Virtual work

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A force F, which may be real (actual) or imaginary (fictitious), acting on a particle is said to do virtual work when the particle is imagined to undergo a real or imaginary displacement component D in the direction of the force. Thus, virtual work is the mathematical product (FD) of unrelated forces and displacements. Since forces and/or displacements need not be real nor related by cause-and-effect, the work done is called virtual work. It is of more general nature than the real work used in the principle of conservation of energy where work is computed by integrating over the loading path.

In practical applications, however, either the forces or the displacements are usually real while the other taken to be virtual. Imaginary forces and displacements are called, respectively, virtual forces and virtual displacements. When the virtual quantities are the independent variables such as external forces or displacements, they are also arbitrary. On the other hand, dependent quantities such as internal stresses or strains, while being virtual (imaginary), cannot be arbitrary if they are constrained by equilibrium or compatibility relations.

Being arbitrary is an essential characteristic that enables us to draw important conclusions from mathematical relations. For example:

  • If we have this matrix relation \mathbf{R}^{*T} \mathbf{r} = \mathbf{R}^{*T} \mathbf{B}^{T} \mathbf{q},
  • and if \mathbf{R}^{*} is an arbitrary vector, then we can conclude that \mathbf{r} = \mathbf{B}^{T} \mathbf{q}. In this way, the arbitrary or virtual quantities will disappear from the final useful results.

Being imaginary or fictitious, virtual quantities allow us to avoid making the unnecessary and confusing assumption that they are sufficiently small so that the body remains undisturbed. It is thus best to consider virtual work taking place only on paper. In variational formulation, virtual quantities are conveniently created as arbitrary infinitesimal variations of the real quantities, but again, only on paper.

In the above discussions, the term displacement may refer to a translation or a rotation, while the term force to a direct force or a moment.

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[edit] Virtual work principle for a particle

The motivation for introducing virtual work can be appreciated by the following simple example from statics of particles. Suppose a particle is in equilibrium under a set of forces F_{x_i}, F_{y_i}, F_{z_i} i = 1,2,...n:

\sum_{i=1}^n F_{x_i} = 0
\sum_{i=1}^n F_{y_i} = 0
\sum_{i=1}^n F_{z_i} = 0 \qquad \mathrm{(a)}

Multiplying the three equations with the respective arbitrary constants Dx, Dy, Dz:

D_x\sum_{i=1}^n F_{x_i} = 0
D_y\sum_{i=1}^n F_{y_i} = 0
D_z\sum_{i=1}^n F_{z_i} = 0 \qquad\mathrm{(b)}

When the arbitrary constants Dx, Dy, Dz are thought of as virtual displacements of the particle, then the left-hand-sides of (b) represent the virtual work. The total virtual work is:

D_x\sum_{i=1}^n F_{x_i} + D_y\sum_{i=1}^n F_{y_i} + D_z\sum_{i=1}^n F_{z_i} = 0 \qquad\mathrm{(c)}

Since the preceding equality is valid for arbitrary virtual displacements, it leads back to the equilibrium equations in (a). The equation (c) is called the principle of virtual work for a particle, or more specifically, the principle of virtual displacements. Its use is equivalent to the use of the equilibrium equations in (a), thus, providing the motivation for using the virtual work principle.

[edit] Virtual work principle for a rigid body

Applied (c) to 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.

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).

[edit] Virtual work principle for a deformable body

Virtual work by stresses.
Virtual work by stresses.

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

  • The \boldsymbol{\sigma}-State (Fig.a): This shows external surface forces T, body forces f, and internal stresses \boldsymbol{\sigma} in equilibrium.
  • The \boldsymbol{\epsilon}-State (Fig.b): This shows continuous displacements \mathbf {u}^* and consistent strains \boldsymbol{\epsilon}^*.

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 \boldsymbol{\sigma}-State undergo the displacements and deformations in the \boldsymbol{\epsilon}-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 FA 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 FA, FB which act on an individual cube, e.g. for the one-dimensional case in Fig.(c):
F_B \big ( u^* + \frac{ \partial u^*}{\partial x} dx \big ) - F_A u^* \approx \frac{ \partial u^* }{\partial x} \sigma dV + u^* \frac{ \partial \sigma }{\partial x} dV = \epsilon^* \sigma dV - u^* f dV
where the equilibrium relation \frac{ \partial \sigma }{\partial x}+f=0 has been used and the second order term has been neglected.
Integrating over the whole body gives:
\int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} \, dV - Work done by the body forces f.

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

\mbox{Total external virtual work} = \int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} dV \qquad \mathrm{(d)}

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

\int_{S} \mathbf{u}^{*T} \mathbf{T} dS + \int_{V} \mathbf{u}^{*T} \mathbf{f} dV = \int_{V} \boldsymbol{\epsilon}^{*T} \boldsymbol{\sigma} dV \qquad \mathrm{(e)}

The right-hand-side of (d,e) is often called the internal virtual work. The principle of virtual work then states: External virtual work is equal to internal virtual work when equilibriated 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.

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.

[edit] 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 \delta\ \mathbf {u} \equiv \mathbf{u}^* and \delta\ \boldsymbol {\epsilon} \equiv \boldsymbol {\epsilon}^*
  • 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 St that do work.

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

\int_{S_t} \delta\ \mathbf{u}^T \mathbf{T} dS + \int_{V} \delta\ \mathbf{u}^T \mathbf{f} dV = \int_{V}\delta\boldsymbol{\epsilon}^T \boldsymbol{\sigma} dV \qquad \mathrm{(f)}

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 St of the surface. Conversely, (f) can be reached, albeit in a non-trivial manner, by starting with the differential equilibrium equations and the stress boundary conditions on St, and proceeding in the manner similar to (a) and (b).

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.(f) would then be written using more complex measures of stresses and strains.

[edit] 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 St of the surface that has prescribed forces, and thus only surface (reaction) forces on Su (where displacements are prescribed) would do work.

The virtual work equation becomes the principle of virtual forces:

\int_{S_u} \mathbf{u}^T \delta\ \mathbf{T} dS + \int_{V} \mathbf{u}^T \delta\ \mathbf{f} dV = \int_{V} \boldsymbol{\epsilon}^T \delta \boldsymbol{\sigma} dV \qquad \mathrm{(g)}

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

[edit] 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.

[edit] See also

[edit] Bibliography

  • Bathe, K.J. "Finite Element Procedures", Prentice Hall, 1996. ISBN 0-13-301458-4
  • Charlton, T.M. Energy Principles in Theory of Structures, Oxford University Press, 1973. ISBN 0-19-714102-1
  • Dym, C. L. and I. H. Shames, Solid Mechanics: A Variational Approach, McGraw-Hill, 1973.
  • Greenwood, Donald T. Classical Dynamics, Dover Publications Inc., 1977, ISBN 0-486-69690-1
  • Hu, H. Variational Principles of Theory of Elasticity With Applications, Taylor & Francis, 1984. ISBN 0-677-31330-6
  • Langhaar, H. L. Energy Methods in Applied Mechanics, Krieger, 1989.
  • Reddy, J.N. Energy Principles and Variational Methods in Applied Mechanics, John Wiley, 2002. ISBN 0-471-17985-X
  • Shames, I. H. and Dym, C. L. Energy and Finite Element Methods in Structural Mechanics, Taylor & Francis, 1995, ISBN 0-89116-942-3
  • Tauchert, T.R. Energy Principles in Structural Mechanics, McGraw-Hill, 1974. ISBN 0-07-062925-0
  • Washizu, K. Variational Methods in Elasticity and Plasticity, Pergamon Pr, 1982. ISBN 0-08-026723-8
  • Wunderlich, W. Mechanics of Structures: Variational and Computational Methods, CRC, 2002. ISBN 0-8493-0700-7
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