2.1 Analytical Method for Construction of Influence Lines 19
Influence lines of reactions are shown in Fig.2.2.IfloadPD 1 is situated at point
D.xDlCd/, then reactionRADd=l. The negative sign means that the reaction
RAis directed downward. The maximum positive reactionRAoccurs if loadPD 1
stands at pointA, the maximum negative reactionRAoccurs if loadPD 1 stands
at pointD.
If loadP D 1 is situated at pointD,thenRBD.lCd/=l. This means, that
reactionRB>PD 1 and is directed upward. The maximum positive reactionRB
occurs if loadPD 1 stands at pointD; the negative reactionRBdoes not be arise.
Equations (2.1)–(2.3) for influence lines of reactions show that overhang does not
change the equations of influence lines; therefore an influence line within the over-
hang is an extension of influence line within the span. This is a common property
of influence lines for any function (reaction, bending moment, and shear). Thus,
in order to construct the influence lines for reaction of a simply supported beam
with overhang, the influence lines for reaction between supports should be extended
underneath the overhang.
2.1.1.3 Cantilevered Beam (Fig.2.3)
At the fixed supportA, the following reactions arise: vertical and horizontal forces
RAandHA, and momentM 0 ; for the given design diagram the horizontal reaction
HAD 0. Positive reactionsRAandM 0 are shown in Fig.2.3.
The Vertical ReactionRA
This reaction may be calculated considering the equilibrium equation in form of the
projections of the all external forces on the vertical axis
RA!X
YD 0 W RAPD 0 !RADP:Fig. 2.3 Cantilevered beam.
Design diagram and influence
lines for reactions
RA lP= 1xM (^0) A B
1 +
Inf. line RA
−
l Inf. line M 0