2.1 Analytical Method for Construction of Influence Lines 17
Fig. 2.1 Simply supported
beam. Influence lines for
reactions
P= 1xRA l RB11Inf. line RA+ Inf. line RB+AB
0.25l0.750.25Loaded contourReactionRAmay be calculated for any locationxof the loadP. Therefore, last
equation should be considered as afunctionRA.x/. This function is called the in-
fluence line forRAand denoted as IL.RA/.SincePD 1 , then equation of influence
line for reactionRAbecomes
IL.RA/Dlx
l: (2.1)Ifx D 0 (at supportA), then ordinate of influence line IL.RA/D 1 .IfxD l
(at supportB), then ordinate of influence line IL.RA/D 0. These two points are
connected by straight line, since function (2.1) is linear. Influence line forRAis
presented in Fig.2.1.
We can see that units of influence line ordinates for reaction is dimensionless. In
general, units of influence line ordinates for any factorZare defined as quotient of
two units, mainly, unit of the factorZand unit of the loadP(kN). Thus, unit of
influence line for reactionsand shear is dimensionless because kN/kN; for bending
moment: kNm/kNDm; for linear deflection m/kN; for angular deflection rad/kN.
Influence line IL.RA/can be used for analysis of reactionRAonly. Positive
ordinates mean that reaction ofRAis directed upward for any position of the con-
centrated load. If loadPD 1 is located above pointA, then reaction ofRAis equal
to 1; it means that loadPcompletely transmits on the supportA.IfloadPD 1 is
located above pointB,thenreactionofRAis equal to zero. If loadPD 1 has, for
example, coordinatexD0:25l,thenreactionofRAis equal to 0.75.
Analytical presentation of equation of influence line allows to avoid many times
repeated computation of functionZfor different location of the forceP;thisisa
huge advantage of analytical approach for construction of influence lines. Note a
following fundamental property: influencelines for reactions and internal forces of
any statically determined structures are always presented by straight lines.