Principles of Virtual Work: Integral Form of the Basic Equations - Mechanics of Structures: Variational and Computational Methods - page 76

work. A linear elastic spring (or rod) that is stretched has acquired potential energy. Work
is done as the energy is stored or diminished due to stretching or relaxing the stretch.
As a solid deforms, the internal forces perform work in moving through displacements
until reaching a final configuration. If the strained elastic solid were permitted to return
slowly to its unstrained state, the solid would be capable of returning the work performed
by the external forces. This capacity of the internal forces to do work in a strained solid is
due to the strain energy or the internal energy stored in the body. Thus, in an elastic body
(with no initial strains) the strain energy ( U i ) is equal to but opposite in sign to the work

done by the internal forces, i.e., U i W i . This kind of energy is also called the potential

energy of the internal forces or the internal energy as it describes the behavior of the structural
system due to the material properties of its members. Another kind of energy is related to
the position of the body with respect to the gravity effect. A rigid solid situated at some
height h above some reference plane has the potential energy (with respect to a given plane)
of h times the weight of the solid.
Under certain conditions, the quantity d

=−

is the exact differential of some functional, say

. In a two-dimensional space, if is the energy stored,

d =− dW =− ( F y dy + F z dz ) (2.2)

where F y and F z are the forces in the y and z directions. The necessary and sufficient condition
for d

to be an exact differential is (see an elementary calculus textbook)

∂ F y

∂ z = ∂

F z
∂

y

Suppose that the force moves along a closed contour, corresponding to an area S . According
to Green's theorem [Appendix II, Eq. (II.3)]

∂ dy dz

T
∂ y − ∂

R

( Rdy + Tdz ) =

∂ z

S

Let R be F y and T be F z , so that the energy stored would be

∂ dy dz

F z
∂ y − ∂

F y
∂

=− W =− ( F y dy + F z dz ) =− (2.3)

z

S

If the necessary and sufficient condition is satisfied, the energy as the forces move along a
closed contour is zero. Equivalently, the integral
B
A (

F y dy + F z dz )

is independent of the path on which the forces move. Furthermore, the energy stored in
moving around a closed path will be zero .

In a system where the force F is moved from point A to B , the potential energy is the
energy stored or possessed as a result of the position of the force. More precisely, the
potential energy is the capacity of a conservative force system to perform work by virtue
of its position with respect to a reference level. The function

( − = 0 )

A A

is the potential energy.
If no net work is done in moving on a closed path, the system is said to be conservative .
Forces associated with pure elastic deformations are conservative, whereas those associated
with friction and with plastic or damped deformations are termed nonconservative.