•

188 CH.6.~HR 4 FORCE SVSTE'-! RESULTANTS

• FUNDAMENTAL PROBLEMS

F"-37. Determine the resultant force and specify where it acts on the beam measured from A .

F4-37

........ lS. Determine the resultant force and specify whe re il acts on the beam measured from A .

ISO lb/ f!

• I I j 0 llTll l n III ± ·s

~." --.1.--.,,--1

F4-39. DeteTnnne the resultant force and spccif)· ,,·here it acts on the beam measured from A .

li tNlm

• A

- 3m-'-- .m---I F ..... 39

F+4fl. Determine the resultant force and specify whe re it :lets on the beam measured from A.

, . .....,

F4-Il. Determine the resuhant force and specify "here it acts on the beam mcasur~d from A .

6 kN/m rn rn I n-i'li'h

f-- 4.5m---' l.5m-;

F.i-42. Delermine the resultant force and 5pccif)· where it acts on the beam measured from , I.

t60Nfm

f4-l2

• PROBLEMS

4-142. Replace Ihe distributed loading 1I';lh an equil'alent resultant fOTC'C. and specify ils location on Ihe beam n>easurcd from point A .

" rob. "- loll

4-143, Replace the dimibulcd loading with an cqui--alent rcsult:lnl force. and specify its location on lhe beam measured from point A

'm--- --'m - -l' r(lb. 4-143

°4-- 1:14. Replace the distributed loading by an cquil'a lcnl rcsultant force and specif)' ils location. measured from point A.

- 2m-- 'm--

I'roh. 4-I44

4.9 REOOC1'ION Of' '" SIMPlE OtsTRIBIJTW l OAOlNG '"

4 - 145. Rcplare the dimibulcd loading wilh an C'quh'lllcni resultant force. and specify its location on Ihe beam measured from pomt A .

t --- t --"'rob.4-145

4-146. The distrihution of soil loading on Ihe bonom of a building slab is sllown. Replace this loading by an cquiwllcnt resul tant force and specify ils localion. measured from POInt O.

0 1-__

.--- tUl

l'rob.4-I46

4-147. I)ctermine the intensitie5 "'I and ".~ of the distributed loading acting on the bottom of tbe slab $0 that tbis loading has an equivalent resultant force tb3t is equal but opposite to the rcsult3nl of the distributed loading ;\(Cting on the top of the plate .

. Q J !'rob.4-147

•

190 CH.6.~HR 4 FORCE SVSTE'-! RESULTANTS

' 4-1411. The bricks on lOP of Ihe beam and the supports at Ihe bottom create the distributed loading sho""n in the seeond figu re. Determine the required intensity ... and dimension 1/ of the right support so that the resultant force and couplc moment about point A of the system arc bolh 1.ero.

•.

Prob. 4-I411

4 ... 149. '!be wind pressure acting on a triangular sign is uniform. Replace this loading by an equivalent .clulwnt force and couple moment 31 point O.

l.2m J O.l m

--, ,-

I.2m

'm "---~, 0-

"rob. 4- 149

4-150. The beam is subjected 10 the dislributed loading. Determine the length b of the uniform load and its position " on the beam such that the resultant force and couple moment acting on the beam arc ~cro.

b --I JOlb/Jl

tOft

601b{11

I

f'roll.4-ISll

4-151. Currently eighty-fivc percent of all neck injuries arc cau~ by rcar-end car collisions. To alle"iatc this problem. an automobile seal restr"1lint has been del'eloped Ihat provides additional p.c5Sure contacI with the cran ium. During dynamic leSr$ the dIstribution of load on rhe cranium has IM:cn plotted and shown to 1M: p.;IT3bolic. Determine the equi\'alent resultant force and ils location. measured from point A.

Prob. 4- 151

-4-152.. Wind lias blown sand o,'Cr a plat form sueh thaI the intensit)· of the load can be approximated by the fundion " . - (o.sxl) NJm. Simplify thi~ distributed lo.1ding 10 an equi"aknt resultant forC<.' and specify its magnitude and location measured from A .

. ' SOON/m

I'rob.4-15l

04-153. Wet ronerete exerts a pressure distribution along the wall of Ihe form. Determine Ihe resullanl force of this distribution and specif)' the height', where Ihe bracing suut should be placed 50 that II lies through the line of aetion of Ihe resultant force. The wall has a widlh of 5 m.

, ---

,.

, ~ ~"''''-----'l--''----'''!'<''.

1 I'roh. 4-I53

4.9 REOOCfION Of' '" SIMPlE OtsTRIBIJTW lOAOlNG '9' 4-154. Replace the dlstribuled looding with an cqui\'a1c1'll resultanl foree. and specif)' its location on the beam measured from poil'll A.

,. l' rOb . .1-154

4- 155. Repbce the loading by an equhoalent resuitant force and couple moment at pointA .

· 4-156. Replace Ihe loading by an equi"alent rcsuhant force and couple momel'll acti ng al point 8 .

lOll fbjfl

I'rolls. 4-13~156

·.4-157. The hfllnS force alonsthe wins of a jet aircraft consists of a uniform d,Slnbulion alons AB. and a sem'p,arabolk dlilnbunon alons BC wilh onsm al 8 . Replace thIs loiIdlll& by a slIlgk' resultant forrc and spccif)' 'Uloc:alion measured from polnl A .

• Zl!I!O Ib II

~ ... .. (!lIIlO _ s.rl) Ib Ie

~J1JLll l I ) ,

1: , "" r

.4- 1.511. 1~ diSlnbuted lood acts on Ihe beam as s.hcw."TI. Delcrmme the ma&nllude of II\(' equi\l1llent resultant fon:c and sp«lfy ",here II acts. measured from point A.

.4- 159. 11Ic distributed load KlS on Ihe hearn as s.hcw."TI. Determme lhe ma~lmum intcM!!y " '_ •. What is the magnitude of the equl\"llenl resultant force? SpeCify ",here It acts..masured from point 8 .

• ... . t - 2:,: + 4.\ - t') rb/fl

r '" I'mhs.. 4- ISIII ISI'

· .4-160. The distributed load acts on the beam as shown. I)ctermine Ihe magnitude of Ihe equi\'a lent resultant forre and spttl fy ,tslocation.measured from POllll A.

I'rob . .4- 160

04-1&1. If Ihe dislributlon of Ihe ground reacllon on the pipe per fOOl of length tan be approximated as shown, delermine the magnitude of the rnultant force dire 10 thIS Ioadmg.

... · ZS(l.f;"OSf)lh/fl ~Ib/ll

Proh. 4- 161

• PROBLEMS

Alillrobl~1II wlll/ialls ""1$/ illrlmir IIIl f"HO.

5- 11. Determine the normal reactions a1 A and H in l'rob.5-I.

-5-11. Determine the Icnsioll in Ihe cord and the horizontal and '-erlic31 components of reaction at support;\ of the beam in I'rob. 5-4 .

• 5- 1). Determine the llori1.omal and "cnical components of reaction a1 C and the tension in Ihe cable A8 for the truss in Prob. 5-S.

5-14. Determine the hO';7.0111al and l'crtic,lI components of reaction al II and the tension in cable He on the boom in !'rob. >-<>. 5-15. Determine Ihe hori1.omal Dnd I'cnical romp')r\cnlS of reaccion a1 A and the normal reaction al H on the spanner 'Hench in I'rob. 5-7.

*5-16. Determine tile normal reactions at II and Hand Ihe force in link CD acting on the nH'mb('r in I'rob. 5-8.

· 5-17. Determine the normal rcaaions a, ,he poims of comac, a, A. 8. Dnd C oflhc bar in Prob.5-9.

5-18. o..-,ermine 'he horizon ,al and I'cnical components o(reac,ion al pin C and 'he force in lhe pawl ofl~ winch in Prob. 5-10.

5- 19. Compare Ihe force e_~ened on Ihe IDC and heel of a 120-lb " ·oman "hen she I§ wearing n:gubr shoes and 51ilc110 heel§. Assume 311 her weight is placed on one foOl and the re<ldions occur al poims A and II as shown.

120lb

l

k-0.75 'D. 3.75 in .

l' mb. 5-I11

227

·5-20. The Imin car has a weight of 24 000 III and a cenler ofgr::lViJy at a.1! is suspended from "S front ~nd rearon the tract by six lires located 31 A. H. Dnd C. Determine the normal reaclions on thcse !lrCS if Ihc \rack Is assumed 10 be a smooth surface and an equal portion of the load is supported al bolh Ihe (ronl and rear li res.

-,-

."

l' rob.5-20

05-21. Determine the horizon tal and wnkal components of reaction 3t the pin A and the Icnsion dc."clopcd in cable HC used 10 suppon Ihc Sleel frame.

60kN

r 'm --'m-..... ) - 1 m ....l

.lOkN · m

c

Prah. 5-21

228 CHA'TU 5 f OUllillliUM OF ... RIGIO BODY

5-11. 1'hc aruculalcd (I1me boom lias a .... eight of 125lb and ccnlCTofgra\"ly at G. lfll supports. load 0(600 Ib,dclcrmirw: lhe force actong al the plD A and the for«' In the hydl1lulK" cyhnder 8C .. lIcn 11M: boom IS in the JnIIuonshovoll.

'" 'If. G

'"

C

I' rob. S-ll

5-lJ. l 11c airs1rokc ~ClUatoral 0 is Ilscd to appl)' a force of F . 200 1'1 on the IllCnlner ~l 8. Delcrm;nc the horizontal and "cnical romponents of reaction :11 lhe pin A :I.Ild the force ofltlc smooth shaft I I Con lhc member.

- S-U '.lle :IlBtrokc actualor al D is used to apply a force of F on lhe member II H. The normal reaction of the smooth shaft al Con illc member is .300 N. Dclcrminc lhe rna&nnudc of F and 11M: homonlal and \'cnK;al romponcnlS of n';I(\1QII al pin A .

I' rob!'. S-l Jll.,j

. 5-25. "The JOO-lbcloc\ncallrnnsfOfTllCr "ilhccnlcrol VoI"II)' II G IS wpporloo by a pm al A and a smOOl1l p;td at 8. Delcnmnc the honlonllli and >"Cruea! components ofreac'IIon II the pinA and the 1l':IC'I;ooOflhc pad 8 on lhe nansformer.

'" "

I'rub. S-15

S-16. A sltelelal diagram ofa hnnd holdIng a load issho ... ·n in the upper figure. lfthc load and the forearm h3\'c mas5CS of 21.:& and 1.21.:g.respcetwcly. and Ihelr «meN of mass are Iocaled al Gt and G~. determine the force dC"doped in Ihe bi«ps CO and Ihe honlOnial and , 'ertieal (omponent$ of reaclion 31 Ihe elbow JOint 8 . lhe forearm supporting S)"Slcm can be modeicd as tho: slruclur.tl S)"Slcm sho",n in the Io"'er figure.

" , I G. ·IOOmm·· ' 135mm

I'roll. S-Z6

3-27. As an airplane's brakes arc applied.lhe nose wheel excrlS IWO forces on the end of the landing gear as $ho"·n. Determinc the horizontal a!ld "crlkal components of reaelion at tile pin C and tile force in 5trul AB.

Prllb.5-27

*3-28. The 1 .4·~ l g drainpipe is held in Ihe tines of the fork lift. DClermine Ihe normal forc-cs al A and 8 as functions of the blade angle 0 and plot tile rcsults of force ("crlical nis) . 'crsus 0 (hori7.o!lwl axis) for 0 :s 0 :s 90".

l' rob. 5-28

229

' 3-29. 'The mass of 700 kg is 5us~ndcd from a trolle)' " 'lIich moves along tbe ernne rail from II - 1.7 m \0 II - 3.5 m. Determine Ihe force alo!lg the pin·oon!lected knee Sirut BC (sllorllink) and tile magnitud.:: of force at pin II as a function of posi tion fl. Plot tlle$C rcsult$ of FII(" and fA ("crtica! axis) wrsus ,/ (lioriwntal a~is).

1--- ' --1

" mh. 5-29

3-3t1. If the force of F ~ 100 Ih is applied 10 the handle of tile bar bender. determine the hornontal and "enical components of reaction al pin A and the rcaction of the roll er 8 on the smooth bar.

5-31. If the force of the smooth roller at 8 on the bar bender is requi red to be 1.5 kip. determIne the homontal and vertical components of reaction at pin It and tile required magnitude of force f applied !O the handle.

-10 ,no

PrnIK. 3-301.\1

230 CH ... prE~ 5 EOVllIB~IU M OF ... R IGID BODY

' 5-32.. The jiberane is supported by a pin at Cand rod liB. Iflhe load has3 mass 0(2 Mg " 'jlh ilseenlc r of mass localed DI G. (\clemline Ihe horizon lal and \"Crtical components of reaction al the pin C and Ihe force developed in rod 118 on Ihe emne when.l .. 5 m.

' 5-33. The jiberane is supponed by a pin at C and rod A8. The rod C"3n " 'ilh5land a maximum lens ion of 40 tN. If the load has;'l mass 0(2 Mg. wilh Ilscenler of masslocaled 31 G. determine its maximum allowable diSlance .r and the corrcspondmg horizontal and ,"ertkal componenl5 of rcaction at C.

3.2m

..-,--- 'm-­A

c 0.2111

I'robs. s-JUJ3

8

G D

5-34. Determine the homontal and ,"ertkal components ofrcaction al the pin;l and the normal force at Ihe smoolh peg 8 on Ihe member.

" roh. 5-34

5-35. the framework is supported by the member ;18 which rC5ts on the smooth OOOt. When loaded. the pressur~ distribution on A H is linear as shown. Delermine Ihe length If of member AB and the intensity II' for this case.

----H, i

1

f' rob.5-35

' 5-36. Outriggers II and B are used to stabilize the crane from o'<crlurning " 'hen lifting large lo~uls. If the load to be lifted is 3 Mg.detcrminc the lIIa.dl/llml boom angle (J so that the crane docs not o,·ertuTll. The cmne has a mass of 5 Mg and center of mass at Gc• whereas the boom has a mass of 0.6 Mg and center of mass at Gil'

2.¥ no

"rob. 5-36

03-.n. The ..... ooden plank resting bet ..... ~en thc buildings deneel$ slightly when iI supports the 5()..kg boy. This dcnection causes a triangular distribution of load ~ t its ends. ha'·,ng maximum inleru.ities of "A and "'B. IXlermine "A and "'~. cadi measured in Nj m. when the boy is sTanding J m from onc end as shown. Neglect the mass or the plank.

8

". ", ~~ ,. T 'm tUSm JU.

Prob. 3-J7

3-JII. Spring CD remains in the horitontal position at all limes due to the roller m O. If the spring i$ unslretchcd ..... hen (I .. 0" and Ihe bracket achie"cs its equilibrium position when 8 - JO". deh.'rminc the stiffness Ii: of the spring and the hori1.ontal and ,"ertical components of reaction al pin II .

3-J9. Spring CD rcmains in the horizontal position at nil limes due to the roUer at D. If the spring is Ilns1TelChed when (I _ O· and the stiffness is Ii: .. 1.5 kN/ m. determine Ihc smallest angle (I for equilibrium and the horizonwl and \"ertical components of reaction at pin A .

,

Probs. 3-J8Il9

231

"5-40. lhe platform assembly has a weighl of 250 lb lind cenler of gm" ity at GI. [f it is intended 10 support a maximum lo:od of -100 [b placed at point G!. determine Ihe smallest counler ..... eight \I' Ihat should be placed al H in order 10 pre"cnl the plalform from tipping o'-er.

"

'" l'rob. 5-40

. 5-41. IXterminc the horizontal and '-ertical components of reaction 3t Ihe pin II and Ihe reaction of Ihe smooth collar B on the rod.

C

"." ~~ Ib

j 1 8

~ A

, + 111 ""1 ' " ,. 2/.

'" Proh. 5-4 1

232 CHAPTER 5 EOU I ll8RIUM OF A R IGIO BOOY

5-42. Deleml;ne tile supporl reaclions of roller A and IIII' smoolh colla r 8 on Ihe rod. The collar is fixed 10 Ihe rod AB. but is allowed 10 slide along rod CD.

.. N

'm--'m B

6(lON ' m , .W~ ~

l' rob. 5-42

5-43. The uniform rod ..18 has a weighl of 151b. Delcmline Ihe (orce in Ihe cable when Ihe rod is in Ihe position shown.

B

T

Prob. 5-43

-5-44. DClermine tile horizonlal and verlical components of (orce at Ihe pin A and Ihe reaclion at Ihe rocker 8 of Ihe eur.·ed beam.

""N

I

A

I ,m "

Prob. 5--44

05-45. ' llIe noor crane and the driver have a tOI,11 weighl of 2500 lb wilh a cenler of gm"sly al G. If Ihe crane is required 10 lifl tbe 5oo·lb drum. delermine tbe normal reaclion on 1101/1 Ihe wbeels al A and bOlh Ibe wheels al 8 when Ihe boom is in Ihe posilion shown .

.5-46. The noor crane and lhe dri"cr ba\"l,' a 10lal weighl of 2500 1b wilh a cenler of gravity at G. Delermine Ihe largesl weighl of lhe drum lhal can be lifted wilhoul causing lhe crane 10 overturn when ils boom is in Ihe posilion shown.

",

5-47. l be mOlor has a weigl1t of &SO lb. Detcnnine the fore<: that each of Ihe chains exerts on Ihe. supporting hooks at A. 8. and C. N"eglecI the sill.' of Ihe hooks and the Ihickness of the beam.

"""

I'rob. 5-47

*5-48.. Delermine the force I' needed 10 pullthe.SO·kgroller owr Ihe smoOlh step. Take 0 - 60".

' 5-49. D1.'lermin~ Ihe magnitude. and dir.."'Clion e of Ihe minimum force r needed 10 pulllhe SO·kg roller ovcr Ihe smOOlh slep.

I'rob!;. S-48149

5.4 Two- AND THRE~·FORCE MEMBERS 233

5-5tl. "fbe "-inch cablc on a lOW ITUCI.;. is subjected 10 3 force of T - 6 k1'1 when Ihe cable ;5 directed me '" t:i.f'. Determine the magnitudes of Ihe 10lal brnke friclional forre F for Ihe remr SCI of ,,·heels Band Ihe 10lal normal forces al /Wlir fronl wheels II and bOlh rear whcels 8 for equilihrium. The Iruck has a 10lal mas5 of 4 il,lg and mass cemer 31 G.

5-51. Determine the minimum cable force T and critical angle 0 which will cause Ihe lOW truck 10 5larltipping. i.e .• for lhe nonnal reaction al A 10 be zero. Assume thallhe Iruck is braked and will not slip at B. The truck has a t01al moss of 4 Mg and masseenl~r at G.x

"'robs. 5-S1.If5 1

' 5-52. Thrce uniform books. each having a weight IV and lenglh <I. arc stacked 35 shown. Determine lhe maximum dislane<: d Ih31 Ihe top book can extend OUI from Ihe bonom onc so lhe Slack docs nOI lopple o\'er.

f---,,---'- II-I

"'rob. 5-52

23 4 CH"'PfE~ 5 EOVllIB~IU M OF ... R IGID BODY

05-.53. D~t~rmin~ t h~ angle 0 al whi~h th~ link ABC is held in equilibrium if member HO mo,'es 2 in. 10 Ihe righ l. lhe springs arc originally unsln.'lched " 'hen fJ .. 0". Each spring has the sliffncss sho\1.'II. 'Ibe springs remain horizonml since lhey arc allached 10 roller guides..

ka - tOO tb/fl

l' rob. 5-53

S-S4. The uniform rod 118 has a weight of 15 Ib and lhe spring is unstrelched when fJ .. 0". If 0 .. 30". delermine th~ Sliffness k orlhe spring.

8

5-55. 'Ibe hotizonlal beam is supported by springs al its ends. Eaeh spring has a stiffness of II. .. .5 kN/ m and is originally unslretched so thai Ihe beam is in lhe horil-onwl posilkm. Dctt!rmint! the angle of lilt of the beam if a load of 800 N is applied al poinl Cas sho"·n.

"5-St>. llle hori1.ontal beam is supported by springs al ils ends.. If lhe sliffness of the spring 3t A is 11." .. .5 kN/ m. determine tht! required stiffness orthe spring at H so that if the beam is loaded Wilh Ihe SOO N il remains in the hori1.0nlal posi lion. The springs arc originall), ~oll$lructed so Ihat Ihe beam is in th~ hon~ontal position when it is unloaded.

c

"

.- J ,-

oS-57. The smoolh disks 0 and E ha"", a " 'Cighl of 200 Ib and 100 lb. rcspccti,· ... I)~ If a horizon131 force of P .. 200 Ib is applied 10 the cenler of disk E. dt!lc rmlne Ihe normal reactions al the points of conlael Wilh the ground 31 A . H. and C.

5-58. The smooth disks D and E h3\'''' a " 'eight of 200 Ib and 100 lb. respccli, .... ly. Dclermine Ihe largest horizonlal for~ f' lhal can be applied 10 the center of disk E: without causing the disk D 10 mow up the incline.

I'robs. S-57158

5-59. A man slands 01,11 at the end of the dwing board . which is supported by two springs A and B. each ha';ng a stiffness of k .. 15 kN/ m. In the posit ion shown the board is honwntal. l f the n13n has a mas.s of 40 kg. determine the angle of till ""hich Ihe boa rd makC$ with the horizontal after he jumps off. Neglectlhe weight of the board and assume il is rigid.

4- tm - - --- 'm

l' roo.5-59

' 5-60. the uniform rod has a length / and weight IV. It is supported at one end A b}' 3 smooth " 'all and the other end by a cord of lenglh s which is attached to the wall as sho",·n. Show thai for equilibrium it is required that " .. [(.I - 1!)!31 '~.

I' rab. S-6O

235

' 5-61. If spring BCis unstretched .... ith 8 .. 0" and the bell cTank achie"C"S ilS equilibrium position when 8 " IS". de temline the force F applied perpendicular to segment tiD and the horizontal and "eTtical CQntponenls of reaction at pin A. Spring BC remains in the horizontal postion 3t all times due to the roller al C.

k - 2kNfm C H

\ ,

"roh. 5-6 1

5-62. The thin rod o/length I is supported by the smooth lube. Delermine Ihe distance u nceded for eq ui librium if the applied Io.~d is P.

....".,--,.

I'rob. 5-62