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

17.9 Deflections of Trusses
by the Virtual-Work Method
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dEFLECTIONS OF TRuSSES by
THE VIRTuAL-wORk METHOd
The method of virtual work is readily applied to structures such as trusses whose members
are axially loaded. To develop the method, consider a truss that is subjected to two external
loads P 1 and P 2 (Figure 17.22a). This truss consists of j = 7 axial members. The vertical
deflection of the truss at joint B is to be determined.
Since the truss is statically determinate, the real internal force F j created in each truss
member by the application of real external loads P 1 and P 2 can be calculated by means of
the method of joints. If F j represents the real internal force in an arbitrary truss member j
(e.g., member CE in Figure 17.22a), then the real internal deformation of the member is
given by
δ =
j
FL
j
AE
j
j
j
in which L, A, and E denote the length, cross-sectional area, and elastic modulus, respectively,
of member j. We will assume that each member has a constant cross-sectional area
and that the load in each member is constant throughout the member’s length.
Next, a virtual load system that is separate and independent from the real load system
is carefully chosen so that the desired joint deflection can be determined. For this truss, the
vertical deflection of joint B is desired. To obtain that deflection, first the real external loads
P 1 and P 2 are removed from the truss and then a virtual external load having a magnitude
of 1 is applied in a downward direction at joint B, as shown in Figure 17.22b. In response
to this unit load, axial forces necessary to maintain equilibrium will be developed in each
of the truss members.
To determine these forces, termed the virtual internal forces f j , imagine that the truss
is initially loaded only by the virtual external load (Figure 17.22b). Then, with the virtual
load still in place, the real external loads P 1 and P 2 shown in Figure 17.22a are applied at
joints D and E, respectively. Equation (17.29) can now be applied to express the virtual
work done by the entire truss. The product of the virtual external load and the real external
deflection D gives the virtual external work W ve :
Wve = 1⋅D
P 2
P1
D
E
Member j
δ =
j
FjLj
A jE j
D
E
Member j
f j
4.0 m
A
B
Δ
C
A
B
C
5.0 m 3.0 m
(a) Real system
FIGUrE 17.22 Statically determinate truss.
1
(b) Virtual system