3.3 Equilibrium of rigid bodies in two and three dimensions

Equilibrium of rigid bodies is all about balance. In 2D, we need zero forces in x and y directions, plus zero total moment. In 3D, it's the same, but we add a z-direction and moments around all three axes.

Free-body diagrams are key to solving these problems. They show all forces and moments acting on an isolated body. Whether forces meet at one point or not, we use equilibrium equations to find unknown forces and keep things steady.

Equilibrium of Rigid Bodies

Conditions for Equilibrium in Two and Three Dimensions

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• Equilibrium of a rigid body occurs when the sum of all forces acting on the body is zero and the sum of all moments about any point is zero
• In two dimensions, the conditions for equilibrium are:
  • ΣFx = 0 (sum of forces in the x-direction equals zero)
  • ΣFy = 0 (sum of forces in the y-direction equals zero)
  • ΣM = 0 (sum of moments about any point equals zero)
• In three dimensions, the conditions for equilibrium are:
  • ΣFx = 0 (sum of forces in the x-direction equals zero)
  • ΣFy = 0 (sum of forces in the y-direction equals zero)
  • ΣFz = 0 (sum of forces in the z-direction equals zero)
  • ΣMx = 0 (sum of moments about the x-axis equals zero)
  • ΣMy = 0 (sum of moments about the y-axis equals zero)
  • ΣMz = 0 (sum of moments about the z-axis equals zero)

Static Equilibrium and Free-Body Diagrams

• A rigid body is considered to be in static equilibrium when it is at rest or moving with a constant velocity, and the net force and net moment acting on the body are zero
• The concept of free-body diagrams is essential in solving equilibrium problems
  • Involves isolating the body of interest
  • Representing all forces and moments acting on the isolated body
• Example: A ladder leaning against a wall can be analyzed using a free-body diagram, showing the forces acting on the ladder (normal force from the wall, normal force from the ground, and the ladder's weight) and the moments they create

Concurrent vs Non-Concurrent Forces

Concurrent Force Systems

• Concurrent force systems are those in which all forces act through a single point
• For concurrent force systems, the equilibrium equations can be solved by considering the forces acting at the point of concurrency
• Example: Multiple cables attached to a single point on a suspended object create a concurrent force system at that attachment point

Non-Concurrent Force Systems

• Non-concurrent force systems have forces that do not pass through a single point
• Non-concurrent force systems require the consideration of both forces and moments to solve for equilibrium
• The moment of a force about a point is the product of the force magnitude and the perpendicular distance from the point to the line of action of the force
• Varignon's theorem states that the moment of a force about a point is equal to the sum of the moments of the force's components about the same point
• Couples are a special case of non-concurrent force systems
  • Consist of two equal and opposite forces with a non-zero moment arm
  • Produce a pure moment
• Example: A beam supported at both ends with a load applied at its midpoint experiences non-concurrent forces from the supports and the applied load

Equilibrium Applications in Structures

Trusses

• Trusses are structural systems composed of connected members, typically forming triangular units
  • Designed to support loads primarily through axial forces (tension or compression) in the members
• The method of joints is used to analyze trusses by considering the equilibrium of each joint, assuming pin connections and no moments at the joints
• The method of sections is an alternative approach to analyzing trusses
  • Involves the imaginary cut of the truss into two sections
  • Considers the equilibrium of one of the sections
• Example: A bridge truss supporting the weight of vehicles and the bridge deck itself can be analyzed using the method of joints or the method of sections

Frames and Machines

• Frames are structures containing multi-force members, which can support loads through axial forces, shear forces, and bending moments
• Machines are devices that transmit or modify forces and can be analyzed using equilibrium principles
  • Consider the forces acting on each component and the connections between them
• Example: A crane frame supporting a heavy load can be analyzed by considering the forces acting on each member and the moments they create at the joints

Moment Equilibrium for Rigid Bodies

Moment Equilibrium Principle

• Moment equilibrium is achieved when the sum of all moments acting on a rigid body about any point is zero
• The choice of the moment center (the point about which moments are calculated) is arbitrary, but selecting a point that simplifies the calculations is often advantageous
• When a rigid body is in equilibrium, the net moment about any point must be zero, regardless of the chosen moment center

Couples and Distributed Loads

• The moment of a couple is independent of the choice of the moment center and is always equal to the product of one of the forces and the perpendicular distance between the two forces
• In problems involving distributed loads, the concept of the center of gravity or centroid is used to determine the equivalent point load for calculating moments
• Example: A rectangular plate subjected to a uniformly distributed load can be analyzed by calculating the moment about a chosen point, considering the total load acting at the plate's centroid