Basic Axioms of Statics, Constraints, and Constraint Forces

Basic Axioms of Statics, Constraints, and Constraint Forces

Basic Axioms of Statics, Constraints, and Constraint Forces

I. Basic Axioms of Statics

Axiom 1: Axiom of Equilibrium of Two Forces

The necessary and sufficient condition for two forces acting on the same rigid body to be in equilibrium is that they are equal in magnitude, opposite in direction, and act along the same straight line.

An object in equilibrium under two forces is called a two-force body. If the object is a component or rod, it is also referred to as a two-force component or two-force rod (simply "two-force rod").

Axiom 2: Axiom of Adding or Removing Equilibrium Force Systems

Adding or removing an equilibrium force system to/from any force system acting on a rigid body does not change the effect of the original force system on the rigid body.

Principle of Transmissibility of Force: A force acting on a rigid body can be transmitted along its line of action to any point within the rigid body without changing its effect on the body. Thus, a force acting on a rigid body is a sliding vector, and its three key characteristics are: magnitude, direction, and line of action.

Axiom 3: Parallelogram Law of Forces

Two forces acting on the same point of an object can be combined into a resultant force acting at that point. The magnitude and direction of the resultant force are represented by the diagonal of the parallelogram formed with the two force vectors as adjacent sides. When finding the resultant of two forces at a point, the triangle law of forces is often used for simplicity.

Theorem of Three-Force Equilibrium Convergence: If a rigid body is in equilibrium under the action of three non-parallel forces in the same plane, the lines of action of these three forces must intersect at a single point.

Axiom 4: Axiom of Action and Reaction

The mutual forces between two objects always exist simultaneously. They are equal in magnitude, opposite in direction, and act along the same straight line, each acting on one of the two objects.

Axiom 5: Principle of Rigidification

If a deformable body is in equilibrium under a given force system, treating the deformable body as a rigid body (rigidification) will not change its equilibrium state.

II. Constraints and Constraint Forces

In engineering, objects are generally classified into two types:

• Free bodies: Objects that can move arbitrarily in space (e.g., airplanes, rockets).
• Non-free bodies: Objects restricted by other bodies and unable to move in certain directions (e.g., a suspended weight limited by a rope). Such restrictions that hinder movement are called constraints.

Since constraints prevent movement in specific directions, when an object tends to move in a direction restricted by a constraint, the constraint exerts a force to resist this movement. This force is called a constraint force (or reaction force). The direction of a constraint force is always opposite to the direction of the movement or movement tendency that the constraint can hinder. Its point of action is at the contact point between the constraint and the constrained object, and its magnitude can be determined through calculation.

In engineering, forces that actively cause movement or movement tendencies in an object are called active forces. Active forces are usually known, while constraint forces are unknown—they depend on both the active forces and the type of constraint.

Common Constraint Types in Engineering and Characteristics of Their Constraint Forces

Engineering applications involve various constraint types, each with distinct constraint force characteristics, determined by the way the constraint restricts movement. These include:

• Flexible constraints (e.g., ropes, cables), which only exert tensile forces along the direction of the flexible member.
• Rigid surface constraints (e.g., supports, contact surfaces), where the constraint force acts perpendicular to the contact surface (normal direction).
• Hinge constraints (e.g., pin joints), which restrict translation in two directions, resulting in a constraint force with unknown magnitude and direction (often represented by two perpendicular components).

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