The principle of virtual work - Static case

✨ 5.6.2025, 12:13

•

✏️ 3.8.2025, 23:44

1. Equilibrium of a mechanical system

Or lets assume a system where external (impressed) forces \(\mathbf{\lforce}_{ i}\) act at points \(\textbf{P}_{ i}\), then the total virtual work is given by

\begin{align} \label{eq:3ybmk5fehp} \delta \work = \mathbf{\lforce}_{ 1} \cdot \delta \mathbf{\pos}_{ 1} + \mathbf{\lforce}_{ 2} \cdot \delta \mathbf{\pos}_{ 2} + \dots + \mathbf{\lforce}_{ i} \cdot \delta \mathbf{\pos}_{ i} \end{align}

where \(\delta \mathbf{\pos}_{ i}\) are the virtual displacements of the points \(\mathbf{P}_{ i}\).

1.1. Virtual displacements

The virtual displacements are in harmony with the given kinematical constraints. What is a kinematical constraint? What does it mean to be in harmony with these kinematical constraints?

1.2. Principle of virtual work - The only postulate of analytical mechanics

The principle of virtual work is now expressed by

\begin{align} \delta \work = 0, \end{align}

i.e. the system is in equilibrium, if, and only if, the total virtual work vanishes for all impressed forces \( \mathbf{\force}_{ i}.\) Using \eqref{eq:3ybmk5fehp} this means

\begin{align} \label{eq:y9v1lehind} \sum \mathbf{\lforce}_{ i} \cdot \delta \mathbf{\pos}_{ i} = 0 \end{align}

1.3. Principle of equilibrium - Newton's third law

Since any particle \(i\) is in equilibrium the resultant force \( \mathbf{\force}_i\) on any particle of the system is is zero

\begin{align} \label{eq:af4ikfk3nw} \mathbf{\force}_i = \mathbf{0} \end{align}

and therefore trivially for the resultant force of the entire system

\begin{align} \label{eq:8ki296qyqw} \mathbf{\force} = \mathbf{\force}_{ 1} + \mathbf{\force}_{ 2} + \dots + \mathbf{\force}_{ i} = \mathbf{0} \end{align}

The resultant force can be decomposed into the impressed force \( \mathbf{\lforce}_{ i}\) and the forces that maintain the kinematical constraints, the so called forces of reaction \( \mathbf{\rforce}_{ i}\) (Zwangskräfte)

\begin{align} \mathbf{\force}_{ i} = \mathbf{\lforce}_{ i} + \mathbf{\rforce}_{ i} \end{align}

Clearly in general what applies to the resulting force in \eqref{eq:af4ikfk3nw} does not hold for impressed or constraint forces alone \(\mathbf{\lforce}_{ i} = - \mathbf{\rforce}_{ i}\neq \mathbf{0}\) and hence from \eqref{eq:8ki296qyqw}

\begin{align} \mathbf{\lforce}_{ 1} + \mathbf{\lforce}_{ 2} + \dots + \mathbf{\lforce}_{ i} + \mathbf{\rforce}_{ 1} + \mathbf{\rforce}_{ 2} + \dots + \mathbf{\rforce}_{ i} &= \textbf{0} \\ \label{eq:0k28pmdm6g} \sum \mathbf{\lforce}_{ i} + \sum \mathbf{\rforce}_{ i} &= \textbf{0} \end{align}

Multiplication by virtual displacements yields

\begin{align} \mathbf{\force}_{ i} \cdot \delta \mathbf{\pos}_{ i} = 0 \end{align}

for any particle \(i\) of the system, where again this does not hold individually for impressed or constraint forces, \(\mathbf{\lforce}_{ i} \cdot \delta \mathbf{\pos}_{ i}= - \mathbf{\rforce}_{ i} \cdot \delta \mathbf{\pos}_{ i}\neq 0\). From \eqref{eq:0k28pmdm6g} Newton's third law leads to

\begin{align} \label{eq:hparzqkryi} \sum \mathbf{\lforce}_{ i} \cdot \delta \mathbf{\pos}_{ i}= - \sum \mathbf{\rforce}_{ i}\cdot \delta \mathbf{\pos}_{ i} \end{align}

As is obvious the principle of virtual work \eqref{eq:y9v1lehind} cannot be deduced by Newton's third law alone which does not include the fundamental postulate of analytical mechanics¹ ^,².

1.4. Alternate form

With \eqref{eq:hparzqkryi} we may derive an alternate form of the principle of virtual work \eqref{eq:y9v1lehind}: the virtual work of the forces of reaction is always zero for any virtual displacements in harmony with the given kinematical constraints.

\begin{align} \label{eq:fhigg9jxzl} \sum \rforce_{ i}\cdot \delta \pos_{ i} = 0 \end{align}

Postulate

The virtual work of the forces of reaction is always zero for any virtual displacement which is in harmony with the given kinematic constraints.

with the remark

Postulate A \eqref{eq:fhigg9jxzl} is actually the only postulate of analytical mechanics, and is thus of fundamental importance. - (1986)

2. Equilibrium for a system of particles

First define the virtual work upon a particle \(i\) by

\begin{align} \delta \work_{ i} := \force_{ i} \cdot \delta \pos_{ i} \end{align}

The decomposition of the net force into constraining (reacting) forces and (impressed) load forces yields

\begin{align} \label{eq:dkzix0jhhy} \force_{ i} = \rforce_{ i} + \lforce_{ i} = \mass_{ i} \, \ddot{\pos}_{ i} \end{align}

Then by multiplying \eqref{eq:dkzix0jhhy} by the variation \(\delta \pos_{ i}\)

\begin{align} \delta \work_{ i} = \left( \rforce_{ i} + \lforce_{ i} \right)\cdot \delta \pos_{ i} = \mass_{ i} \, \ddot{\pos}_{ i}\cdot \delta \pos_{ i} \end{align}

Then for one particle \(i\) we can rearrange the last two terms

\begin{align} \left( \lforce_{ i} - \mass_{ i} \, \ddot{\pos}_{ i} \right) \cdot \delta \pos_{ i} + \rforce_{ i} \cdot \delta \pos_{ i} = 0 \end{align}

3. Notation

Notation for continuum mechanics and thermodynamics notes

4. References

Lanczos, C. (1986). The Variational Principles of Mechanics, Dover Publications.