Advanced Methods of Structural Analysis

10.2 Construction of Influence Lines by the Displacement Method 353

All reactions should be calculated as follows

RAD

X 1

l

D0:096P .#/; RBD 2

X 1

l

C0:6PD0:792P; RCD

X 1

l

C0:4PD0:304P

Using the above reactions, we can calculate the bending moments for all sections of

the beam

M 2 D0:096P

l

5

D0:0192P l;

M 3 D0:096P

2l

5

D0:0384P l; : : :

Thus, having the influence lines we can easily construct the internal force diagrams

for any fixed loads.

10.2.2 Redundant Frames............................................

Construction of influence lines for statically indeterminate frames may be effec-

tively performed using the displacement method. As in case of the fixed load, the

displacement method is more effective than the force method for framed structures

with high degree of the static indeterminacy. Construction of influence lines for in-

ternal forces at any section of the frame starts from construction of influence lines

for primary unknowns. The following example illustrates the construction of influ-

ence line for primary unknown of redundant frame. As have been shown early, this

influence line should be treated as key influence line.

Figure10.15a presents a design diagram of statically indeterminate frame. Bend-

ing stiffnessEIis constant for all portions of the frame. We need to construct the

influence line for angle of rotation of the rigid joint.

The primary system is presented in Fig.10.15b. The primary unknownZ 1 is the

angle of rotation of the rigid joint.

Equation of influence line for primary unknownZ 1 is IL.Z 1 /Dr^111 IL.r1P/.

1.Calculation of unit reactionr 11. Bending moment diagramM 1 caused by unit
rotation of introduced constraint 1 and free body diagram of this constraint is
shown in Fig.10.15c. Equilibrium condition of rigid joint leads tor 11 D10i.
2.Calculation of free termr1P. It is necessary to consider two positions of moving
loadPD 1 : the load is traveling along the left and right spans. The position of
loadPD 1 in the each span is indicated by parametersuand.uCD1/.
Bending moment diagrams are shown in Fig.10.15d. Ordinates of bending mo-
ment diagrams for pinned–fixed and fixed–fixed beams are taken from TablesA.3
andA.4, respectively. Note, in our case, the left span of the frame is a mirror
of those presented in TableA.3, however a distance from the left pinned sup-
port is labeled asul. This should be taken into account, i.e., in formula forMA