STATICS. Bodies. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Design of a support

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1 4 Equilibrium CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University of Rigid Bodies 2010 The McGraw-Hill Companies, Inc. All rights reserved Design of a support 1. Identify the main structure. 2. Where is the load on the structure? 3. How is the structure fixed to the ground? 4. How do we decide on the dimensions of the ground support? 2-2 1

2 Contents and Objectives Draw Free-Body Diagram Identify Reactions at Supports for a Two-Dimensional Structure Solve Problems of Equilibrium of a Rigid Body in Two Dimensions Identify Statically Indeterminate Reactions Recognize a Two-Force Body Recognize a Three-Force Body Solve Problems of Equilibrium of a Rigid Body in Three Dimensions Recognize Reactions at Supports and Connections for a Three-Dimensional Structure 4-3 Introduction For a rigid body in static equilibrium, the external forces and moments are balanced and will impart no translational or rotational motion to the body. The necessary and sufficient condition for the static equilibrium of a body are that the resultant force and couple from all external forces form a system equivalent to zero. F 0 MO r F 0 Resolving each force and moment into its rectangular components leads to 6 scalar equations which also express the conditions for static equilibrium, Fx 0 M 0 x Fy 0 M 0 y Fz 0 M 0 z 4-4 2

3 Free-Body Diagram Geometry Geometry + Loads The truck ramp has a weight of 1800 N. The ramp is pinned to the body of the truck and held in the position by the cable. How can we determine the cable tension and support reactions? How are the idealized model and the free body diagram used to do this? Which diagram above is the idealized model? 2-5 Free-Body Diagram First step in static equilibrium analysis of a rigid body is identification of all forces acting on the body with a free-body diagram. Draw an outlined shape. Cut the body free form its constraints. Indicate point of application, magnitude, and direction of external forces, including the rigid body weight and couple moments. Indicate point of application and assumed direction of unknown applied forces. These usually consist of reactions through which the ground and other bodies oppose the possible motion of the rigid body. Label loads and dimensions on FBD necessary to compute the moments of the forces

4 Reactions at Supports and Connections for a Two-Dimensional Structure Reactions equivalent to a force with known line of action. 4-7 Reactions at Supports and Connections for a Two-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

5 Quiz 1 1. The beam and the cable (with a frictionless pulley at D) support an 80 kg load at C. In a FBD of only the beam, there are how many unknowns? A) 2 forces and 1 couple moment B) 3 forces and 1 couple moment C) 3 forces D) 4 forces 2-9 Quiz 2 2. If the directions of the force and the couple moments are both reversed, what will happen to the beam? A) The beam will lift from A. B) The beam will lift at B. C) The beam will be restrained. D) The beam will break

6 Quiz 3 Draw a FBD of member ABC, which is supported by a smooth collar at A, roller at B, and link CD Quiz 4 How many unknown support reactions are there in this problem? A) 2 forces and 2 couple moments B) 1 force and 2 couple moments C) 3 forces D) 3 forces and 1 couple moment

7 Practical Example A 4000 N of engine is supported by three chains, which are attached to the spreader bar of a hoist. You need to check to see if the breaking strength of any of the chains is going to be exceeded. How can you determine the force acting in each of the chains? 2-13 Equilibrium of a Rigid Body in Two Dimensions For all forces and moments acting on a twodimensional structure, F z 0 M M 0 x y M z M O Equations of equilibrium become F F 0 M 0 x 0 y A where A is any point in the plane of the structure. The 3 equations can be solved for no more than 3 unknowns. The 3 equations can not be augmented with additional equations, but they can be replaced. Fx 0 M A 0 M B

8 STEPS FOR SOLVING 2-D EQUILIBRIUM PROBLEMS 1. If not given, establish a suitable x - y coordinate system. 2. Draw a free body diagram (FBD) of the object under analysis. 3. Apply the three equations of equilibrium to solve for the unknowns Sample Problem 4.1 SOLUTION: Create a free-body diagram for the crane. Determine B by 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 fixed crane has a mass of 1000 kg and is used to lift a 2400 kg crate. It is held in place by a pin at A and a rocker at B. The center of gravity of the crane is located at G. Determine the components of the reactions at A and B. Determine the reactions at A by solving the equations for the sum of all horizontal force components and all vertical force components. Check the values obtained for the reactions by verifying that the sum of the moments about B of all forces is zero

9 Sample Problem 4.1 Determine B by solving the equation for the sum of the moments of all forces about A. M A 0 : B1.5m 9.81kN2m B 107.1kN 23.5 kn 6m 0 Create the free-body diagram. Determine the reactions at A by solving the equations for the sum of all horizontal forces and all vertical forces. F x 0 : A B 0 x A 107.1kN x F y 0 : A 9.81kN 23.5kN 0 A 33.3 kn y y Check the values obtained Sample Problem 4.3 SOLUTION: Create a free-body diagram for the car with the coordinate system aligned with the track. Determine the reactions at the wheels by solving equations for the sum of moments about points above each axle. A loading car is at rest on an inclined track. The gross weight of the car and its load is 25 kn, and it is applied at at G. The cart is held in position by the cable. Determine the tension in the cable and the reaction at each pair of wheels. Determine the cable tension by solving the equation for the sum of force components parallel to the track. Check the values obtained by verifying that the sum of force components perpendicular to the track are zero

10 Sample Problem 4.3 Determine the reactions at the wheels. M A 0 : 10.5 kn625 mm R 1250 mm kn 150 mm R2 8 kn M B 0 : 10.5 kn625 mm R 1250 mm kn 150 mm Create a free-body diagram x y W 25 kn cos kn W 25 kn sin kn R1 2.5 kn Determine the cable tension. Fx 0: kn T 0 T 22.7 kn 4-19 Sample Problem 4.4 SOLUTION: Create a free-body diagram for the telephone cable. Solve 3 equilibrium equations for the reaction force components and couple at A. A 6-m telephone pole of 1600 N is used to support the wires. Wires T 1 = 600 N and T 2 = 375 N. Determine the reaction at the fixed end A

11 Sample Problem 4.4 Create a free-body diagram for the frame and cable. Solve 3 equilibrium equations for the reaction force components and couple. F x 0: A (375N) cos 20 (600N) cos10 0 A x N x 600Nsin Nsin 20 0 F y 0: A 1600N A y A 0 : y N M 600Ncos10(6m) 375N M A M A (6m) N.m cos Notes 1. If there are more unknowns than the number of independent equations, then we have a statically indeterminate situation. We cannot solve these problems using just statics. 2. The order in which we apply equations may affect the simplicity of the solution. For example, if we have two unknown vertical forces and one unknown horizontal force, then solving F X = 0 first allows us to find the horizontal unknown quickly. 3. If the answer for an unknown comes out as negative number, then the sense (direction) of the unknown force is opposite to that assumed when starting the problem

12 Equilibrium of a Two-Force Body Consider a plate subjected to two forces F 1 and F 2. For static equilibrium, the sum of moments about A must be zero. The moment of F 2 must be zero. It follows that the line of action of F 2 must pass through A. Similarly, 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 any direction be zero leads to the conclusion that F 1 and F 2 must have equal magnitude but opposite sense Applications of 2-Force Bodies In the cases above, members AB can be considered as two-force members, provided that their weight is neglected. This fact simplifies the equilibrium analysis of some rigid bodies since the directions of the resultant forces at A and B are thus known (along the line joining points A and B)

13 Equilibrium of a Three-Force Body Consider a rigid body subjected to forces acting at only 3 points. Assuming that their lines of action intersect, the moment of F 1 and F 2 about the point of intersection represented by D is zero. Since the rigid body is in equilibrium, the sum of the moments of F 1, F 2, and F 3 about any axis 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. The lines of action of the three forces must be concurrent or parallel Sample Problem 4.6 SOLUTION: Create a free-body diagram of the joist. Note that the joist is a 3 force body acted upon by the rope, its weight, and the reaction at A. A man raises a 10 kg joist, of length 4 m, by pulling on a rope. Find the tension in the rope and the reaction at A. The three forces must be concurrent for static equilibrium. Therefore, 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

14 Sample Problem 4.6 Create a free-body diagram of the joist. Determine the direction of the reaction force R. AF AB cos 45 4m cos m CD AE 1 AF m BD CD cot(45 20) m tan m CE BF BD m m CE tan AE Sample Problem 4.6 Determine the magnitude of the reaction force R. T sin 31.4 R sin N sin 38.6 T 81.9 N R N

15 Statically Indeterminate Reactions Redundant Constraints: When a body has more supports than necessary to hold it in equilibrium, it becomes statically indeterminate. More unknowns than equations A problem that is statically indeterminate has more unknowns than equations of equilibrium. Are statically indeterminate structures used in practice? Why or why not? 4-29 Improper Constraints Here, we have 6 unknowns but there is nothing restricting rotation about the AB axis. Equal number unknowns and equations but improperly constrained

16 Improper Constraints In some cases, there may be as many unknown reactions as there are equations of equilibrium. However, if the supports are not properly constrained, the body may become unstable for some loading cases. M A Equilibrium of a Rigid Body in Three Dimensions Six scalar equations are required to express the conditions for the equilibrium of a rigid body in the general three dimensional case. Fx 0 M 0 x Fy 0 M 0 y Fz 0 M 0 z These equations can be solved for no more than 6 unknowns which generally represent reactions at supports or connections. The scalar equations are conveniently obtained by applying the vector forms of the conditions for equilibrium, F 0 MO r F

17 Applications Ball-and-socket joints and journal bearings are often used in mechanical systems. To design the joints or bearings, the support reactions at these joints and the loads must be determined Applications The tie rod from point A is used to support the overhang at the entrance of a building. It is pin connected to the wall at A and to the center of the overhang B. If A is moved to a lower position D, will the force in the rod change or remain the same? By making such a change without understanding if there is a change in forces, failure might occur

18 Application The crane, which weighs 1550 N, is supporting a oil drum. How do you determine the largest oil drum weight that the crane can support without overturning? 2-35 Reactions at Supports and Connections for a Three-Dimensional Structure

19 Reactions at Supports and Connections for a Three- Dimensional Structure 4-37 Universal joint in transmission shaft of a truck Exchange arm for a suspension

20 Bearing for trolley wheels 2-39 Sample Problem 4.8 SOLUTION: Create a free-body diagram for the sign. Apply the conditions for static equilibrium to develop equations for the unknown reactions. A sign of uniform density weighs 1200 N and is supported by a balland-socket joint at A and by two cables. Determine the tension in each cable and the reaction at A

21 Sample Problem 4.8 Create a free-body diagram for the sign. Since there are only 5 unknowns, the sign is partially constrain. It is free to rotate about the x axis. It is, however, in equilibrium for the given loading. T T BD EC T T T T T T BD BD BD EC EC EC rd rb rd rb 2.4i 1.2 j 2.4k i j k rc re rc re 6i 3 j 2k 7 i j k Sample Problem 4.8 Apply the conditions for static equilibrium to develop equations for the unknown reactions. F A TBD TEC 1200 Nj i : Ax 3 TBD 7 TEC j : Ay 3 TBD 7 TEC 1200 N k : Az 3 TBD 7 TEC 0 M A rb TBD re TEC j : 1.6TBD 0.514TEC 0 k : 0.8T 0.771T 1440 N. m 0 BD TBD 451 N T A EC EC 1402 N 1.2 mi 1200 N Solve the 5 equations for the 5 unknowns, 1502 Ni 419 Nj Nk j

22 Quiz 5 1. The rod AB is supported using two cables at B and a ball-andsocket joint at A. How many unknown support reactions exist in this problem? A) 5 force and 1 moment reaction B) 5 force reactions C) 3 force and 3 moment reactions D) 4 force and 2 moment reactions If an additional couple moment in the vertical direction is applied to rod AB at point C, then what will happen to the rod? A) The rod remains in equilibrium as the cables provide the necessary support reactions. B) The rod remains in equilibrium as the ball-and-socket joint will provide the necessary resistive reactions. C) The rod becomes unstable as the cables cannot support compressive forces. D) The rod becomes unstable since a moment about AB cannot be restricted

23 Quiz 6 1. A plate is supported by a ball-andsocket joint at A, a roller joint at B, and a cable at C. How many unknown support reactions are there in this problem? A) 4 forces and 2 moments B) 6 forces C) 5 forces D) 4 forces and 1 moment What will be the easiest way to determine the force reaction B Z? A) Scalar equation F Z = 0 B) Vector equation M A = 0 C) Scalar equation M Z = 0 D) Scalar equation M Y =