._1 [ ii hç 'J 11 I ,.,"1 l-aI --*_l

I i. .r'r 'l

:-<TàJri' €ï 14r \r ! y-[r,.44,r. vJ Frø7.t

=----t)'F. :-Ò

L___=a_\tI rí., T it/\ *. --.*\

--=En'!J

tr:| /-

I' /ì

-)NAI't

/ \) \/ \t ¿/At/ /\ 'Â.tf-.^ n L\ | L)1¡_t'', ./i

--:-j ",

.,1-,Y?q¿ ',^,,t .¡n r-¡

-,,;/- -, r¡' /'- i (u

,2\ I/)\ - V \ ¿i¿r,\ (-.,^\

\ t\

\^"1 h t' ä, Ê, =o ) Z l4r. .lres wrtJryr.l) ,nry -ir

@ .^, /

)\ /r)r,

Ir0

-) /"(is, / \/\

Wû- "art ¿L,o¿-t.,-

t, ,, l¡ IV,S AqvtJ a þ,ccSq

Cl tlrl e'JI

t'ltt { (zMc4ltø^f é;ìc_

! \¡\, rr\rl r¡rlvrùwvl <l u\Jllllllull aÀlù allu ¿Ic Ilu r ljararrer [O One AnO

ffi'' --'tr'-ëffiF,..,Æ ,: : ,. .,-:, . _.:r:: .'.. . .. ..... . . . :. ,. :,:.. .':,..t..; |.]:..:i:1.::.-:i:.;:.-i Three-dimensional equilibrium problemsj solved using the following pro""åur".

i::ir,l- Ii:lj ! i.,i :,y.j i¡, i:.'.e Draw an outlined shape of the body.,,r Show all the forces and couple moments acting on the body.,,' Establish the origin of the i, y, , axes at a convenient point andorient the axes so thar they are paralrel ,o ur'nuny årìrr5ã*t..nurforces and moments as possible.'¡; Label alr the l0adings and specify the.ir directions. In generar,tå:î,.',

,t!:::known components having a positíve,rn.i, utong

" Indicate the dimensions of the body necessary for computing themoments of forces.l

If the x, L z force and moment components seem easy todetermine, then apply the six scalar equations of *[,riUürirr;otherwise use the vector equations." It is not necessary that the set of axes chosen for force summationcoincide with the set of axes chosen for moment summation.Actually, an axis in any arbitrary direction may be

"frãr"n fo,summing forces and moments." choose the direction of an axis for moment summation such thatit intersects the lines of action of as many unknown tbrces aspossible' Realize that the moments of foices puuring-tt rougr,points on rhis axis and the moments of forces *iri"r, uå fäLailetto the axis will then be zero.lf the sorution of rhe eq.uiribrium equations yierds a negarivc

:,:"lit for.a force or couple moment nìagnitude, it indicates thaithe sense is opposite to that assumed on ih" free_bodf ã*;";

Iypes of Connection Reaction Numbei of Unknowns ffiJ) ="*ffi

'-m\.v

Fg,--èa\ 1:l\- t_+\\..iii

\.'i,

One unknown. The reaction is a force rvhich acts ariàofrom.the member.in the kno\ n ditection of the cable.. :'.,

One nnknorvn. The ¡eaction is a force which achperpendicular to the surface at the point of contact. .

'7) .,*Yþ

,.+'ffi'Wnooth sulface supporr

/ .,'f.,1i':!f¡ïnI

One unknorvn. The reactio¡ is a fo¡ce whìch acrsperpendicular to the surface at the poìnt of contâct-

F, ,fììFiV,,r/--'*oFf -,

Three unknowns, 'lhe reactions arc three rectangulârforce comþonents.

hail ¿nd socket

Four unknowns. The reactions are t\.vo Torce and twocouple-moment componenls whicb act perpendicular eo

the shaft. Note: The cou¡ie moments ere senerâlìv no¡applied if the body is supported elsewlrere. See theexamples.

single journaì bearing

Types of Gonnedt'-' Rèaction Nunìber,óf. Unknoú¡s):-&.br/"'%.--(-

cabie

One unknown. The reaction is a tension force ryhicbaway Jrom thè member in the direcr.ion of the cable.

ñ, o; ;j{f one unknown. The reacrion is a force v/h¡ch âcrs âlo

tÞ ù. ,' \) the axis o{ the link'

\J'l {Ê f7

'F ,//Þr f,#,rolle¡ F

One unftnolvn. The reaction is a force whìch áctsperpendicuìar ro the surface at the point of contact.

".#ê ,, - n#F one unknown. The rc:crion ìs a.[o¡ce which a*s¡ ¿l r L\ pcrpendrcular to the slot.

roller or þin in:confíned smoÒth slct

r5) ffi'f.Ê-

-'.3[r*'..'(fl r/------/'-

F

One unknown. 'Ihe reaction is a force which actsperpendicu.lar to the surface at the point of contact-

(6)

#Éæfl---

smooth cortactingsurlace

One unknown. The reacfion is a fol.ce which ãctsperpendicrìlar to the surlãce at tlte point of contact.

.. /-lo¡ tx .--' -t,t Y. .--

{¡.One unknown. The reaction is a fo¡ce rvhich actsperpendicular to the rod.

mcrnber pin conncctedto collar on smooth rod

5-2 Fnre-Booy Dlncnqvs 203

Types ofConnection

8)&fñ¡e

:Iè.i

. . .,.: .,' .... ..;._¡ .:...,;1.ì¡,ii;riil.,iTwo unknowns,.,.Tlrg; .re4ctions are frvo components oIlorce, or Lbe magni[ude ahd direction r] of tbe rcsultantforce..Note thar¿ and 6 aig not necessarily equal fusuallynoL, unless tbe rod sbown,is a.tink as in {2)1.

arooth Pin or hinge

Ð ¿;p.K¡a-

lber fixed connectedcolia¡ on smrioth îoä

'lwo unlçnow¡s. Thé.Éáðtiii4S are the couple momentand the.force which acts perpendiculâr to the rod.

ffir0)

':

FÀ.. Ã,^ fhree u¡knowni. The reactions are the couple momenr

or ( , Ê+:Å arìd the two-.fo. rce:co. $ji-o. henti, or the ccuple moment andM\- the mâgnirudè ânr!.dirþ_ction.14,of the resultânt force.

ixed support

ypical examples cf aciual süpports are shcwn in the follouiirg seqùence of photos. The numbers reÍe¡ to thenection types in Täble 5-1.

This concrete girderrests on the ledgo thatis assumed to act asa smooth contactingsurlace. (ó)

câblc exerts a forcc on lhe brackete dilection of the cable. (1)

Thc rccker supoort for this hr icLrgirder allows horízont¿ìl movemenlso the bridge is lree to expand andc{)orract due to a change intemperâtùre. (5)

This utility building ispin supported at rhe tÐpof the column. {8)

"fhe floor be¿ms of this buildingare rvelded together and thusform fixed connections. (1û)

l'r'li:;:,:" :j,- " j!:¡|::¡:.;

S-El" The sign has a mass of 100 kg with center of mass atG. Determine the x,l, z components of reaction at the ball-and-socket joint A and the tension in wires tsC and tsÐ,

i, l,'..; i,r, _i,*ti i

Ilm- i

1

tN*rn Pl-Ë

¿= ¿cr^,t)1,'.r (zrn),t

^'/"

ìy et*r-:=\-r n- -.i., )ide ì V/

For¡¿e-5

-> , A t, \-,- = l)/-1 | -7t "i- t/ I' r) .-J \ LJJ',L^'.4y-= C ( '' -.J î2-L _1

-\ ¡ ^ ¡ t" , l;"F^ = /-f*1 -.4'v I */}-=-["\1

/.,,.], /.s - r,.. \ (G'., .,/,2\ t)\Al * _ (Tlt\ r<{1- ), f..-Ih1{S) [

ús- z1)ß,"[À q^J su--r<el-3ì# ..{-o h,*

\n,t tgo,. .n. lt/va.nev,Å\,

{ - 6-zrr)i' * crø ¡,t'-/F'Ð

D

r"

-'-nnfrLrtcnnt )

\N¡.', U/\4{-14

(¿oúIr0 Y-

ny

'a

',t)\^

I

N

Irþt-tu

Krs

-f

fiK

/.,(2n¿ ) {' -'/o

t')/\G,l n3"= G'rJcj, .*-ì ,^,øt- hS = (Z'"'14'

J-\iì.-\ ü

r/. L:- - (q'd ' îi)il¿ ú/ \

'.?

-trÍ-- .:'.8

t"- a

B

KV.

¿ \¡/

= ,'l vn\ 4

r'- - -Z rn| - J(

ô\L^'¿_ w=/{o*- ar*ctJlrw"x øì(it ho-ustJ S Jd*.rn,r, Ax,Åyr4¿søi¡r,.L \^^d_,,/t-{?i$ øþdi"^-f {r c C ,D

Yr.a s,¿u..b "L*rf ø t

-,-l 4,î2,'\.. l,N,oÐ -- (zv 11' x r) (-ri -{ +Ji ) -luir t?r"1t o-- (zr^ )j x e (î -.¿ t uL) -z cÎa raclì'

tra =o

tr,* = (ru)$ x ( llt u) i. *- - (qar N'(J á

7D r4¿-îAi I'J --oTl À¡Z-t iìlrt'tt:o-.i) \ Wlit-: -,^( t I \,7ì- =-\/._l

(r - ç¡\,--,/ 4 t¿. )

Tu-S¿-,, TO=SO

5Bq N

zq+ N

Qi¡--

'|l./

w-LÞ U

trl-*lJ

Lfx--o *.¿D+C-t"A<--o

Fç =o *lD -7 L È¡{v = oII

h-h-¿- =u A*tD l ZC - I gL{ = Ò

A< "- aD"'Õ -20-20 "- Ð4y = aÐ flc- = {p = 6 (Tg ir,r)

Az = ? rcry * Ð--zc- = 1Fl n{ - tsl 6't i N/) _I

/\/ \ x_- tl

\ = 58?'{4+ = 4lo r,{

--_1_

è)1_

ì

SrrttIe- r+<ci,ril{¿ * r[ryi*'. uJAiì

n I t i t L r-ì{,w1-. Cc.Â( id.,¡¡ W,Õ¡Iqt,'l\ ,.,¡l:,1r1 ij",'=. )

-J ^¡ I' t,/t*. laX = 0+ l"'61. -tr

¿i MBx' o Å+ = tfi'J = 41c Iv

L_'

î

,¡¿crøø,.Ñ ^l"rrÅ-C>

=1-i t.l. --\

!- {'I' f /lr,+= ù -r i- v< --1t. .' 1T'o : ;TL.-) r_ìc-ì--l r, f-l .Õ''l -La:Ll'--I I i/\/tI n l\' loK - \-' I O,>.¡ ¡ì - n r-', 4

- L- -;'\ \l_l\ - , l, +j_, l,r =- =¿'l:,L ¡i*c'a-.t ='t)

AT ={90 u

5a1m *'>- €x-a r,rr¡ lg

ìr.st a-þ r¡"cu,act/,Í o//,"1

i t-1, .1

^.ì- /- I\ ltì

t,t t tl,[y' tt n,\:¡ \wvv \tL.,r ll) vv' ¿

Ev¿,14'er

¡t rut

Lr{ttfd

/¿:(au,'ün^,s

Q k-1 r.l

o

O

C)

Q8-lu,i:

?

L

r)I

(

.- ¿l(_

L

LT

L-

,]_

L,

/-

c

O

ô

Ò

.1l

L-

-D

Å

/\^l

Æ-

18tv

oG,qL

OÒOCIr)O1dc

t)o

/V

--f)Jô)1\¡1 -

5-79" The bent rod is supported at A, E,and c by smootrrjournal bearings. Compute the x, y, 7 components ofreaction at the bearings if the rod is subjected to rorcesF1 : 3001b and Fz: Z50lb. F.1 lies in the y_z plane"T.hebearings are in proper alignment and exãrt ånly forcereaetions on the rod"*S-8&, The bent rod ís supported at,,4, B,and C by smoothjournal boarings" Determine the magnitude of w,, wtrich witieause the reaction Ð, at the bearing c to be equal to zero.T'he bearings are in proper alignment and exeri only forcereactions on the rod. Set Fr : 300lb.

1

l

5-8L" The sign has a mass of 100 kg with center of mass atG" Determíne the x,l, z comporlents of reaction at the bail-and-socket joint ,4 and the tension in wires EC and AÐ.

zI

ruts5

0Õ+tå)

ÕÕ

Õ

IA*

Õf..-/1t-J\:.i'

il

þ

þx

'àÕÕ

rú

l,

-.to\)o(¡

oo

C¡

.lo

o

HX

OrJo(n ll-r(Dcil--!

.DPô-r :l-i-Joo5(nC

ttPFi (nO-p

. c-l.

(D

-u) l-Ì!c+ \JÐ- Fl

( l,) .-l-

,.1 P):JH.r-: .l-J

;.H

.DÊDr¡o,'l -'-..i

-l_ l--l¡;\ r,rJ (n

'Ú

-to¡Jalrl

o(D

C¡

-lo

ofL)

Tt^,"-"g So* r^reffilcer

<ltt¡i(il;1¡;11.¡' '\

-, ¿-lTr, -- ô ) ¿_" G, xF, = (,

QSSt¡,r¡,,<_-

t.-. t) ta)' A,'l-È ,l--i J .2

c4r"e- f.otpøcr"l4øL

114.

T-{/_

t

-/.e- [J..r.- fu,*. ,;*J¿s \î,/,\\9+ 'r,u- vr, l-{rc 5 at^,ae plAa.e-

----:. --à-t-l¡R[=^,F-- ':let, \ It-) ., - ,v\{ ( plørr<-.--.>\' --a\ *-=) 'J ' Itr- :- - , F. *Ë--\ -i "r (1Y-r P'ot-tt (/ - -',e t\ y/

L / \ !.1nó | la. (n llt,a- Se'/,1p._

ti¡tLrn, {Y,fû lt€ I {ttOt,c!

I 1 i /.'fh ç r.'r- t,.¡^n' r rls:

t/.J-/ lA'n<'- u-xtt

1 t ^ilryt- 7-Ä,[1r'r il e+4

t, "-L i^ ,n*J t'i1Lu t u.'¿

a-

n.

Cl,,uo:.- "¡ur.Jiro,!,r.ç S, úÅ'of tlr,t*ç.fts ù ¿e7¿^ rl,¡r,r(ø,-

þ *14 o( {L.ose p(4ft1þ( planes

tll rto p Vr tU lcT (iil.¿l

".[c iVn 6{ Z - q^t :t)Li

vwove fun.n: on ]4isìr- 1'tso{ *dtd {. fiqirìiffi-,1;1+w,

-j-1-<ot^.lrttu't#^i-: ú *l'r

_)

irt,ilo Ê r.r r",,-o-ztÅs ,,,ú'^i A \tl :t-17 t I o ¡4t ! \

. -lt .n -l ¡/llÉs twrY- r'{ ',_ = ¡t

sìÁe vìçc, ia-okrn{ otort.''*. /-*X',

_ ß--

v iew I*uust b¿

't,Ålu¿t{sl Va1lßl^'t--t3^ifs.i=e' t1 ^ , -^r

',Jre\al. l.¡L l'{ .,q \

*of

1,1laq 6.¡1.tt.,,.tJ o,l"¿.,,,-tr|

lF-Il-.rx I \cÂl.-.lr

sU)l t (av^ ,tçr¡u t,.,orXJz^J, ei'fio. f=o..

'--Jr E.il

- Fìey lrJ*'*7 .

\ t)-s i.=- ò

e'+

þy o'iF**

- i= r=-' /r-K ì rJ¡

lr¿a i :l_ >..I '- '--'

rt Ô ..r^ {{w_ #r1-3\ .-:tþ* Fn \A. \ \5 ..,_'\r.,c1., wr_ (qvp*

ffiþçm{ n,c] \< n u(-- ff"*- €-r.S-1-J\.^1-r'lrq tt ç_¡\ I -trlrr\:L

-rþ.<- lr,,r<-e-. \i*Ls u& a"{v^", 6r-€- ù +{-- *0,,4,1e*,

tf \a^"* ol,NJ uvJn"su.rt u,f 0, (r-nr,iu 6- , VÒ,,4^I

l" o*- cr'rLtçl'*,'-ù)

I

\æ* 4 @,8 *"\

ItIr@--*-å "q"

-t\\, ¡ \*

\,\ $\'\0'

æ\m n o.\-¡* i ru&\*x

-----i-- --14 ./rl ,/-\\-

\ \-@

\ "!Çff \,w crçp

\\

q

*eø tù' = \e*þ L S .^{e .æ-----_- Tf S *rlnÞ æ-

Ø-= *+"^(tød-,) =