Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

17.8 method of Virtual Work
The method of virtual work is probably the most direct, versatile, and foolproof conservationof-energy
method for calculating deflections. The method may be used to determine deformations
or deflections, at any location in a structure, that are caused by any type or combination
of loads. The only limitation to the theory is that the principle of superposition must
apply.
The principle of virtual work was first stated by Johann Bernoulli (1667–1748) in
1717. The term “virtual,” as used in the context presented here, refers to an imaginary or
hypothetical force or deformation, either finite or infinitesimal. Accordingly, the resulting
work is only imaginary or hypothetical in nature. Before we discuss the virtual-work
method in detail, some further discussion of work will be helpful.
Further Discussion of Work
As discussed in Section 17.2, work is defined as the product of a force that acts on a body
and the distance that the body moves in the direction of the force. Work can be either a
positive or a negative quantity. Positive work occurs when the body moves in the same direction
as the force acts. Negative work occurs when the body moves opposite to the direction
in which the force acts.
Consider the simple axial rod shown in Figure 17.19a. The rod is subjected to a load
P 1 . If applied gradually, the load increases in magnitude from zero to its final intensity P 1 .
The rod deforms in response to the increasing load, with each load increment dP producing
an increment of deformation, dδ. When the full magnitude of P 1 has been applied, the deformation
of the rod is δ 1 . The total work done by the load as it increases in magnitude from
zero to P 1 can be determined from
W
=
∫
δ
0
1
Pdδ
(17.28)
As indicated by Equation (17.28), the work is equal to the area under the load–deformation
diagram shown in Figure 17.19b. If the material behaves in a linear-elastic manner,
the deformation varies linearly with the load as shown in Figure 17.19c. The work for
linear-elastic behavior is given by the triangular area under the load–deformation diagram
and is expressed as
W
=
1 2
P1δ
1
P
P
P 1
δ
W = 0
1P dδ
P 1
1
W= P δ
2 1 1
δ 1
O
δ
1
δ
O
δ
1
δ
P 1
(a)
(b)
FIGURE 17.19 Work done by a single load on an axial rod.
(c)
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