Equilibrium and Free Body Diagrams

Equilibrium

• Examines the conditions under which an object remains at rest or in a state of constant motion.
• Makes use of concepts that were introduced in earlier chapters.
• Requires the application of free body diagrams and equations of equilibrium.

Free Body Diagram (FBD)

• It's a sketch that isolates the particle of interest from its environment.
• Displays all forces (known and unknown) acting on the particle.
• It is the most important step in solving force-related problems.

FBD Conceptualization

• Visualize the particle detached from its surroundings.
• Indicate all forces on the particle, categorizing them as active or reactive.
  • Active forces initiate motion.
  • Reactive forces prevent motion through constraints or supports.
• Clearly label each force with its magnitude and direction, using letters for unknown magnitudes.
• Define force directions using angles or rectangular components.

Support Reactions

• Forces exerted by supports on a body, crucial for equilibrium analysis.

Types of Supports:

• Spring: Exerts a tension force proportional to its deformation, defined by ( F = ks ), where ( k ) is the spring constant and ( s ) is the deformation.
• Cables and Pulleys: Feature uniform cable tension directed away from the particle
• Smooth Surfaces: Apply a normal force perpendicular to the surface.
• Pin or Hinge: Provide a force in any planar direction, represented by ( F_x ) and ( F_y ) components.
• Roller: Applies a perpendicular force to the surface.
• Fixed Support: Exerts both force (components ( F_x ) and ( F_y )) and a moment on the member.

Equations of Equilibrium

2D Problems:

• The equations are:
  • ( \sum F_x = 0 ) (sum of forces in the x-direction equals zero)
  • ( \sum F_y = 0 ) (sum of forces in the y-direction equals zero)
  • ( \sum M_O = 0 ) (sum of moments about point O equals zero).

3D Problems:

• The equations are:
  • ( \sum F_x = 0 ), ( \sum F_y = 0 ), ( \sum F_z = 0 ) (sum of forces in each direction equals zero)
  • ( \sum M_x = 0 ), ( \sum M_y = 0 ), ( \sum M_z = 0 ) (sum of moments about each axis equals zero).

Procedure for Analysis:

• Establish a coordinate system.
• Draw a Free-Body Diagram (FBD).
• Apply the equations of equilibrium.
  • Resolve forces into rectangular components.
  • Compute moments about a strategic point to eliminate unknown forces.
• Solve the equations for the unknowns.
• A negative result indicates the force direction is opposite to what was initially assumed in the FBD.