Virtual Work

Why it matters

The principle of virtual work is a powerful tool in engineering mechanics, allowing engineers to analyze complex mechanical systems efficiently. It simplifies the process of determining equilibrium and stability in structures, which is crucial for designing safe and reliable infrastructure.

Key ideas

• Virtual Work Principle: This principle states that for a system in equilibrium, external virtual work equals internal virtual work for compatible virtual displacements. For an ideal rigid system, zero-work constraint reactions can be eliminated, leaving zero total virtual work of applied forces.
• Virtual Displacement: A hypothetical, infinitesimally small change in the configuration of a system, consistent with the constraints.
• Applications: Used in structural analysis, particularly for determining deflections and internal forces in beams, frames, and trusses.
• Advantages: Simplifies calculations by reducing the number of equations needed, especially in statically indeterminate structures.

Formulas

For a linearly elastic Euler–Bernoulli beam with small deflections, constant or known EI, and negligible shear deformation, the unit-load method gives displacement in a chosen direction:

Δ = ∫ M(x)m(x)/(EI) dx

M is the actual bending moment; m is the moment per unit force from a virtual unit load in the displacement direction. Use compatible signs. For a truss the corresponding expression is Δ = Σ N_i n_i L_i/(A_i E_i).

Worked example

A simply supported prismatic beam has span L = 6 m, a central downward point load P = 10 kN, E = 200 GPa and I = 8×10⁻⁵ m⁴. Find its midspan deflection, neglecting shear deformation. Stiffness data are essential; loads and span alone cannot determine deflection.

For 0 ≤ x ≤ L/2, M = Px/2 and the virtual moment per unit central load is m = x/2. By symmetry,

Δ = 2∫₀^(L/2) (Px/2)(x/2)/(EI) dx = PL³/(48EI)

Δ = 10000 × 6³ / (48 × 200×10⁹ × 8×10⁻⁵) = 0.0028125 m

Answer: 2.8125 mm downward. Multiplying just the peak moment by an assumed rotation does not account for the distributed strain energy and cannot replace the integral.

Common mistakes

• Confusing real and virtual displacements.
• Incorrectly applying the principle to non-equilibrium systems.
• Neglecting constraints in virtual displacements.

For GATE CE

Questions often involve calculating deflections or internal forces using virtual work. Practice problems on beams, frames, and trusses, focusing on setting up virtual displacements and applying the principle correctly.

Quick check

1. What is the principle of virtual work?
2. How is virtual displacement defined?
3. Why is virtual work useful in structural analysis?

Answers: 1. The total virtual work done by all forces during a virtual displacement is zero. 2. A hypothetical, infinitesimally small change in the configuration of a system. 3. It simplifies calculations, especially in statically indeterminate structures.