Building Design and Construction Handbook - Merritt - Ventech!

5.72 SECTION FIVE
where v � shearing unit stress
G � modulus of rigidity
A � cross-sectional area
Truss Deflections. The dummy unit-load method may also be adapted for the
determination of the deformation of trusses. As indicated by Eq. (5.30), the strain
energy in a truss is given by
2 SL
U � � (5.97)
2AE
which represents the sum of the strain energy for all the members of the truss. S
is the stress in each member caused by the loads. Applying Castigliano’s first
theorem and differentiating inside the summation sign yield the deformation:
� AE �P
SL �S
d � (5.98)
The partial derivative in this equation is the rate of change of axial stress with the
load P. It is equal to the axial stress u in each bar of the truss produced by a unit
load applied at the point where the deformation is to be measured and in the
direction of the deformation. Consequently, Eq. (5.98) can also be written
� AE
Sul
d � (5.99)
To find the deflection of a truss, apply a vertical dummy unit load at the panel
point where the deflection is to be measured and substitute in Eq. (5.99) the stresses
in each member of the truss due to this load and the actual loading. Similarly, to
find the rotation of any joint, apply a dummy unit moment at the joint, compute
the stresses in each member of the truss, and substitute in Eq. (5.99). When it is
necessary to determine the relative movement of two panel points, apply dummy
unit loads in opposite directions at those points.
It is worth noting that members that are not stressed by the actual loads or the
dummy loads do not enter into the calculation of a deformation.
As an example of the application of Eq. (5.99), let us compute the deflection of
the truss in Fig. 5.53. The stresses due to the 20-kip load at each panel point are
shown in Fig. 5.53a, and the ratio of length of members in inches to their crosssectional
area in square inches is given in Table 5.5. We apply a vertical dummy
unit load at L 2, where the deflection is required. Stresses u due to this load are
shown in Fig. 5.53b and Table 5.5.
The computations for the deflection are given in Table 5.5. Members not stressed
by the 20-kip loads or the dummy unit load are not included. Taking advantage of
the symmetry of the truss, we tabulate the values for only half the truss and double
the sum.
SuL 2 � 13.742,000
d � � � 0.916 in
AE 30,000,000
Also, to reduce the amount of calculation, we do not include the modulus of
elasticity E, which is equal to 30,000,000, until the very last step, since it is the
same for all members.