1. EQUILIBRIUM OF RIGID BODIES
CHAPTER 6A
EQUILIBRIUM OF RIGID BODIES
Engineering Mechanics
Part I: Statics
Dr. El-Sayed Awad Sayed-Ahmed

2. CHAPTER 6 - Contents
EQUILIBRIUM OF RIGID BODIES
6.1 Introduction
6.2 Free-body Diagram
SEC. 1: EQUILIBRIUM IN TWO DIMENSIONS
6.3 Equilibrium Equations of a Rigid Body in Two Dimensions
SEC. 2: EQUILIBRIUM IN THREE DIMENSIONS
6.4 Introduction
6.5 Free-Body Diagrams
6.6 Equilibrium Equations of a Rigid Body in Three Dimensions
6.7 Constraints for a Rigid Body

3. EQUILIBRIUM OF RIGID BODIES
A rigid body is said to be in equilibrium when the sum of all
the external forces acting on the body is equal to zero to
prevent the body from translating with accelerated motion,
and the sum of the moments of the external forces about any
point is equal to zero to prevent the body from rotating .
The vector equations for the equilibrium of a rigid body:
R=∑F = MR0 = ∑(r × F) =0

4. EQUILIBRIUM IN TWO DIMENSIONS
The three scalar equations for the equilibrium of a rigid body
∑Fx = 0, ∑Fy = 0,
∑Mz = 0

5. EQUILIBRIUM IN TWO DIMENSIONS
Alternative sets of equilibrium equations:
∑F a = 0, ∑F A = 0, ∑MB =0
Where, points A and B do not lie on a line that is perpendicular to the a axis.
∑M A = 0, ∑MB = 0, ∑MC =0
Where, points A, B, and C do not lie on the same line.

6. EQUILIBRIUM IN TWO DIMENSIONS
Reactions at Supports and Connections for a Two-
Dimensional Rigid Body:
Forces and couples exerted on an object by its supports
or connections are called reactions. The other forces
and couples on the object are the loads.

7. EQUILIBRIUM IN TWO DIMENSIONS
General rule for supports:
If a support prevents translation of a body in a particular direction, then the support exerts a force on the body in that direction.
If rotation is prevented by the support, then the support exerts a couple moment on the body

8. EQUILIBRIUM IN TWO DIMENSIONS
Reactions Equivalent to a Force with Known Line of Action
Rollers
Rockers
Frictionless
surfaces
Short links
Short cables
Collars on
frictionless rods
Frictionless pins
in slots.

9. EQUILIBRIUM IN TWO DIMENSIONS
Reactions Equivalent to a Force with unKnown Line of Action
Frictionless pins in fitted holes.
Hinges
Rough surfaces.

10. EQUILIBRIUM IN TWO DIMENSIONS
Reactions Equivalent to a Force and a Couple
Fixed supports

11. Free-body diagram for a rigid body
Construction of a free-body diagrams for a rigid body
A clear decision is made concerning the choice of the free body to be isolated . You should always be certain that you have completely isolated the body by drawing
its outline shape.
All the external forces and couple moments that act on the isolated body as applied by the removed contacting and attracting bodies are representing in their proper positions on the diagram of the isolated body. Recall, these forces are due to (1) applied loadings, (2) reactions occurring at the supports or at points of contact with other bodies, the weight of the body. The internal forces should not be included .

12. Free-body diagram for a rigid body
The dimensions of the body necessary for computing
the moments of forces should be clearly marked on
the free-body diagram. The forces and couple moments
that are known should be labeled with their proper
magnitudes and directions. Letters are used to
represent the magnitudes and direction angles of
forces and couple moments that are unknown.
Establish an x, y coordinate system .

13. Free-body diagram for a rigid body

14. Free-body diagram for a rigid body

15. Equilibrium under two force members
If the object in equilibrium, the two forces are equal in
magnitude, opposite in direction and collinear i.e.
FA=-FB, since and ∑MO =0.

16. Equilibrium under three force members
If the object is in equilibrium, three forces are coplanar
and either parallel or concurrent

17. Draw the free body diagram of member ABC which is supported by a smooth collar at A, roller at B, and short link CD. Explain the significance of each force acting on the diagram. Determine the reaction on the member A,B,C

21. Determine the magnitude of force at the pin A in cable BC needed to support the 500 –lb load. Neglect the weight of the boom AB

23. Determine the reactions at the pin A and B
Determine the reactions at the pin A and B . The spring has an stretched length of 80mm.