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

764
ENERgy METHOdS
Obtain the total virtual internal work done on the beam by integrating Equation (d)
over the length of the beam:
W
vi
=
∫
0
L
⎛ ⎞
m⎜
M ⎟ dx
⎝ EI ⎠
(17.34)
This expression represents the amount of virtual strain energy that is stored in the beam.
Finally, the virtual external work [Equation (a)] can be equated to the virtual internal
work [Equation (17.34)], giving the virtual-work equation for beam deflections:
1⋅D =
∫
0
L
⎛ ⎞
m⎜
M ⎟ dx
⎝ EI ⎠
(17.35)
The principle of virtual work can also be used to determine the angular rotation of a
beam. Note that the slope of a beam can be expressed in terms of its angular rotation θ
(measured in radians) as
dv
dx
= tanθ
If the beam deflections are assumed to be small, as is typically the case, then tan θ ≅ θ
and the slope of the beam is equal to
dv
dx
The terms angular rotation and slope are thus effectively synonymous, provided that
the beam deflections are small.
Now consider again a beam subjected to an arbitrary loading as shown in Figure 17.23a.
Assume that the angular rotation θ of the beam at point C is desired. To determine θ, a virtual
external unit moment will first be applied to the beam at C in the direction of the anticipated
slope, as shown in Figure 17.23d. If this beam (as illustrated in Figure 17.23d) is then
subjected to the deformations created by the real external loads (in Figure 17.23a), the
virtual external work W ve performed by the virtual external moment as the beam rotates
counterclockwise through the real beam angular rotation θ is
≅ θ
Wve = 1⋅θ
(e)
The expression for the virtual internal work developed in Equation (17.34) remains
the same as before, with the exception that m now represents the virtual internal moment
created by the load of Figure 17.23d. Thus, the virtual-work equation for beam slopes is
1⋅ θ =
∫
0
L
⎛ ⎞
m⎜
M ⎟ dx
⎝ EI ⎠
(17.36)
In deriving Equation (17.34) for the virtual internal work performed in the beam, the
internal work performed by virtual shear forces acting through real shear deformations has
been neglected. Consequently, the virtual-work expressions in Equations (17.35) and
(17.36) do not account for shear deformations in beams. However, shear deformations are
very small for most common beams (with the exception of very deep beams), and they can
be neglected in ordinary analyses.
In evaluating the integrals in Equations (17.35) and (17.36), a single integration over
the entire length of the beam may not be possible. Concentrated forces or moments, or