Travail virtuel

The principle of virtual work states that for a body in static equilibrium, the algebraic sum of virtual work done by all forces and moments must equal zero for any virtual displacement or rotation. Virtual work is calculated as the product of force and virtual displacement, or moment and virtual angular displacement. This principle allows engineers to analyze equilibrium conditions without solving complex force equations directly.

Virtual work done by a force equals the dot product of the force and virtual displacement in the direction of the force. For a couple moment, virtual rotational work is determined by multiplying the couple moment by its respective virtual rotation. Both calculations follow the same fundamental principle: work equals the product of the action (force or moment) and the corresponding displacement or rotation.

Reaction forces at supports do not contribute to virtual work because the supports prevent displacement at those points. When a supported beam undergoes virtual rotation, only the forces acting away from the fixed support perform work. Since virtual displacement at a support is zero, the product of reaction force and displacement is zero, regardless of the force magnitude.

When a ball undergoes virtual downward displacement, its weight does positive virtual work because the force and displacement are in the same direction. Conversely, the normal force does negative virtual work because it acts upward while displacement is downward. For equilibrium, these opposing virtual works sum to zero, satisfying the principle of virtual work.

Three separate equations are required to analyze rigid bodies subjected to coplanar forces. These equations account for translation in the x and y directions and rotation about an axis perpendicular to the x-y plane. Each equation represents a different type of virtual displacement, allowing complete analysis of rigid body equilibrium under coplanar loading conditions.

Virtual work applies to systems of connected rigid bodies by considering the virtual displacements of all bodies simultaneously. The principle states that the sum of virtual work done by all external forces and couple moments across the entire system must equal zero for equilibrium. This approach simplifies analysis of complex multi-body systems compared to analyzing each body independently.

Virtual work and equilibrium are fundamentally linked: a body is in equilibrium if and only if the sum of virtual work done by all forces and moments equals zero for any virtual displacement. This relationship provides a powerful alternative method to traditional force and moment equations for determining equilibrium. The principle applies to both static and dynamic systems, making it versatile for engineering analysis.