Virtual Work - Theory & Concepts

The method of Virtual Work is an alternative, powerful approach to solving equilibrium problems. Instead of considering the forces acting on a static, rigid body using ΣF=0\Sigma F = 0 and ΣM=0\Sigma M = 0, we imagine the body or system of interconnected bodies undergoing a small, hypothetical displacement (a virtual displacement) and analyze the work done by the active forces.

Definition of Work

Before discussing virtual work, we must define mechanical work.

Work of a Force

The work UU done by a force F\mathbf{F} moving through a displacement drd\mathbf{r} is the dot product of the force and displacement vectors:
$ dU = \mathbf \cdot d\mathbf = F ds \cos \theta
WhereWhere\theta$ is the angle between the force vector and the direction of displacement.
  • If the force and displacement are in the same direction (θ=0∘\theta = 0^\circ), work is positive (+F⋅ds+F \cdot ds).
  • If they are in opposite directions (θ=180∘\theta = 180^\circ), work is negative (−F⋅ds-F \cdot ds).
  • If they are perpendicular (θ=90∘\theta = 90^\circ), work is zero.
The work done by a couple moment MM during a small rotation dθd\theta is:

Work of a Couple Moment Differential

Calculates the differential work done by a moment.

$$ dU = M \, d\theta $$
Where dθd\theta is measured in radians. Work is positive if the moment and rotation are in the same sense (both clockwise or both counter-clockwise).

Principle of Virtual Work

A virtual displacement (δs\delta s or δθ\delta \theta) is a purely imaginary, infinitesimal displacement given to a system that is assumed to be in equilibrium. It must be consistent with the physical constraints of the system (e.g., a point on a flat floor can only move horizontally, not vertically down into the floor).

Important

Principle of Virtual Work: For a rigid body or a system of connected rigid bodies to be in equilibrium, the total virtual work (δU\delta U) done by all external active forces during any virtual displacement consistent with the constraints must be zero.
$ \delta U = 0
$

Active vs. Reactive Forces

The main advantage of Virtual Work is that we can often ignore reaction forces entirely.
  • Active Forces: Forces that do work during the virtual displacement (e.g., applied external loads, gravity/weight).
  • Reactive Forces: Forces that do no work because their point of application does not move in the direction of the force (e.g., the normal force from a fixed support, the tension in an inextensible cable, the internal forces at a frictionless pin connecting two members).

Degrees of Freedom

A system's degrees of freedom (DOF) is the number of independent coordinates required to completely specify the position of all its parts.

Checklist

Potential Energy and Stability

When a system is subjected only to conservative forces (like gravity or linear elastic springs), the work done is independent of the path taken and depends solely on the initial and final positions.

Potential Energy (VV)

The capacity of a conservative force to do work is measured by its Potential Energy (VV).
  • Gravitational Potential Energy (VgV_g): Vg=WyV_g = Wy where WW is the weight and yy is the vertical elevation of its center of gravity relative to a defined datum. (Positive if above datum, negative if below).
  • Elastic Potential Energy (VeV_e): The energy stored in a deformed spring. Ve=12ks2V_e = \frac{1}{2}ks^2 where kk is the spring stiffness and ss is its deformation. VeV_e is always positive.
The Total Potential Energy of the system is V=Vg+VeV = V_g + V_e.

Important

Criterion for Equilibrium and Stability: According to the potential energy theorem, a system is in equilibrium if the first derivative of the total potential energy with respect to its independent coordinate (e.g., θ\theta) is zero: dVdθ=0\frac{dV}{d\theta} = 0
Furthermore, the stability of that equilibrium position is determined by the second derivative evaluated at the equilibrium angle:
  • Stable Equilibrium: d2Vdθ2>0\frac{d^2V}{d\theta^2} > 0 (Potential energy is at a minimum. If slightly disturbed, the system will return to this position).
  • Unstable Equilibrium: d2Vdθ2<0\frac{d^2V}{d\theta^2} < 0 (Potential energy is at a maximum. If disturbed, it will move further away from this position).
  • Neutral Equilibrium: d2Vdθ2=0\frac{d^2V}{d\theta^2} = 0 (The system remains in equilibrium even when disturbed).

Mechanical Efficiency

In real-world machines, the work input is never fully converted to work output due to non-conservative forces like friction, which dissipate energy as heat. The Mechanical Efficiency (ϵ\epsilon) of a machine is the ratio of useful work output to total work input. ϵ=UoutUin\epsilon = \frac{U_{\text{out}}}{U_{\text{in}}} Where ϵ\epsilon is always less than 1 (or <100%< 100\%). When using the principle of virtual work to analyze a machine with friction, the virtual work done by friction must be included as negative work, meaning: δUin+δUfriction=δUout\delta U_{\text{in}} + \delta U_{\text{friction}} = \delta U_{\text{out}}.
Key Takeaways
  • The Principle of Virtual Work states that a system is in equilibrium if the total work done by active forces during any virtual displacement is zero (δU=0\delta U = 0).
  • A virtual displacement is an imaginary, infinitesimal change in position consistent with the system's constraints.
  • This method is highly efficient for analyzing mechanisms (like linkages) because it allows you to ignore internal reactive forces at pins and external reactive forces at fixed supports.
  • The standard procedure involves defining coordinates in terms of a single variable (e.g., θ\theta), differentiating to find virtual displacements (δx,δy\delta x, \delta y), and setting the sum of Force ×\times virtual displacement to zero.
PreviousMoments of Inertia - Examples & Applications
Quiz Me
NextVirtual Work - Examples & Applications