Chapter 4 EQUILIBRIUM OF RIGID BODIES In the study of the equilibrium of rigid bodies, i.e. the situation when the external forces acting on a rigid body form a system equivalent to zero, we have
r ∑F =0
r r r ∑ MO = ∑ ( r × F ) = 0
Resolving each force and each moment into its rectangular components, the necessary and sufficient conditions for the equilibrium of a rigid body are expressed by six scalar equations:
ΣFx = 0 ΣMx = 0
ΣFy = 0 ΣMy = 0
ΣFz = 0 ΣMz = 0
These equations can be used to determine unknown forces applied to the rigid body or unknown reactions exerted by its supports.
When solving a problem involving the equilibrium of a rigid body, it is essential to consider all of the forces acting on the body. Therefore, the first step in the solution of the problem should be to draw a free-body diagram showing the body under consideration and all of the unknown as well as known forces acting on it.
Free-Body Diagram First step in the static equilibrium analysis of a rigid body is identification of all forces acting on the body with a free-body diagram. • Select the extent of the free-body and detach it from the ground and all other bodies. • Indicate point of application, magnitude, and direction of external forces, including the rigid body weight. • 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. • Include the dimensions necessary to compute the moments of the forces.
Reactions at Supports and Connections for a TwoDimensional Structure • Reactions equivalent to a force with known line of action.
Reactions at Supports and Connections for a TwoDimensional Structure • Reactions equivalent to a force of unknown direction and magnitude.
• Reactions equivalent to a force and a couple.
Equilibrium of a Rigid Body in Two Dimensions • For all forces and moments acting on a twodimensional structure,
Fz = 0 M x = M y = 0 M z = MO • Equations of equilibrium become
∑ Fx = 0 ∑ Fy = 0 ∑ M A = 0 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 ∑MA = 0 ∑MB = 0
Statically Indeterminate Reactions
• More unknowns than equations
• Fewer unknowns than • Equal number unknowns equations, partially and equations but constrained improperly constrained