EQUILIBRIUM OF RIGID BODIES CHAPTER 6A EQUILIBRIUM OF RIGID BODIES Engineering Mechanics Part I: Statics Dr. El-Sayed Awad Sayed-Ahmed elsayed_awad46@yahoo.com

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

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 = 0 MR0 = ∑(r × F) =0

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

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.

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.

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

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.

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

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

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 .

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 .

Free-body diagram for a rigid body

Free-body diagram for a rigid body

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.

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

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

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

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.