Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 4/1 Rgd bod n equlbum gd bod s sad to be n equlbum the esultant of all foces actng on t s zeo. hus, the esultant foce R and the esultant couple ae both zeo, and we hae the equlbum equatons R. 4.1 4/ ee od Dagam 1. Select the etent of the fee-bod and detach t fom the gound and all othe bodes.. Indcate pont of applcaton, magntude, and decton of etenal foces, ncludng the gd bod weght.. Indcate pont of applcaton and assumed decton of unnown appled foces. hese usuall consst of eactons though whch the gound and othe bodes oppose the possble moton of the gd bod. 4. Include the dmensons necessa to compute the moments of the foces. o eample: 44
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 4/ Reactons at Suppots and Connectons fo a wo-dmensonal Stuctue 45
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes Eamples Of ee od Dagam: gue / ges fou eamples of mechansms and stuctues togethe wth the coect fee-bod dagams. g. (4-) 46
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 4/4 Equlbum of a Rgd od n wo Dmensons o all foces and moments actng on a two-dmensonal stuctue, z z O Equatons of equlbum become whee s an pont n the plane of the stuctue. he equatons can be soled fo no moe than unnowns. EXPLE 1. fed cane has a mass of 1 g and s used to lft a 4 g cate. It s held n place b a pn at and a oce at. he cente of gat of the cane s located at G. Detemne the components of the eactons at and. SOLUION: 1. Ceate a fee-bod dagam fo the cane.. Detemne b solng the equaton fo the sum of the moments of all foces about. : ( 1.5m) 9.81N( m).5 N( 6m) 1.1 N. Detemne the eactons at b solng the equatons fo the sum of all hozontal foces and all etcal foces. : 1.1N 4
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes : 9.81N.5 N. N EXPLE. loadng ca s at est on an nclned tac. he goss weght of the ca and ts load s 55 lb, and t s appled at G. he cat s held n poston b the cable. Detemne the tenson n the cable and the eacton at each pa of wheels. SOLUION: 1. Ceate a fee-bod dagam fo the ca wth the coodnate sstem algned wth the tac. W W ( 55 lb) 498 lb ( 55 lb) lb cos 5 sn 5 o o. Detemne the eactons at the wheels. : ( lb) 5n. ( 498 lb) R ( 5n. ) 6n. R 158 lb : ( lb) 5n. ( 498 lb) R ( 5n. ) 1 6n. R 1 56 lb 48
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes. Detemne the cable tenson. : 498 lb 498 lb 4/5 Categoes of Equlbum he categoes of foce sstems actng on bodes n two-dmensonal equlbum ae summazed n g. (4-). g. (4-) 49
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 4/6 Equlbum of a wo-oce od Consde a plate subected to two foces 1 and as shown n g. (4-4a). o statc equlbum, the sum of moments about must be zeo. he moment of must be zeo. It follows that the lne of acton of must pass though see g. (4-4b). Smlal, the lne of acton of 1 must pass though fo the sum of moments about to be zeo. Requng that the sum of foces n an decton be zeo leads to the concluson that 1 and must hae equal magntude but opposte sense see g. (4-4c). g. (4-4) 4/ Equlbum of a hee-oce od Consde a gd bod subected to foces actng at onl ponts as shown n g. (4-5a). ssumng that the lnes of acton ntesect, the moment of 1 and about the pont of ntesecton epesented b D s zeo see g. (4-5b). Snce the gd bod s n equlbum, the sum of the moments of 1,, and about an as must be zeo. It follows that the moment of about D must be zeo as well and that the lne of acton of must pass though D see g. (4-5c). he lnes of acton of the thee foces must be concuent o paallel. 5
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes g. (4-5) EXPLE 1. man ases a 1 g ost, of length 4 m, b pullng on a ope. nd the tenson n the ope and the eacton at. SOLUION: 1. Ceate a fee-bod dagam of the ost. Note that the ost s a foce bod acted upon b the ope, ts weght, and the eacton at.. he thee foces must be concuent fo statc equlbum. heefoe, the eacton R must pass though the ntesecton of the lnes of acton of the weght and ope foces. Detemne the decton of the eacton foce R. cos 45 CD E CD cot(45 tanα 1 CE o α 58.6 CE E ( 4 m) 1.414 m cos 45.88 m ) ( 1.414 m) (.88.515).1 1.66 1.414 tan.515 m m.1 m 51
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes. Utlze a foce tangle to detemne the magntude of the eacton foce R. sn 1.4 81.9 N R 14.8 N o R sn11 o 98.1 N sn 8.6 o 4/8 Equlbum of a Rgd od n hee Dmensons S scala equatons ae equed to epess the condtons fo the equlbum of a gd bod n the geneal thee dmensonal case. z z hese equatons can be soled fo no moe than 6 unnowns whch geneall epesent eactons at suppots o connectons. he scala equatons ae conenentl obtaned b applng the ecto foms of the condtons fo equlbum, ( ) O 4/9 Reactons at Suppots and Connectons fo a hee-dmensonal Stuctue 5
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes g. (4-6): Reactons at Suppots and Connectons. 5
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 54 EXPLE 1. sgn of unfom denst weghs lb and s suppoted b a ball-and-socet ont at and b two cables. Detemne the tenson n each cable and the eacton at. SOLUION: 1.Ceate a fee-bod dagam fo the sgn. E C E C E C D D D 6 6 6 1 1 8 4 8 8 8 4.ppl the condtons fo statc equlbum to deelop equatons fo the unnown eactons. ( )...() lb :...(1) : lb 1 6
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes : z ( 4 ft) ( lb) E 8...() 1 6 6 (4 ) ( ) : : 5..66 1.14.51.14...(4) 18lb...(5) Substtuton Eq.(4) n Eq.(5) we get 11. lb 15 lb ( 8 lb) ( 11. lb) (.5 lb) EXPLE. 48-n. boom s held b a ball-and-socet ont at C and b two cables and DE; cable DE passes aound a fctonless pulle at. o the loadng shown, detemne the tenson n each cable and the eacton at C. SOLUION: Substtuton these alues n Eqs.(1,,) we get 1.Ceate a fee-bod dagam fo the sgn. D E D 48 D D D 48 D 5 n. 16 n. z 48 n. D D E W - ( Ib) E C C C n. C z n. 55
Chapte fou.lectue Saddam K. Kwas Equlbum Of Rgd odes 5 1 D 1 1 E 48 5 1 E E E E 5 E 1 1 16 16 8 15 4 1 1.ppl the condtons fo statc equlbum to deelop equatons fo the unnown eactons. C D E ( lb) 5 8 : C D 1 1... 5 : C E 1 lb... 1 1 15 : Cz D E 1 1 1 ( 1) ( )...( ) C D 48 E D ( n. ) ( lb) 48 5 1 1 1 5 1 1 1 8 1 ( ) ( ) E 15 1 : : 18.46 18.46 D E 96 14.11 E 5 Ib D sub. n eq.() C 68 Ib sub. neq.(1) C 1 Ib 1 Ib Sub. the alues of D, E, and n eq.() we get C z 156Ib C 1 1 156 ns. 56