Internal virtual work represents the work done by internal forces within a body when it undergoes a virtual displacement. In essence, it's a measure of the internal stresses resisting deformation during a hypothetical, infinitesimally small movement. The principle of virtual work provides a condition for establishing equilibrium.
Understanding Internal Virtual Work
Here's a breakdown of the key aspects:
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Virtual Displacement: This is a hypothetical, infinitesimally small displacement imposed on the body. It's "virtual" because it doesn't necessarily represent an actual displacement the body is undergoing.
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Internal Forces: These are the forces that hold the body together, arising from the stresses within the material. They resist the deformation caused by external loads.
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Virtual Work Done by Internal Forces (δWint): This is the work done by these internal forces as the body undergoes the virtual displacement. It is negative when internal forces resist the virtual displacement.
The Principle of Virtual Work
The principle of virtual work states that a body is in equilibrium when the virtual work done by the internal forces (δWint) equals the virtual work done by the external forces (δWext):
δWint = δWext (From provided reference)
This principle provides a powerful tool for solving structural mechanics problems.
Example
Consider a simple beam subjected to a load.
- Apply a Virtual Displacement: Imagine applying a small, virtual deflection to the beam.
- Identify Internal Forces: Internal bending moments and shear forces will arise within the beam to resist this deflection.
- Calculate Virtual Work: The internal virtual work (δWint) would be the integral of the internal moments multiplied by the virtual curvature along the beam. The external virtual work (δWext) would be the external load multiplied by its virtual displacement.
- Apply the Principle: By setting δWint = δWext, we can derive equations to determine the equilibrium state of the beam (e.g., bending moment distribution).
Why is Internal Virtual Work Important?
- Equilibrium Determination: It provides a direct method to determine the equilibrium conditions of a structure.
- Solving Complex Problems: It simplifies the analysis of statically indeterminate structures where equilibrium equations alone are insufficient.
- Finite Element Analysis (FEA): The principle of virtual work is fundamental to the formulation of many finite element methods used in engineering simulations.
Table summarizing Internal Virtual Work
| Concept | Description |
|---|---|
| Virtual Displacement | Hypothetical, infinitesimally small displacement. |
| Internal Forces | Forces within a body resisting deformation. |
| δWint | Virtual work done by internal forces during a virtual displacement; negative when internal forces resist displacement. |
| Equilibrium Condition | δWint = δWext (Virtual work done by internal forces equals virtual work done by external forces). |
