.7 will be devotcil
`to tli«
`determine
`the ~naxinn»n
`clef7ectioi»
`solution of'roblems
`staticallit
`involving
`in&leterininate
`1&eanrs
`530
`
`-, 'l
`
`i'articularly
`
`Costco Exhibit 1007, p. 12
`
`

`
`9.2 MOMENT-AREA THEOREMS
`
`531
`
`'2. MOME'&)7-Ail)IEA ll HEOAEOI)S
`
`Coz&sider a beam AB sub)ected to soznc arbitrary loading (I'ig. 9.1(z).
`&'I'e shall draw the diagram n prescnttng
`along thc beatn of
`the variation
`IIIIEI ol&taincd l&y clividing
`th» bending moment M by the
`th«&Iuantity
`rigiclity El (Fig. 9.11&). &Ve note that, except for a dif'ferencc
`flcxur;tl
`in
`th«scales of ordin'ttcs,
`this diagram «ill bc the sazne
`as the bencliztg-
`rigidity of the beam is constant.
`zi&&m&ent cliagram if th» flexural
`Recalling Eq. (8.4) of Scc. I&.3, a cl the f, ct th; t dtjI 4 = 0,
`
`e vrite
`
`AI
`
`(
`
`)
`
`Oz
`
`&10
`
`&1-'p
`
`III
`&lx &1x-'l
`&10 = —&lx
`
`III
`El
`
`(9 1)t
`
`poizzts C and D on the beam and integrat-
`t«(o 'zrbitrary
`Considering
`l&oth zn& mbers of E&I. (9,1) from C to D, we write
`
`ing(
`
`&10 = —,&b
`El
`
`oz
`
`Ot& —0&; =
`
`I"&
`
`) —dx
`
`El
`
`(9 -')
`
`Fig. 9.1
`
`~
`
`C D~
`
`IB
`
`C
`
`D B
`
`D
`
`C
`
`e
`
`/&
`
`B
`
`T e&
`
`B
`
`D
`
`(I'ig. 9.1c).
`wlz»r«0& and 0» denote the slope at C;md D, respectively
`nzenzbcr of E&1. (9.o) reprc.scnts
`thc area under
`right-ha»d
`the
`Bttt
`thc'.
`(IIIIEI) diagran& bet«'ccn C ancl D, and the left-hancl member
`thc angle
`to the clastic curve at C and D (I'ig. 9.1d). Denot-
`between the tangc.nts
`ing this angle by Ot»c, wc have
`
`Ot&(c = al'ea tlllclcz'tl(IIEI) diagl"lln
`between C ancl D
`
`(9.3)
`
`This is the first
`t)zo»zezzt-a(ca
`theo& e&zz.
`the area under
`the angle Ot&(& ancl
`'&'«e note that
`the (I(IlEI) diagram
`a positive area (i.e., an ar.ea located
`have th«same sign,
`In oth( r words,
`rotation of the t.uzgent
`to a counter«foe&vise
`ahov« the x. axis) corresponds
`area corre-
`to thc elastic curve as we move f'rozn C to D, and a negative
`to a clock«&lse I 0t&ltzozl ~
`spol&&.ls
`
`i I'1»s (elation niay also be dcnvcd &vithout
`rcfernng to the results obtained in Scc. 9 3, by
`the illlgli.'Ie fornied by th(.'an &'i&ts to tlic elasnc curve at P and P's also the
`»&&t»ig tliat
`to that c&irve (Fig, 9,»), We thus have (Ie =
`a»gic I&&iined by the corn seondir&g nornials
`. I,
`«'i&'i & (Is is the length of the arc PP and p the radius of curvatui
`& at I'. Substituting for
`wl
`II
`'p I &»ii E&I
`(-I,gl). and noting that, since the slop&. at P is very sinai),
`I'r&
`in first
`(Is is c&nial
`dist.»ice (Ix l&ct&v«(n P and P', wc wnte
`aI'I's»»,&t&o» to the honzontal
`
`'
`
`&&&I
`
`(le = —,(Ix
`EI
`
`(9.I)
`
`(Ie
`
`P'f)
`
`(Is
`
`Fig. 9.2
`
`Costco Exhibit 1007, p. 13
`
`

`
`DEFiEOTION OF BEAMS BY MOMENT-AREA METHOD
`
`Ffg. 9.3
`
`(
`A
`Fig. 9.4
`
`2)
`
`txvo points P and P'ocated hr-tween
`l.et us now conslclcr
`6) and D
`(I" ig. 9.3). The tangents
`dx from each other
`to thc
`,md. at a distance
`elastic curve drawn at P and P'ntercept
`a segmc»t of le»gfh df on thr
`thl'ough pol»t c,
`s luce thc slope". 0 at P 'Ind thc «1»glc d0 fo)'IH('rl
`vcl'tie'll
`llt P a»el P,'u'c hoth sn»111 cflllll')f.1tlcs, wc HIBy 'llssulnc
`hy thc tangents
`tl'I,'lf
`to the are of'circle ofradlus
`thc angle d0.
`xl suhtcndlng
`dt ls equal
`the)'cfol c,
`have,
`
`or, s»hsfituting
`
`for d0 f'ronl
`
`frrp (CJ.l)«
`
`df=x)
`
`I")I
`El
`
`dx
`
`(9.qj
`
`«ltc &~(p (9.4) fr0171 C t0 D.
`)Vc»ot(« that, as po)»f
`EVc shBll nolv lntcgl
`thc cl,Istic curve from C to D, thc ta»gc»t;lt P ssvccps
`thc
`P describes
`through C from C to E. Thc integral of'hc lcf't-h.uld memhcr
`vertical
`is
`(if. D.
`fl ol» t) to thc t«ulgc)lt
`f his
`to tl'lc vcl'tlcal
`«l)stance
`thus
`c($(lal
`(Ie(:iation of C II:itll
`distance is clcnotcd hy tr»& an(i is calle(1 thc tangential
`to D. M)c have,
`therefore,
`)est&ect
`
`fr II& =
`
`f-«
`f
`
`«
`
`xl—dx
`
`LI
`
`(9.5)
`
`lows: Tile tangent'Icll deolatlon f(»& 0jCi Io)fl) I especf to D ls e(/ «a/
`
`an clement of'area unrlcr
`KVC now observe that (M/EI) dx represents
`and x)(AI/LI) dx thc first momnlt
`of th«lf. clcnlcnt
`the (M/EI) diagram,
`to a vertical axis through C (1'ig. 9.4). The right-hand mcnl-
`with respect
`bcr»l Lrq. (9.5), thus,
`thc fn st n&0»l(.nt w)th Icspc(*t to th«lf.
`I cplcscnts
`rliagram hctwcen C and D.
`axis of thc area located under
`thc (I)I/EI)
`the) cfolc,
`st«lfc thc second n)onlent-a)e(I
`f lien) eP)l «Is fol-
`IVC nlav,
`to
`to a oertiral axis fhl-o«~«12 C oj fl)e area «»der
`respect
`)no«lent
`Ieitl)
`I&et(ace» t", and D.
`the& (III/EI) diagra)n
`lrlr)n'lent of an ale'I with rcspcct.
`P(ccaf ling thl(t. thc fll'st
`to lln llxls
`Is
`to the product of'hc area and of the disfa»cc from its centroid to
`erfual
`svc IH;ly «llso exp).css thc scconcl Pion)cnt-.u.ca tl)co).cnl as fol-
`th«lt «lxls,
`lows:
`
`fl«'irst
`
`tcjl& = (Bl.c«l hchvccn C luld D) xl
`
`(CJ. (l)
`
`I)'here thc area refers to the area under
`
`2&'
`
`I
`
`and where x)
`the (III/EI) diagram,
`is the distance from the centroid of thc area to the vertical axis through t
`(f«)g 9 5a)
`between the tangential
`deviation
`Care should hc taken to distinguish
`to D, rlenoted hy tr!II&, «uld the tangential deviation of D
`of C with respect
`hy f)&,r, The tangential
`deviation
`with respect
`to C, which is rlenoterl
`
`Fig. 9.5
`
`(I&)
`
`Costco Exhibit 1007, p. 14
`
`

`
`93 CANTILEVER BEAMS AND BEAMS WITH SYMMETRIC LCIADINGS
`
`633
`
`the vertical distance from D to the tai&gent
`r«presents
`t»,
`to the elastic
`cur&e at C, at&cl
`is obtainccl b) n&ultipl&ing
`the area under
`the ()tl/EI)
`cjj,'&grain by the clistance rz f'rom its centroicl
`to the x crtical axis through D
`(Fig. 9.5/&):
`
`t„/, ——(a&ca between C ancl D) Xg
`
`(9 /)
`
`We note that,
`if an area uncler
`the (/»I/FI) cliagram is located above
`its f'irst mo&nent with respect
`to a s ertical axis will be positive;
`thc x axi»,
`if
`i» locatecl below the x axis,
`its first moment will bc negative.
`it
`As we
`n&'r) check from Fig. 9.5, a point xvith a positive
`tangential
`deviat&on
`is
`the corresponding
`located a/&ave
`svith a»egatiue
`tangent, while a point
`deviation woukl be located f&etc&&a that
`tangent.
`t,u(grential
`
`of'lopes
`
`We recall
`the first moment-area
`that
`theorem derivecl
`in the preced-
`section clcfhlcs
`'&ctracan thc tangcults
`thc allglc 0
`i»
`at hvo pohlts C
`anrl D of the elastic curve. Thus,
`thc angle 0D that
`the tangent «t D forms
`i.e., the slope at D, may be obtainecl only if the slope
`&vith thc horizontal,
`at C is know»,
`the second moment-area
`Shnilarly,
`thcorcn& defines the
`&cr tical distance of one point of the elastic curve from the t,u&gent
`at
`Thc tangclltial
`clcviation t»/('herefore,
`al&othe& point.
`will help us lo-
`cate point D only if the tangent at C is known. We conclucle that
`the hvo
`theorems may be appliecl effectivel
`to the determination
`n&on&ent-area
`only if a certain refe/ence
`and deflections
`to thc elastic
`ta»g&e»t
`curve has first hec» detevtninccl.
`In the case of a cantilever
`(Fig. 9.6), the tangent
`to the elastic
`/&ea»r
`curve at
`the fixed end A is known
`and may be used as the reference
`Since 0a = 0, the slope of the beam at any point D is 0r& = 0/&/,,
`tangent.
`and n&ay be obtainccl by the first n&oment-area
`theorem.
`On the other
`the deflection t//& of'point D is eclual
`to the tangential
`deviation tr&/,
`h;u&d,
`reference tangent at A and may be obtarnecl
`n&easured f'&om the horizontal
`1&y the second moment-area
`theorem.
`In the case of a simply supportccl beam AB with a slJuunctric
`lonrfiug
`(I'ig. 9.7a) ov in the case of an overhanging
`symmetric
`beaten with a sym-
`inctric loacling (see Sample Prob. 9.2), the tangent
`at the center C of the
`by reason of symn&etry
`beam must be horizontal
`and may be used as thc
`(Fig. 9.7t&). Since 0c = 0, the slope at the support 8 is
`rc fcrence tangent
`0/; = 0/&/c.
`a&&el n& ly be obtained by the first nlonlellt-area
`theo&'cun
`(Ve
`also note that gtI„,„ is equal
`to the tangential
`cleviation
`ta/c and may,
`therefore, be obtainecl by the second moment-area
`The slope
`theorem.
`at any other point D of the bean& (Fig. 9.7c) is found in a similar
`fashion,
`as lj/& = t/&/c
`the deflection at D may be expressecl
`'u&cl
`
`D
`
`ll(t&J&('(lf.
`
`ilr
`
`t&
`
`II I) = /I& (
`
`0/& = 0/&/,
`Fig. 9.6
`
`IC( ((('( (((( u(»r&(nt
`
`(a)
`
`1 tn('(/(&lit((I
`
`(&n
`
`(n)
`Fig. 9.7
`
`Costco Exhibit 1007, p. 15
`
`

`
`Appendix 0
`
`Elastic Curve
`
`Maximum
`Deflection
`
`Slope at End
`
`l/
`
`t)
`
`0
`
`IJ
`
`0
`
`L~
`
`r
`
`PI s
`3EI
`
`u:L'EI
`
`MLs
`2EI
`
`raI
`6EI
`
`PLs
`16EI
`
`—(x'
`ra
`2-HEI
`
`v
`
`4Lx + 6L-'x
`3
`
`For x & ~tI.:
`(4x" —3L'-'x)
`p
`48EI
`
`gl ~a~
`
`For a p b:
`Pb(L2
`
`b2)s/'&
`
`For x Q a:
`
`For x = a'a'b'EIL
`
`b'-')
`
`as)
`
`Pb(L'
`6EIL
`Pa(I 2
`6LPL
`
`5u L4
`384EI
`
`(x& —OLxs + I.sx)
`
`2,4EI
`
`ML
`6I'I
`
`M
`6EIL
`
`Costco Exhibit 1007, p. 16