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

internal forces f 1 and f 2 acting in members (1) and (2), respectively, must satisfy the following
two equilibrium equations:
753
METHOd OF VIRTuAL wORk
Σ F = P′ − f cosθ
− f cosθ
= 0
x
Σ F = f sinθ
− f sinθ
= 0
y
1 1 2 2
1 1 2 2
(d)
Next, we will assume that pin B is given a small real (as opposed to virtual) displacement
D in the horizontal direction. Note that the displacement D is shown greatly exaggerated,
for clarity, in Figures 17.21a and 17.21b. The real displacement D should be assumed
to be small enough so that the displaced geometry of the two-bar assembly is essentially the
same as the equilibrium geometry. Furthermore, the deformation of the two-bar assembly
is compatible, meaning that bars (1) and (2) remain connected together at joint B and
attached to their respective supports at A and C.
Since supports A and C do not move, the virtual forces f 1 and f 2 acting at these joints
do not perform any work. The total virtual work for the two-bar assembly is thus equal to
the algebraic sum of the separate bits of work performed by all the forces acting at joint B.
The horizontal virtual external force P′ moves the body it acts on through a real displacement
D; thus, the work it performs is P′D. Recalling that work is defined as the product of
a force acting on a body and the distance that the body moves in the direction of the force,
we observe that the virtual internal force f 1 in member (1) makes the body it acts on move
through a distance D cos θ 1 in a direction opposite to the direction of force f 1 . Therefore, the
virtual work done by the internal force in member (1) is negative, equal to −f 1 (D cos θ 1 ).
Similarly, the virtual internal force f 2 in member (2) makes the body it acts on move through
a distance D cos θ 2 in a direction opposite that of the force. Therefore, the virtual work done
by the internal force in member (2) is −f 2 (D cos θ 2 ). Consequently, the total work W v done
by the virtual forces acting at joint B is
which can be restated as
W = P′D - f ( Dcos θ ) - f ( Dcos θ )
v 1 1 2 2
W = ( P′ − f cosθ
− f cos θ ) D (e)
v 1 1 2 2
when D is factored out on the right-hand side.
The term in parentheses on the right-hand side of Equation (e) also appears in the
equilibrium equation (d) for the sum of forces in the x direction; therefore, from Equation (d),
we can conclude that the total virtual work for the two-bar assembly is W v = 0. Then, from
this observation, Equation (e) can be rewritten as
P′∆ = f ( D cos θ ) + f ( D cos θ )
(f)
1 1 2 2
The term on the left-hand side of Equation (f) represents the virtual external work W ve
done by the virtual external load P′ acting through the real external displacement D. On the
right-hand side of Equation (f), the terms D cos θ 1 and D cos θ 2 are equal to the real internal
deformations of bars (1) and (2), respectively. Consequently, the right-hand side of Equation
(f) represents the virtual internal work W vi of the virtual internal forces acting through
the real internal displacements. As a result, Equation (f) can be restated as
W
ve
= W
(g)
vi