Lecture 12

1. ENGR-1100 Introduction to Engineering Analysis Lecture 12

2. Lecture Outline • Rigid body equilibrium. • FBD- Free Body Diagram.

3. Rigid Bodies • A rigid body is an ideal object that has dimensions andmassbut does not deform under loading. • The sizeand shape does significantly affect the response (reaction at supports) to applied force/s - A force applied to a rigid body may be translated along its line of action without altering its effects (principle of transmissibility)

4. Particle equilibrium Rigid body equilibrium Those are vector equations. How many scalar equations in the 2D? in the 3D? Necessary and Sufficient Conditions of Equilibrium of a Rigid Body The necessary AND sufficient condition for a rigid body to be in equilibrium is that the resultant force and couple acting on the rigid body must be zero.

5. 6 independent equations in 3D 3 independent equations in 2 D

6. Isolation of bodies: Free Body Diagram -In a system of interacting bodies, to be able to apply Newton’s laws properly, a systematic technique is to: • isolate each of the bodies or a collection of bodies • identify the forces acting on each of the bodies, and then • apply the equilibrium equations to each. -The sketch of the isolated body or system of bodies considered as a single body, with all the external forces acting on it by mechanical contact with other bodies by gravitational attraction (weight) when the rest of the bodies are imagined to be replaced by their actions, it is known as a free body diagram.

7. Isolation of bodies: Free Body Diagram The free body diagram is the most important single step in the solution of problems in mechanics. • Remember: • The forces on the isolated body (or system of bodies) are to be considered. • 2. Apply Newton’s Third Law (every action has an equal, opposite and collinear reaction) carefully

8. Steps of Drawing a FBD • Clearly identify the body (or system of bodies) to be isolated (the FREE body). • Draw a diagram of this “free body” completely isolated from the rest of the bodies. • Traverse the boundary of this “free body” and indicate ALL forces acting ON the free body (contact forces with other bodies forces) • Known forces: Show vector arrows with proper magnitude (UNITS!!), direction and sense. • Unknown magnitude but known direction of force: Show vector arrows with magnitude assumed as positive (if calculations show that the magnitude is negative, the minus sign indicates that the sense is opposite to the one assumed) • Unknown magnitude and direction of force: Show x- and y- components of the vector with unknown magnitudes. • 4. Show coordinate directions on the diagram

9. 1. Gravitational attraction 2. Flexible cord, rope, chain, or cable Two-dimensional reaction at supports and connections

10. 3. Rigid link 4. Ball, roller, or rocker 6. Smooth pin 5. Smooth surface

11. 8. Pin in a smooth pin guide 7. Rough surface

12. 9a. Collar on a smooth shaft Pin connection 9b. Collar on a smooth shaft: Fixed connection

13. 1. Ball and socket 2. Hinge Three-dimensional reaction at supports and connections

14. 3. Ball bearing 4. Journal bearing

15. 6. Smooth pin and bracket 5. Thrust bearing 7. Fixed support

16. Example P6-2 Draw complete free-body diagram of the beam shown in Fig. P6-2, which has a mass m.

17. B Ay y P Ax 450 x Known forces mg Unknown forces Solution

18. Class Assignment:Exercise set 6-1, 6-3, 6-4, 6-10 please submit to TA at the end of the lecture y p Ay Ax x MA W y x N1 N2 W p T y Ay Ax x y p p W N1 x F1 F1 N1 N2 p 6-1 6-3 p 6-4 6-10

19. Example P6-25 Draw complete free-body diagram of the bent bar shown in Fig. P6-25. The support at A is a journal bearing and the supports at B and C are ball bearings.

20. z Cz P1 Cx By Bx Az Mz P2 My y P3 Ay x Solution Known forces Unknown forces

21. p1 p2 Az T MAZ MAy p1 Az Ay MAx Ax Ax Ay p2 Class Assignment:Exercise set 6-22 & 6-24 please submit to TA at the end of the lecture p1 p2 p1 p2