do work

Virtual work

Virtual work on a system is the work resulting from either virtual forces acting through a real displacement or real forces acting through a virtual displacement. In this discussion, the term displacement may refer to a translation or a rotation, and the term force to a force or a moment. When the virtual quantities are independent variables, they are also arbitrary. Being arbitrary is an essential characteristic that enables one to draw important conclusions from mathematical relations. For example, in the matrix relation

mathbf{R}^{*T} mathbf{r} = mathbf{R}^{*T} mathbf{B}^{T} mathbf{q},

if mathbf{R}^{*} is an arbitrary vector, then one can conclude that mathbf{r} = mathbf{B}^{T} mathbf{q} . In this way, the arbitrary quantities disappear from the final useful results.

Principle of virtual work for applied forces

Consider a system of particles, i, in static equilibrium. The total force on each particle is

mathbf {F}_{i}^{(T)} = 0.

Summing the work exerted by the force on each particle that acts through an arbitrary virtual displacement, delta mathbf r_i, of the system leads to an expression for the virtual work that must be zero since the forces are zero:

delta W = sum_{i} mathbf {F}_{i}^{(T)} cdot delta mathbf r_i = 0.

At this point it should be noted that the original vector equation could be recovered by recognizing that the work expression must hold for arbitrary displacements. Separating the forces into applied forces, mathbf F_i, and constraint forces, mathbf C_i, yields

delta W = sum_{i} mathbf {F}_{i} cdot delta mathbf r_i + sum_{i} mathbf {C}_{i} cdot delta mathbf r_i = 0.

If arbitrary virtual displacements are assumed to be in directions that are orthogonal to the constraint forces, the constraint forces do no work. Such displacements are said to be consistent with the constraints. This leads to the formulation of the principle of virtual work for applied forces, which states that forces applied to a static system do no virtual work:

delta W = sum_{i} mathbf {F}_{i} cdot delta mathbf r_i = 0.

There is also a corresponding principle for accelerating systems called D'Alembert's principle, which forms a theoretical basis for Lagrangian mechanics.

Virtual work principle for a rigid body

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.

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).

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 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 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:

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.

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:

  • 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

References

Bibliography

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  • 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
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  • Wunderlich, W. Mechanics of Structures: Variational and Computational Methods, CRC, 2002. ISBN 0-8493-0700-7

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