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

SolutioN
Calculate the member lengths and record them in a column. The axial deformation of a
truss member due to a change in temperature is given by δ = α DTL. With this expression,
calculate the real internal deformation produced in each member by the temperature
change and record the values in a second column.
For this example, there are no real external loads that perform work on the truss.
Since the vertical deflection of the truss at joint D is to be determined, apply a vertical
virtual load of 1 kip in a downward direction at that joint. Perform a truss analysis and
compute the corresponding virtual internal forces f. Record these results in a column.
Following is the table produced by the preceding instructions for the given truss:
Member
L
(in.)
αDTL
(in.)
f
(kips)
f (αDTL)
(kip ⋅ in.)
AB 192 -0.2264 0.550 -0.125
AC 187 -0.2210 -0.716 0.158
AD 301 -0.3545 2.088 -0.740
BC 129.244 -0.1524 -1.481 0.226
CD 144 -0.1698 -1.833 0.311
∑ f( αD TL) = -0.170
Equation (17.31) can now be applied:
∑
1 ⋅D = f ( α DT L )
j
j j j j
The left-hand side of this equation represents the external work performed by the virtual
external load as the member it acts on moves through the real joint deflection at D. The
right-hand side of the equation represents the internal work performed by the virtual internal
forces f as the members they act on move through the real internal deformations
that occur in them in response to the temperature change.
From the tabulated results,
( 1kip)⋅D =-0.170 kip⋅in.
D
D = - 0.170 in. = 0.170 in. ↑
D
Ans.
The virtual load at D was applied in a downward direction. The negative value obtained
here means that joint D actually moves in the opposite direction—that is, upward.
17.10 Deflections of Beams
by the Virtual-Work Method
The principle of virtual work can be used to determine the deflection of a beam. Consider a
beam subjected to an arbitrary loading, as shown in Figure 17.23a. Assume that the vertical
deflection of the beam at point B is desired. To determine this deflection, a virtual external
unit load will first be applied to the beam at B in the direction of the desired deflection, as
shown in Figure 17.23b. If the beam (as shown in Figure 17.23b) is then subjected to the
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