In mechanics, virtual work arises in the application of the principle of least action to the study of forces and movement of a mechanical system. The work of a force acting on a particle as it moves along a displacement is different for different displacements. Among all the possible displacements that a particle may follow, called virtual displacements, one will minimize the action. This displacement is therefore the displacement followed by the particle according to the principle of least action. <blockquote>The work of a force on a particle along a virtual displacement is known as the virtual work.</blockquote>

Historically, virtual work and the associated calculus of variations were formulated to analyze systems of rigid bodies, but they have also been developed for the study of the mechanics of deformable bodies.

History

The principle of virtual work had always been used in some form since antiquity in the study of statics. It was used by the Greeks, medieval Arabs and Latins, and Renaissance Italians as "the law of lever". The idea of virtual work was invoked by many notable physicists of the 17th century, such as Galileo, Descartes, Torricelli, Wallis, and Huygens, in varying degrees of generality, when solving problems in statics. In 1768, Lagrange presented the virtual work principle in a more efficient form by introducing generalized coordinates and presented it as an alternative principle of mechanics by which all problems of equilibrium could be solved. A systematic exposition of Lagrange's program of applying this approach to all of mechanics, both static and dynamic, essentially D'Alembert's principle, was given in his Mécanique Analytique of 1788. shows that these generalized forces can also be formulated in terms of the ratio of time derivatives. That is,

<math display="block"> Q_k = \sum_{i=1}^m \mathbf{F}_i \cdot \frac{\partial \mathbf{v}_i}{\partial \dot{q}_k} + \sum_{j=1}^n \mathbf{M}_j \cdot \frac{\partial \mathbf{\omega}_j}{\partial \dot{q}_k} , \quad k = 1, 2, \dots, f . </math>

The principle of virtual work requires that the virtual work done on a system by the forces F<sub>i</sub> and moments M<sub>j</sub> vanishes if it is in equilibrium. Therefore, the generalized forces Q<sub>k</sub> are zero, that is

<math display="block"> \delta W=0 \quad \Rightarrow \quad Q_k = 0 \quad k =1, 2, \dots, f . </math>

Constraint forces

An important benefit of the principle of virtual work is that only forces that do work as the system moves through a virtual displacement are needed to determine the mechanics of the system. There are many forces in a mechanical system that do no work during a virtual displacement, which means that they need not be considered in this analysis. The two important examples are (i) the internal forces in a rigid body, and (ii) the constraint forces at an ideal joint.

Lanczos

Gear train

A gear train is formed by mounting gears on a frame so that the teeth of the gears engage. Gear teeth are designed to ensure the pitch circles of engaging gears roll on each other without slipping, this provides a smooth transmission of rotation from one gear to the next. For this analysis, we consider a gear train that has one degree-of-freedom, which means the angular rotation of all the gears in the gear train are defined by the angle of the input gear.

thumb|right|300px|Illustration from Army Service Corps Training on Mechanical Transport, (1911), Fig. 112 Transmission of motion and force by gear wheels, compound train

The size of the gears and the sequence in which they engage define the ratio of the angular velocity ω<sub>A</sub> of the input gear to the angular velocity ω<sub>B</sub> of the output gear, known as the speed ratio, or gear ratio, of the gear train. Let R be the speed ratio, then

<math display="block"> \frac{\omega_A}{\omega_B} = R.</math>

The input torque T<sub>A</sub> acting on the input gear G<sub>A</sub> is transformed by the gear train into the output torque T<sub>B</sub> exerted by the output gear G<sub>B</sub>. If we assume, that the gears are rigid and that there are no losses in the engagement of the gear teeth, then the principle of virtual work can be used to analyze the static equilibrium of the gear train.

Let the angle θ of the input gear be the generalized coordinate of the gear train, then the speed ratio R of the gear train defines the angular velocity of the output gear in terms of the input gear, that is

<math display="block"> \omega_A = \omega, \quad \omega_B = \omega/R.</math>

The formula above for the principle of virtual work with applied torques yields the generalized force

<math display="block"> Q = T_A \frac{\partial\omega_A}{\partial\omega} - T_B \frac{\partial \omega_B}{\partial\omega} = T_A - T_B/R = 0.</math>

The mechanical advantage of the gear train is the ratio of the output torque T<sub>B</sub> to the input torque T<sub>A</sub>, and the above equation yields

<math display="block"> MA = \frac{T_B}{T_A} = R.</math>

Thus, the speed ratio of a gear train also defines its mechanical advantage. This shows that if the input gear rotates faster than the output gear, then the gear train amplifies the input torque. And, if the input gear rotates slower than the output gear, then the gear train reduces the input torque.

Dynamic equilibrium for rigid bodies

If the principle of virtual work for applied forces is used on individual particles of a rigid body, the principle can be generalized for a rigid body: When a rigid body that is in equilibrium is subject to virtual compatible displacements, the total virtual work of all external forces is zero; and conversely, if the total virtual work of all external forces acting on a rigid body is zero then the body is in equilibrium.

If a system is not in static equilibrium, D'Alembert showed that by introducing the acceleration terms of Newton's laws as inertia forces, this approach is generalized to define dynamic equilibrium. The result is D'Alembert's form of the principle of virtual work, which is used to derive the equations of motion for a mechanical system of rigid bodies.

The expression compatible displacements means that the particles remain in contact and displace together so that the work done by pairs of action/reaction inter-particle forces cancel out. Various forms of this principle have been credited to Johann (Jean) Bernoulli (1667–1748) and Daniel Bernoulli (1700–1782).

<!--

totally the same as in static equilibrium

Generalized active forces

The static equilibrium of a mechanical system of rigid bodies is defined by the condition that the virtual work of the applied forces is zero for any virtual displacement of the system. This is known as the principle of virtual work. This is equivalent to the requirement that the generalized forces for any virtual displacement are zero, that is Q<sub>i</sub>&nbsp;=&nbsp;0.

Let a mechanical system be constructed from n rigid bodies, B<sub>i </sub>, i = 1, ..., n, and let the resultant of the applied forces on each body be the force–torque pairs, F<sub>i</sub> and T<sub>i </sub>, i = 1, ..., n. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity V<sub>i</sub> and angular velocities ω<sub>i </sub>, i = 1, ..., n, for each rigid body, are defined by a single generalized coordinate q. Such a system of rigid bodies is said to have one degree of freedom.

The virtual work of the forces and torques, F<sub>i</sub> and T<sub>i </sub>, applied to this one degree of freedom system is given by

:<math display="block"> \delta W = \sum_{i=1}^n \left(\mathbf{F}_i\cdot \frac{\partial \mathbf{V}_i}{\partial \dot{q + \mathbf{T}_i\cdot\frac{\partial \vec{\omega}_i}{\partial \dot{q\right)\delta q = Q\,\delta q,</math>

where

:<math display="block"> Q = \sum_{i=1}^n \left(\mathbf{F}_i\cdot \frac{\partial \mathbf{V}_i}{\partial \dot{q + \mathbf{T}_i\cdot\frac{\partial \vec{\omega}_i}{\partial \dot{q\right),</math>

is the generalized force acting on this one degree of freedom system.

If the mechanical system is defined by m generalized coordinates, q<sub>j </sub>, j = 1, ..., m, then the system has m degrees of freedom and the virtual work is given by,

:<math display="block"> \delta W = \sum_{j=1}^m Q_j\,\delta q_j,</math>

where

:<math display="block"> Q_j = \sum_{i=1}^n \left(\mathbf{F}_i\cdot \frac{\partial \mathbf{V}_i}{\partial \dot{q}_j} + \mathbf{T}_i\cdot\frac{\partial \vec{\omega}_i}{\partial \dot{q}_j}\right),\quad j=1, \ldots, m.</math>

is the generalized force associated with the generalized coordinate q<sub>j</sub>. The principle of virtual work states that static equilibrium occurs when these generalized forces acting on the system are zero, that is

:<math display="block"> Q_j = 0,\quad j=1, \ldots, m.</math>

These m equations define the static equilibrium of the system of rigid bodies.

-->

Generalized inertia forces

Let a mechanical system be constructed from n rigid bodies, B<sub>i</sub>, i=1,...,n, and let the resultant of the applied forces on each body be the force-torque pairs, F<sub>i</sub> and T<sub>i</sub>, i = 1,...,n. Notice that these applied forces do not include the reaction forces where the bodies are connected. Finally, assume that the velocity V<sub>i</sub> and angular velocities ω<sub>i</sub>, i=1,...,n, for each rigid body, are defined by a single generalized coordinate q. Such a system of rigid bodies is said to have one degree of freedom.

Consider a single rigid body which moves under the action of a resultant force F and torque T, with one degree of freedom defined by the generalized coordinate q. Assume the reference point for the resultant force and torque is the center of mass of the body, then the generalized inertia force Q* associated with the generalized coordinate q is given by

<math display="block"> Q^* = -(M\mathbf{A}) \cdot \frac{\partial \mathbf{V{\partial \dot{q - ([I_R]\alpha+ \omega\times[I_R]\omega) \cdot \frac{\partial \boldsymbol{\omega{\partial \dot{q.</math>

This inertia force can be computed from the kinetic energy of the rigid body,

<math display="block"> T = \frac{1}{2} M \mathbf{V} \cdot \mathbf{V} + \frac{1}{2} \boldsymbol{\omega} \cdot [I_R] \boldsymbol{\omega},</math>

by using the formula

<math display="block"> Q^* = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q -\frac{\partial T}{\partial q}\right).</math>

A system of n rigid bodies with m generalized coordinates has the kinetic energy

<math display="block">T = \sum_{i=1}^n \left(\frac{1}{2} M \mathbf{V}_i \cdot \mathbf{V}_i + \frac{1}{2} \boldsymbol{\omega}_i \cdot [I_R] \boldsymbol{\omega}_i\right),</math>

which can be used to calculate the m generalized inertia forces

<math display="block"> Q^*_j = -\left(\frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j}\right), \quad j=1, \ldots, m.</math>

D'Alembert's form of the principle of virtual work

D'Alembert's form of the principle of virtual work states that a system of rigid bodies is in dynamic equilibrium when the virtual work of the sum of the applied forces and the inertial forces is zero for any virtual displacement of the system. Thus, dynamic equilibrium of a system of n rigid bodies with m generalized coordinates requires that

<math display="block"> \delta W = (Q_1 + Q^*_1)\delta q_1 + \dots + (Q_m + Q^*_m)\delta q_m = 0,</math>

for any set of virtual displacements δq<sub>j</sub>. This condition yields m equations,

<math display="block"> Q_j + Q^*_j = 0, \quad j=1, \ldots, m,</math>

which can also be written as

<math display="block"> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = Q_j, \quad j=1,\ldots,m.</math>

The result is a set of m equations of motion that define the dynamics of the rigid body system, known as Lagrange's equations or the generalized equations of motion.

If the generalized forces Q<sub>j</sub> are derivable from a potential energy V(q<sub>1</sub>,...,q<sub>m</sub>), then these equations of motion take the form

<math display="block"> \frac{d}{dt} \frac{\partial T}{\partial \dot{q}_j} -\frac{\partial T}{\partial q_j} = -\frac{\partial V}{\partial q_j}, \quad j=1,\ldots,m.</math>

In this case, introduce the Lagrangian, , so these equations of motion become

<math display="block"> \frac{d}{dt} \frac{\partial L}{\partial \dot{q}_j} - \frac{\partial L}{\partial q_j} = 0 \quad j=1,\ldots,m.</math>

These are known as the Euler-Lagrange equations for a system with m degrees of freedom, or Lagrange's equations of the second kind.

Virtual work principle for a deformable body

Consider now the free body diagram of a deformable body, which is composed of an infinite number of differential cubes. Let's define two unrelated states for the body:

  • The <math> \boldsymbol{\sigma} </math>-State <!-- No Fig.a in the page. -->: This shows external surface forces T, body forces f, and internal stresses <math> \boldsymbol{\sigma} </math> in equilibrium.
  • The <math> \boldsymbol{\epsilon} </math>-State <!-- No Fig.b in the page. -->: This shows continuous displacements <math> \mathbf {u}^* </math> and consistent strains <math> \boldsymbol{\epsilon}^* </math>.

The superscript * emphasizes that the two states are unrelated. Other than the above stated conditions, there is no need to specify if any of the states are real or virtual.

Imagine now that the forces and stresses in the <math> \boldsymbol{\sigma} </math>-State undergo the displacements and deformations in the <math> \boldsymbol{\epsilon} </math>-State: We can compute the total virtual (imaginary) work done by all forces acting on the faces of all cubes in two different ways:

  • First, by summing the work done by forces such as <math> F_A </math> which act on individual common faces (Fig.c): Since the material experiences compatible displacements, such work cancels out, leaving only the virtual work done by the surface forces T (which are equal to stresses on the cubes' faces, by equilibrium).
  • Second, by computing the net work done by stresses or forces such as <math> F_A </math>, <math> F_B </math> which act on an individual cube, e.g. for the one-dimensional case in Fig.(c): <math display="block"> F_B \left( u^* + \frac{ \partial u^*}{\partial x} dx \right ) - F_A u^* \approx \frac{ \partial u^* }{\partial x} \sigma dV + u^* \frac{ \partial \sigma }{\partial x} dV = \epsilon^* \sigma dV - u^* f dV </math> where the equilibrium relation <math> \frac{ \partial \sigma }{\partial x}+f=0 </math> has been used and the second order term has been neglected. Integrating over the whole body gives: <math display="block">\int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} \, dV </math> – Work done by the body forces f.

Equating the two results leads to the principle of virtual work for a deformable body:

where the total external virtual work is done by T and f. Thus,

The right-hand-side of (,) is often called the internal virtual work. The principle of virtual work then states: External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains. It includes the principle of virtual work for rigid bodies as a special case where the internal virtual work is zero.

Proof of equivalence between the principle of virtual work and the equilibrium equation

We start by looking at the total work done by surface traction on the body going through the specified deformation:

<math display="block"> \int_{S} \mathbf u \cdot \mathbf T dS = \int_{S} \mathbf u \cdot \boldsymbol \sigma \cdot \mathbf n dS </math>

Applying divergence theorem to the right hand side yields:

<math display="block"> \int_S \mathbf{u \cdot \boldsymbol \sigma \cdot n} dS = \int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol \sigma \right) dV </math>

Now switch to indicial notation for the ease of derivation.

<math display="block">\begin{align}

\int_V \nabla \cdot \left( \mathbf{u} \cdot \boldsymbol \sigma \right) dV

&= \int_V \frac{\partial}{\partial x_j} \left( u_i \sigma_{ij} \right) dV \\

&= \int_V \left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij{\partial x_j}\right) dV

\end{align}</math>

To continue our derivation, we substitute in the equilibrium equation <math> \frac{\partial \sigma_{ij{\partial x_j} + f_i = 0 </math>. Then

<math display="block">\int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} + u_i \frac{\partial \sigma_{ij{\partial x_j}\right) dV

= \int_V \left(\frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV</math>

The first term on the right hand side needs to be broken into a symmetric part and a skew part as follows:

<math display="block">\begin{align}

\int_V\left( \frac{\partial u_i}{\partial x_j} \sigma_{ij} - u_i f_i\right) dV

&= \int_V\left( \frac12 \left[ \left( \frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right)

+ \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i \right) dV \\

&= \int_V \left( \left[ \epsilon_{ij}

+ \frac12 \left( \frac{\partial u_i}{\partial x_j} - \frac{\partial u_j}{\partial x_i} \right) \right] \sigma_{ij} - u_i f_i\right) dV \\

&= \int_V\left( \epsilon_{ij} \sigma_{ij} - u_i f_i \right) dV\\

&= \int_V \left( \boldsymbol\epsilon : \boldsymbol\sigma - \mathbf u \cdot \mathbf f \right) dV

\end{align}</math>

where <math> \boldsymbol\epsilon </math> is the strain that is consistent with the specified displacement field. The 2nd to last equality comes from the fact that the stress matrix is symmetric and that the product of a skew matrix and a symmetric matrix is zero.

Now recap. We have shown through the above derivation that

<math display="block"> \int_{S} \mathbf{u \cdot T} dS = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV - \int_V \mathbf u \cdot \mathbf f dV </math>

Move the 2nd term on the right hand side of the equation to the left:

<math display="block"> \int_{S} \mathbf{u \cdot T} dS + \int_V \mathbf u \cdot \mathbf f dV = \int_V \boldsymbol\epsilon : \boldsymbol\sigma dV </math>

The physical interpretation of the above equation is, the External virtual work is equal to internal virtual work when equilibrated forces and stresses undergo unrelated but consistent displacements and strains.

For practical applications:

  • In order to impose equilibrium on real stresses and forces, we use consistent virtual displacements and strains in the virtual work equation.
  • In order to impose consistent displacements and strains, we use equilibriated virtual stresses and forces in the virtual work equation.

These two general scenarios give rise to two often stated variational principles. They are valid irrespective of material behaviour.

Principle of virtual displacements

Depending on the purpose, we may specialize the virtual work equation. For example, to derive the principle of virtual displacements in variational notations for supported bodies, we specify:

  • Virtual displacements and strains as variations of the real displacements and strains using variational notation such as <math> \delta\ \mathbf {u} \equiv \mathbf{u}^* </math> and <math> \delta\ \boldsymbol {\epsilon} \equiv \boldsymbol {\epsilon}^* </math>
  • Virtual displacements be zero on the part of the surface that has prescribed displacements, and thus the work done by the reactions is zero. There remains only external surface forces on the part <math> S_t </math> that do work.

The virtual work equation then becomes the principle of virtual displacements:

This relation is equivalent to the set of equilibrium equations written for a differential element in the deformable body as well as of the stress boundary conditions on the part <math> S_t </math> of the surface. Conversely, () can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on <math> S_t </math>, and proceeding in the manner similar to () and ().

Since virtual displacements are automatically compatible when they are expressed in terms of continuous, single-valued functions, we often mention only the need for consistency between strains and displacements. The virtual work principle is also valid for large real displacements; however, Eq.() would then be written using more complex measures of stresses and strains.

Principle of virtual forces

Here, we specify:

  • Virtual forces and stresses as variations of the real forces and stresses.
  • Virtual forces be zero on the part <math> S_t </math> of the surface that has prescribed forces, and thus only surface (reaction) forces on <math> S_u </math> (where displacements are prescribed) would do work.

The virtual work equation becomes the principle of virtual forces:

This relation is equivalent to the set of strain-compatibility equations as well as of the displacement boundary conditions on the part <math> S_u </math>. It has another name: the principle of complementary virtual work.

Alternative forms

A specialization of the principle of virtual forces is the unit dummy force method, which is very useful for computing displacements in structural systems. According to D'Alembert's principle, inclusion of inertial forces as additional body forces will give the virtual work equation applicable to dynamical systems. More generalized principles can be derived by:

  • allowing variations of all quantities.
  • using Lagrange multipliers to impose boundary conditions and/or to relax the conditions specified in the two states.

These are described in some of the references.

Among the many energy principles in structural mechanics, the virtual work principle deserves a special place due to its generality that leads to powerful applications in structural analysis, solid mechanics, and finite element method in structural mechanics.

See also

  • Flexibility method
  • Unit dummy force method
  • Finite element method in structural mechanics
  • Calculus of variations
  • Lagrangian mechanics
  • Müller-Breslau's principle
  • D'Alembert's principle

References

  • Examples applications of the virtual work principle

Bibliography

  • Bathe, K.J. "Finite Element Procedures", Prentice Hall, 1996.
  • Charlton, T.M. Energy Principles in Theory of Structures, Oxford University Press, 1973.
  • Dym, C. L. and I. H. Shames, Solid Mechanics: A Variational Approach, McGraw-Hill, 1973.
  • Greenwood, Donald T. Classical Dynamics, Dover Publications Inc., 1977,
  • Hu, H. Variational Principles of Theory of Elasticity With Applications, Taylor & Francis, 1984.
  • Langhaar, H. L. Energy Methods in Applied Mechanics, Krieger, 1989.
  • Reddy, J.N. Energy Principles and Variational Methods in Applied Mechanics, John Wiley, 2002.
  • Shames, I. H. and Dym, C. L. Energy and Finite Element Methods in Structural Mechanics, Taylor & Francis, 1995,
  • Tauchert, T.R. Energy Principles in Structural Mechanics, McGraw-Hill, 1974.
  • Washizu, K. Variational Methods in Elasticity and Plasticity, Pergamon Pr, 1982.
  • Wunderlich, W. Mechanics of Structures: Variational and Computational Methods, CRC, 2002.