NationMaster - Encyclopedia: Virtual work

FACTOID # 131: United we stand? The United Kingdom and United States are both in the top ten for Gross Domestic Product - and for child poverty.

Interesting economy facts »

Home

Encyclopedia

WHAT'S NEW

RECENT ARTICLES

More Recent Articles »

SEARCH ALL

FACTS & STATISTICS Advanced view

Search encyclopedia, statistics and forums:

(* = Graphable)

Encyclopedia > Virtual work

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 physics, a net force acting on a body causes that body to accelerate; that is, to change its velocity. ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... It has been suggested that this article be split into multiple articles accessible from a disambiguation page. ... Conservation of energy states that the total amount of energy in an isolated system remains constant, although it may change forms (for instance, friction turns kinetic energy into thermal energy). ...

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. Calculus of variations is a field of mathematics that deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is smaller in absolute value than any positive real number. ...

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. In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... In physics, force is an influence that may cause a body to accelerate. ...

1 Virtual work principle for a particle
2 Virtual work principle for a rigid body
3 Virtual work principle for a deformable body
- 3.1 Principle of virtual displacements
- 3.2 Principle of virtual forces
4 Alternative forms
5 See also
6 Bibliography

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 $D x$ , $D y$ , $D z$ :

$D_xsum_{i=1}^n F_{x_i} = 0$

$D_ysum_{i=1}^n F_{y_i} = 0$

$D_zsum_{i=1}^n F_{z_i} = 0 qquadmathrm{(b)}$

When the arbitrary constants $D x$ , $D y$ , $D z$ 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:

$D_xsum_{i=1}^n F_{x_i} + D_ysum_{i=1}^n F_{y_i} + D_zsum_{i=1}^n F_{z_i} = 0 qquadmathrm{(c)}$

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.

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. In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. ...

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). Johann Bernoulli (Basel, July 27, 1667 - January 1, 1748) was a Swiss mathematician. ... Daniel Bernoulli Daniel Bernoulli (Groningen, January 29, 1700 â€“ Basel, March 17, 1782) was a Dutch-born mathematician who spent much of his life in Basel, Switzerland. ...

Virtual work principle for a deformable body

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: Image File history File links VirtualWork-01. ... Image File history File links VirtualWork-01. ... To meet Wikipedias quality standards, this article may require cleanup. ...

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: In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ... In engineering mechanics, deformation is a change in shape due to an applied force. ...

First, by summing the work done by forces such as $F A$ 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 $F A$ , $F B$ 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: In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...

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

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.

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 $S t$ that do work.

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

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

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 $S t$ 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 $S t$ , 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. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... A single-valued function is an emphatic term for a mathematical function in the usual sense. ...

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 $S t$ of the surface that has prescribed forces, and thus only surface (reaction) forces on $S u$ (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 $S u$ . 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: The Unit dummy force method provides a convenient means for computing displacements in structural systems. ... DAlembert DAlemberts-Lagrange principle is a statement of the fundamental classical laws of motion. ...

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. Fig. ...

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. Energy principles in structural mechanics express the relationships between stresses, strains or deformations, displacements, material properties, and external effects in the form of energy or work done by internal and external forces. ... Structural analysis comprises the set of physical laws and mathematics required to study and predict the behavior of structures. ... Solid mechanics is the branch of physics and mathematics that concern the behavior of solid matter under external actions (e. ... Finite element method (FEM) is a powerful technique originally developed for numerical solution of complex problems in structural mechanics, and it remains the method of choice for complex systems. ...

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

Categories: Structural engineering | Classical mechanics | Dynamical systems

Results from FactBites:

Virtual work - definition of Virtual work in Encyclopedia (362 words)
	virtual, the work done is termed virtual work.
	The equation (c) is called the principle of virtual work for a particle.
	The D'Alembert's principle is a form of the virtual work principle applied to the dynamics of a particle.

More results at FactBites »

COMMENTARY

Share your thoughts, questions and commentary here

Want to know more?
Search encyclopedia, statistics and forums:

Feeds | Contact
The Wikipedia article included on this page is licensed under the GFDL.
Images may be subject to relevant owners' copyright.
All other elements are (c) copyright NationMaster.com 2003-5. All Rights Reserved.
Usage implies agreement with terms.

Contents