Building Design and Construction Handbook - Merritt - Ventech!

5.74 SECTION FIVE
line for a reaction is to be found; that is, we wish to plot the reaction R as a unit
load moves over the structure, which may be statically indeterminate. For the loading
condition A, we analyze the structure with a unit load on it at a distance x from
some reference point. For loading condition B, we apply a dummy unit vertical
load upward at the place where the reaction is to be determined, deflecting the
structure off the support. At a distance x from the reference point, the displacement
in d xR and over the support the displacement is d RR. Hence W AB ��1(D xR) �
Rd RR. On the other hand, W BA is zero, since loading condition A provides no displacement
for the dummy unit load at the support in condition B. Consequently,
from the reciprocal theorem,
FIGURE 5.54 Reaction-influence line for a
continuous beam.
FIGURE 5.56 Moment-influence line for a
continuous beam.
d xR
R � dRR
Since d RR is a constant, R is proportional to d xR. Hence the influence line for a
reaction can be obtained from the deflection curve resulting from a displacement
of the support (Fig. 5.54). The magnitude of the reaction is obtained by dividing
each ordinate of the deflection curve by the displacement of the support.
Similarly, the influence line for shear can be obtained from the deflection curve
produced by cutting the structure and shifting the cut ends vertically at the point
for which the influence line is desired (Fig. 5.55).
The influence line for bending moment can be obtained from the deflection curve
produced by cutting the structure and rotating the cut ends at the point for which
the influence line is desired (Fig. 5.56).
And finally, it may be noted that the deflection curve for a load of unity at some
point of a structure is also the influence line for deflection at that point (Fig. 5.57).
5.10.6 Superposition Methods
The principle of superposition applies when the displacement (deflection or rotation)
of every point of a structure is directly proportional to the applied loads. The
FIGURE 5.55 Shear-influence line for a continuous
beam.
FIGURE 5.57 Deflection-influence line for a
continuous beam.