4 Equilibium CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Fedinand P. Bee E. Russell Johnston, J. of Rigid Bodies Lectue Notes: J. Walt Ole Texas Tech Univesity
Contents Intoduction Fee-Body Diagam Reactions at Suppots and Connections fo a Two-Dimensional Stuctue Equilibium of a Rigid Body in Two Dimensions Statically Indeteminate Reactions Sample Poblem 4.1 Sample Poblem 4.3 Sample Poblem 4.4 Equilibium of a Two-Foce Body Equilibium of a Thee-Foce Body Sample Poblem 4.6 Equilibium of a Rigid Body in Thee Dimensions Reactions at Suppots and Connections fo a Thee-Dimensional Stuctue Sample Poblem 4.8 4-2
Intoduction Fo a igid body in static equilibium, the extenal foces and moments ae balanced and will impat no tanslational o otational motion to the body. The necessay and sufficient condition fo the static equilibium of a body ae that the esultant foce and couple fom all extenal foces fom a system equivalent to zeo, F 0 M F 0 O ( ) Resolving each foce and moment into its ectangula components leads to 6 scala equations which also expess the conditions fo static equilibium, F 0 x M 0 x F 0 y M 0 y F 0 z M 0 z 4-3
Fee-Body Diagam Fist step in the static equilibium analysis of a igid body is identification of all foces acting on the body with a fee-body diagam. Select the extent of the fee-body and detach it fom the gound and all othe bodies. Indicate point of application, magnitude, and diection of extenal foces, including the igid body weight. Indicate point of application and assumed diection of unknown applied foces. These usually consist of eactions though which the gound and othe bodies oppose the possible motion of the igid body. Include the dimensions necessay to compute the moments of the foces. 4-4
Reactions at Suppots and Connections fo a Two- Dimensional Stuctue Reactions equivalent to a foce with known line of action. 4-5
Reactions at Suppots and Connections fo a Two- Dimensional Stuctue Reactions equivalent to a foce of unknown diection and magnitude. Reactions equivalent to a foce of unknown diection and magnitude and a couple.of unknown magnitude 4-6
Equilibium of a Rigid Body in Two Dimensions Fo all foces and moments acting on a twodimensional stuctue, F 0 M M 0 M M z x Equations of equilibium become F F 0 M 0 x y 0 y A whee A is any point in the plane of the stuctue. The 3 equations can be solved fo no moe than 3 unknowns. The 3 equations can not be augmented with additional equations, but they can be eplaced F M 0 M 0 x 0 A B z O 4-7
Statically Indeteminate Reactions Moe unknowns than equations Fewe unknowns than equations, patially constained Equal numbe unknowns and equations but impopely constained 4-8
Sample Poblem 4.1 SOLUTION: Ceate a fee-body diagam fo the cane. Detemine B by solving the equation fo the sum of the moments of all foces about A. Note thee will be no contibution fom the unknown eactions at A. A fixed cane has a mass of 1000 kg and is used to lift a 2400 kg cate. It is held in place by a pin at A and a ocke at B. The cente of gavity of the cane is located at G. Detemine the components of the eactions at A and B. Detemine the eactions at A by solving the equations fo the sum of all hoizontal foce components and all vetical foce components. Check the values obtained fo the eactions by veifying that the sum of the moments about B of all foces is zeo. 4-9
Sample Poblem 4.1 Detemine B by solving the equation fo the sum of the moments of all foces about A. 0 : + B( 1.5m) 9.81kN( 2m) M A B +107.1kN 23.5 kn 6m ( ) 0 Ceate the fee-body diagam. Detemine the eactions at A by solving the equations fo the sum of all hoizontal foces and all vetical foces. F A x x 0 : A + B 0 x 107.1kN F y 0 : A 9.81kN 23.5 kn A y y +33.3 kn 0 Check the values obtained. 4-10
Sample Poblem 4.3 SOLUTION: Ceate a fee-body diagam fo the ca with the coodinate system aligned with the tack. Detemine the eactions at the wheels by solving equations fo the sum of moments about points above each axle. A loading ca is at est on an inclined tack. The goss weight of the ca and its load is 5500 lb, and it is applied at at G. The cat is held in position by the cable. Detemine the tension in the cable and the eaction at each pai of wheels. Detemine the cable tension by solving the equation fo the sum of foce components paallel to the tack. Check the values obtained by veifying that the sum of foce components pependicula to the tack ae zeo. 4-11
Sample Poblem 4.3 Detemine the eactions at the wheels. M A 0 : R 2 1758 lb ( 2320 lb) 25in. ( 4980 lb) + R ( 50in. ) 0 2 6in. Ceate a fee-body diagam W W x y + ( 5500 lb) + 4980 lb ( 5500 lb) 2320 lb cos 25 sin 25 o o M B 0 : R 1 562 lb + ( 2320 lb) 25in. ( 4980 lb) R ( 50in. ) 0 1 Detemine the cable tension. F x T 0 : + 4980 lb T 0 +4980 lb 6in. 4-12
Sample Poblem 4.4 SOLUTION: Ceate a fee-body diagam fo the fame and cable. Solve 3 equilibium equations fo the eaction foce components and couple at E. The fame suppots pat of the oof of a small building. The tension in the cable is 150 kn. Detemine the eaction at the fixed end E. 4-13
Sample Poblem 4.4 Solve 3 equilibium equations fo the eaction foce components and couple. 4.5 F x 0 : Ex + 7.5 90.0 kn E x ( 150kN) 0 6 F y 0 : E y 4 7.5 +200 kn E y ( 20kN) ( 150kN) 0 Ceate a fee-body diagam fo the fame and cable. M E + 20kN( 7.2m) + 20kN( 5.4m) + 20kN( 3.6m) + 20kN( 1.8m) 0 : M E 180.0kN 6 7.5 m ( 150kN) 4.5m + 0 M E 4-14
Equilibium of a Two-Foce Body Conside a plate subjected to two foces F 1 and F 2 Fo static equilibium, the sum of moments about A must be zeo. The moment of F 2 must be zeo. It follows that the line of action of F 2 must pass though A. Similaly, the line of action of F 1 must pass though B fo the sum of moments about B to be zeo. Requiing that the sum of foces in any diection be zeo leads to the conclusion that F 1 and F 2 must have equal magnitude but opposite sense. 4-15
Equilibium of a Thee-Foce Body Conside a igid body subjected to foces acting at only 3 points. Assuming that thei lines of action intesect, the moment of F 1 and F 2 about the point of intesection epesented by D is zeo. Since the igid body is in equilibium, the sum of the moments of F 1, F 2, and F 3 about any axis must be zeo. It follows that the moment of F 3 about D must be zeo as well and that the line of action of F 3 must pass though D. The lines of action of the thee foces must be concuent o paallel. 4-16
Sample Poblem 4.6 SOLUTION: Ceate a fee-body diagam of the joist. Note that the joist is a 3 foce body acted upon by the ope, its weight, and the eaction at A. A man aises a 10 kg joist, of length 4 m, by pulling on a ope. Find the tension in the ope and the eaction at A. The thee foces must be concuent fo static equilibium. Theefoe, the eaction R must pass though the intesection of the lines of action of the weight and ope foces. Detemine the diection of the eaction foce R. Utilize a foce tiangle to detemine the magnitude of the eaction foce R. 4-17
Sample Poblem 4.6 Ceate a fee-body diagam of the joist. Detemine the diection of the eaction foce R. AF CD CE tanα AB cos 45 AE 1 2 BD CD cot(45 + CE AE o 58.6 α AF BF BD 2.313 1.414 ( 4m) 1.414 m cos 45 20) ( 1.414 m) ( 2.828 0.515) 1.636 2.828m m tan 20 2.313 m 0.515 m 4-18
Sample Poblem 4.6 Detemine the magnitude of the eaction foce R. T sin 31.4 o R sin110 o 98.1 N sin 38.6 o T R 81.9 N 147.8 N 4-19
Equilibium of a Rigid Body in Thee Dimensions Six scala equations ae equied to expess the conditions fo the equilibium of a igid body in the geneal thee dimensional case. F 0 x M 0 x F 0 y M 0 y F 0 z M 0 These equations can be solved fo no moe than 6 unknowns which geneally epesent eactions at suppots o connections. The scala equations ae conveniently obtained by applying the vecto foms of the conditions fo equilibium, F 0 M F 0 O ( ) z 4-20
Reactions at Suppots and Connections fo a Thee- Dimensional Stuctue 4-21
Reactions at Suppots and Connections fo a Thee- Dimensional Stuctue 4-22
Sample Poblem 4.8 SOLUTION: Ceate a fee-body diagam fo the sign. Apply the conditions fo static equilibium to develop equations fo the unknown eactions. A sign of unifom density weighs 270 lb and is suppoted by a ball-andsocket joint at A and by two cables. Detemine the tension in each cable and the eaction at A. 4-23
Sample Poblem 4.8 Ceate a fee-body diagam fo the sign. Since thee ae only 5 unknowns, the sign is patially constain. It is fee to otate about the x axis. It is, howeve, in equilibium fo the given loading. T T BD EC T T T T T T BD BD BD EC EC EC D B D B 8i + 4 j 8k 12 ( 2 i + 1 j 2 k ) 3 C E C E 6i + 3 j + 2k 7 ( 6 i + 3 j + k ) 7 3 7 3 7 2 4-24
Sample Poblem 4.8 Apply the conditions fo static equilibium to develop equations fo the unknown eactions. F i : j : k : M A j : k : TBD A A + T A A x y B + BD 2 3 1 3 2 3 T Az T T 5.333T T 2.667T + T BD BD BD BD BD BD 101.3 lb v EC + + 6 7 3 7 7 2 + E T T 1.714T + 2.571T T ( 270 lb) EC EC TEC T EC 0 0 + ( 4 ft) i ( 270 lb) 0 0 270 lb EC EC EC j 315 lb 0 1080 lb Solve the 5 equations fo the 5 unknowns, ( 338 lb) i + ( 101.2 lb) j ( 22.5 lb)k 0 j 0 4-25