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

The right-hand sides of Equations (17.30), (17.31), and (17.32) can be merged to
consider trusses with combinations of external loads along with temperature changes or
fabrication errors in some or all of their members:
757
dEFLECTIONS OF TRuSSES by
THE VIRTuAL-wORk METHOd
⎛ FL
⎞
j j
⋅D = ∑ f ⎜ + α jD TL j j + DL
j
⎝ AE
⎟
j j
⎠
(17.33)
1 j
j
procedure for Analysis
The following procedure is recommended for calculating truss deflections by the virtualwork
method:
1. Real System: If real external loads act on the truss, use the method of joints or the
method of sections to determine the real internal forces in each truss member. Take care
to be consistent in the signs associated with truss member forces and deformations. It is
strongly recommended that tensile axial forces and elongation deformations be considered
as positive quantities. In that case, a positive member force corresponds to an
increase in member length. If this convention is followed, then increases in temperature
and increases in member length due to fabrication errors should also be taken as positive
quantities.
2. Virtual System: Begin by removing all real external loads that act on the truss. Then
apply a single virtual unit load at the joint at which the deflection is desired. This unit
load should act in the direction of the desired deflection. With the unit load in place and
all real loads removed, analyze the truss to determine the member forces f j produced in
response to the virtual external load. The sign convention used for the member forces
must be the same as that adopted in step 1.
3. Virtual-Work Equation: Apply the virtual-work equation, Equation (17.30), to determine
the deflection at the desired joint due to real external loads. It is important to
retain the algebraic sign for each of the f j and F j forces when these terms are substituted
into the equation. If the right-hand side of Equation (17.30) turns out to be positive, then
the displacement D is in the direction assumed for the virtual unit load. A negative result
for the right-hand side of Equation (17.30) means that the displacement D actually acts
opposite to the direction assumed for the virtual unit load.
If the truss deflection is caused by temperature changes, then Equation (17.31) will
be used. If the truss deflection is caused by fabrication errors, then Equation (17.32) is
called for. Equation (17.33) can be used when a combination of real external loads,
temperature changes, and fabrication misfits must be considered.
The application of these virtual-work expressions can be facilitated by an arrangement
of the real and virtual quantities into a tabular format, which will be demonstrated
in subsequent examples.
ExAmpLE 17.10
Compute the vertical deflection at joint B for the truss shown in Figure 17.22a. Assume
that P 1 = 10 kN and P 2 = 40 kN. For each member, the cross-sectional area is A = 525 mm 2
and the elastic modulus E = 70 GPa.
Plan the Solution
Calculate the length of each truss member. Determine the real internal forces F j in all of
the truss members, using an appropriate method, such as the method of joints. Remove