Advanced Methods of Structural Analysis

346 10 Influence Lines Method

4.Solve the canonical equations; since the free terms are functions, the primary
unknowns will be also present thefunctionsof location of unit load.
5.Construct the influence lines for primary unknowns. This step presents the central
and most important part of the procedure.
6.Construct the influence lines for reactions and internal forces (bending moment
and shear) at specified section.
The detailed procedure of construction of influence line for primary unknown and
internal forces for continuous beam and frame will be illustrated below.

10.2.1 Continuous Beams............................................

Let us demonstrate the above procedure for uniform two-span continuous beamABC

with equal spansl. The purpose of analysis is construction of influence lines for

bending moment and shear at a specified sectionk(Fig.10.11a).

Primary systemThe structure is kinematically indeterminate of the first degree. The

primary system of the displacement method is shown in Fig.10.11b. The primary

unknownZ 1 is the angle of rotation of additional constraint at the middle support.

10.2.1.1 Influence Line for Primary UnknownZ 1

For construction of this influence line we will have to apply the formula (10.17)

1.For calculation ofr 11 it is necessary to plot the bending moment diagram due to
primary unknownZ 1 D 1 (Fig.10.11c). It is obvious thatr 11 D 6 EI=l.
2.For calculation ofr1Pit necessary to consider the loadPD 1 located within the
left and right span separately. Bending moment diagrams due to traveling load
PD 1 are shown in Fig.10.11d.
Expressions for reactive momentr1P for different beams are presented in
Ta b l eA.3. Note that lengthlis measured from the pinned (roller) support. Or-
dinates of influence line forr1Pin terms of positionof the loadPD 1 in the
left and right spans are presented in Table10.3.

Table 10.3 Calculation ofr1P

Moving loadPD 1 at the left span Moving loadPD 1 at the right span

Point  r1PDl 2 



1 ^2



Point  r1PDl 2 



1 ^2



1 0 0 6 1.0 0

2 0.2 0:096l 7 0.8 0:144l

3 0.4 0:168l 8 0.6 0:192l

4 0.6 0:192l 9 0.4 0:168l

5 0.8 0:144l 10 0.2 0:096l

6 1.0 0 11 0 0