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

both P 1 and P 2 from the truss, apply a unit load downward at joint B, and perform a second
truss analysis to determine the member forces f j created by the unit load. Construct a
table of results from the two truss analyses, and then apply Equation (17.30) to determine
the downward deflection D of joint B.
SolutioN
A tabular format is a convenient way to organize the calculations. Compute the member
lengths and record them in a column. Perform a truss analysis, using the real loads P 1 =
10 kN and P 2 = 40 kN, and record the real internal forces F (i.e., the forces produced in
the truss members by the real loads) in a second column. Note that tension member forces
are assumed to be positive values here. These real internal forces will be used to calculate
the real internal deformations. Accordingly, a positive force corresponds to elongation of
the member.
Remove the real loads P 1 and P 2 from the truss. Since the downward deflection
of the truss at joint B is to be determined, apply a downward virtual load of 1 kN at
joint B, as shown in Figure P17.22b, and perform a second truss analysis. Again, use
the sign convention that tension forces are positive. The member forces obtained
from this second analysis are the virtual internal forces f. Record these results in
a column.
Multiply the virtual internal force f by the real internal force F and the member
length L for each truss member, and record the product in a final column. Sum all of the
values for f that you have recorded, as well as all of the values in the final column, taking
care to note the units that have been used.
Note that the cross-sectional area A and the elastic modulus E are the same for all
members in this particular example. Therefore, they can be included after the two summations.
If A and E differ for any truss members, additional columns will need to be
added to the tabular format to account for the differences.
Following is the table produced by the preceding instructions for the truss shown in
Figure 17.22a:
Member
L
(m)
F
(kN)
f
(kN)
f (FL)
(kN 2 ⋅ m)
AB 5.0 10.0 0.0000 0.000
AD 4.0 -10.0 -0.3750 15.000
BC 3.0 22.5 0.4688 31.644
BD 6.403 16.008 0.6003 61.530
BE 4.0 -10.0 0.6250 -25.000
CE 5.0 -37.5 -0.7813 146.494
DE 5.0 -22.5 -0.4688 52.740
∑ f( FL)
= 282.408
Equation (17.30) can now be applied:
⎛ FL ⎞
j j 1
1⋅D = ∑f
j ⎜ ∑ fj
( FjL
j)
AE
⎟ =
⎝ j j ⎠ AE
j
Recall that 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
j
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