Equilibrium of Rigid Bodies

Transcription

1 Equilibrium of Rigid Bodies 1

2 2 Contents Introduction Free-Bod Diagram Reactions at Supports and Connections for a wo-dimensional Structure Equilibrium of a Rigid Bod in wo Dimensions Staticall Indeterminate Reactions Sample Problem 1 Sample Problem 2 Sample Problem 3 Equilibrium of a wo-force Bod Equilibrium of a hree-force Bod Sample Problem 4 Equilibrium of a Rigid Bod in hree Dimensions Reactions at Supports and Connections for a hree-dimensional Structure Sample Problem 5

3 3 Introduction For a rigid bod in static equilibrium, the eternal forces and moments are balanced and will impart no translational or rotational motion to the bod. he necessar and sufficient condition for the static equilibrium of a bod are that the resultant force and couple from all eternal forces form a sstem equivalent to zero, F M r F O Resolving each force and moment into its rectangular components leads to 6 scalar equations which also epress the conditions for static equilibrium, F F F z M M M z

4 4 Free-Bod Diagram First step in the static equilibrium analsis of a rigid bod is identification of all forces acting on the bod with a free-bod diagram. Select the etent of the free-bod and detach it from the ground and all other bodies. Indicate point of application, magnitude, and direction of eternal forces, including the rigid bod weight. Indicate point of application and assumed direction of unknown applied forces. hese usuall consist of reactions through which the ground and other bodies oppose the possible motion of the rigid bod. Include the dimensions necessar to compute the moments of the forces.

5 5 Reactions at Supports and Connections for a wo- Dimensional Structure Reactions equivalent to a force with known line of action.

6 6 Reactions at Supports and Connections for a wo- Dimensional Structure Reactions equivalent to a force of unknown direction and magnitude. Reactions equivalent to a force of unknown direction and magnitude and a couple.of unknown magnitude

7 7 Equilibrium of a Rigid Bod in wo Dimensions For all forces and moments acting on a twodimensional structure, F z M M M z M O Equations of equilibrium become F F M A where A is an point in the plane of the structure. he 3 equations can be solved for no more than 3 unknowns. he 3 equations can not be augmented with additional equations, but the can be replaced F M M A B

8 8 Staticall Indeterminate Reactions More unknowns than equations Fewer unknowns than equations, partiall constrained Equal number unknowns and equations but improperl constrained

9 9 Sample Problem 1 SOLUION: Create a free-bod diagram for the crane. Determine B b solving the equation for the sum of the moments of all forces about A. Note there will be no contribution from the unknown reactions at A. A fied crane has a mass of 1 kg and is used to lift a 24 kg crate. It is held in place b a pin at A and a rocker at B. he center of gravit of the crane is located at G. Determine the components of the reactions at A and B. Determine the reactions at A b solving the equations for the sum of all horizontal force components and all vertical force components. Check the values obtained for the reactions b verifing that the sum of the moments about B of all forces is zero.

10 1 Sample Problem 1 Determine B b solving the equation for the sum of the moments of all forces about A. M A : B 1. 5 m kn 2 m kn 6 m B kn Create the free-bod diagram. Determine the reactions at A b solving the equations for the sum of all horizontal forces and all vertical forces. F : A B A kn F : A 9.81 kn kn A kn Check the values obtained.

11 11 Sample Problem 2 SOLUION: Create a free-bod diagram for the car with the coordinate sstem aligned with the track. Determine the reactions at the wheels b solving equations for the sum of moments about points above each ale. A loading car is at rest on an inclined track. he gross weight of the car and its load is 55 lb, and it is applied at at G. he cart is held in position b the cable. Determine the tension in the cable and the reaction at each pair of wheels. Determine the cable tension b solving the equation for the sum of force components parallel to the track. Check the values obtained b verifing that the sum of force components perpendicular to the track are zero.

12 12 Sample Problem 2 Determine the reactions at the wheels. M A : 232 lb 25 in lb 6 in. R 5 in. 2 R lb M B : 232 lb 25 in lb 6 in. R 5 in. 1 Create a free-bod diagram W 55 lb 498 lb cos 25 R lb Determine the cable tension. F : 498 lb W 55 lb sin lb 232 lb

13 13 Sample Problem 3 SOLUION: Create a free-bod diagram for the frame and cable. Solve 3 equilibrium equations for the reaction force components and couple at E. he frame supports part of the roof of a small building. he tension in the cable is 15 kn. Determine the reaction at the fied end E.

14 14 Sample Problem 3 Solve 3 equilibrium equations for the reaction force components and couple. F 4. 5 : E kn E 9. kn F 6 : E kn 15 kn E 2 kn Create a free-bod diagram for the frame and cable. : M 2 kn 7.2 m 2 kn 5. 4 m E 2 kn 3. 6 m 2 kn 1. 8 m kn 4. 5 m M E M E 18. kn m

15 15 Equilibrium of a wo-force Bod Consider a plate subjected to two forces F 1 and F 2 For static equilibrium, the sum of moments about A must be zero. he moment of F 2 must be zero. It follows that the line of action of F 2 must pass through A. Similarl, the line of action of F 1 must pass through B for the sum of moments about B to be zero. Requiring that the sum of forces in an direction be zero leads to the conclusion that F 1 and F 2 must have equal magnitude but opposite sense.

16 16 Equilibrium of a hree-force Bod Consider a rigid bod subjected to forces acting at onl 3 points. Assuming that their lines of action intersect, the moment of F 1 and F 2 about the point of intersection represented b D is zero. Since the rigid bod is in equilibrium, the sum of the moments of F 1, F 2, and F 3 about an ais must be zero. It follows that the moment of F 3 about D must be zero as well and that the line of action of F 3 must pass through D. he lines of action of the three forces must be concurrent or parallel.

17 17 Sample Problem 4 SOLUION: Create a free-bod diagram of the joist. Note that the joist is a 3 force bod acted upon b the rope, its weight, and the reaction at A. A man raises a 1 kg joist, of length 4 m, b pulling on a rope. Find the tension in the rope and the reaction at A. he three forces must be concurrent for static equilibrium. herefore, the reaction R must pass through the intersection of the lines of action of the weight and rope forces. Determine the direction of the reaction force R. Utilize a force triangle to determine the magnitude of the reaction force R.

18 18 Sample Problem 4 Create a free-bod diagram of the joist. Determine the direction of the reaction force R. AF AB cos 45 4 m cos m CD AE 1 2 AF m CD cot( 45 2 ) m tan m CE BF m m tan CE AE

19 19 Sample Problem 4 Determine the magnitude of the reaction force R. sin sin R 11 sin N N R N

20 2 Equilibrium of a Rigid Bod in hree Dimensions Si scalar equations are required to epress the conditions for the equilibrium of a rigid bod in the general three dimensional case. F F F z M M M z hese equations can be solved for no more than 6 unknowns which generall represent reactions at supports or connections. he scalar equations are convenientl obtained b appling the vector forms of the conditions for equilibrium, F M r F O

21 Reactions at Supports and Connections for a hree- Dimensional Structure 21

22 Reactions at Supports and Connections for a hree- Dimensional Structure 22

23 23 Sample Problem 5 SOLUION: Create a free-bod diagram for the sign. Appl the conditions for static equilibrium to develop equations for the unknown reactions. A sign of uniform densit weighs 27 lb and is supported b a ball-andsocket joint at A and b two cables. Determine the tension in each cable and the reaction at A.

24 Sample Problem 5 Create a free-bod diagram for the sign. Since there are onl 5 unknowns, the sign is partiall constrain. It is free to rotate about the ais. It is, however, in equilibrium for the given loading. k j i k j i r r r r k j i k j i r r r r EC EC E C E C EC EC B D B D

25 25 Sample Problem 5 Appl the conditions for static equilibrium to develop equations for the unknown reactions. F i j k M j k : : : : : A A A A A z r B EC 6 7 r E EC EC EC 27 EC EC EC lb 27 4 ft i 27 lb j lb 18 Solve the 5 equations for the 5 unknowns, A lb EC 338 lb i 11.2 lb j 22.5 lb k 315 lb lb j