Advanced Methods of Structural Analysis

358 10 Influence Lines Method

more preferable; if the rolled and pinned supports of this frame will be substituted
by fixed ones, then advantages of the displacement method are obvious.
4.For construction of influence lines for arches and trusses the force method is
more preferable.

10.4 Kinematical Method for Construction of Influence Lines

The shape of influence lines allows finding the most unfavorable position of the

load. The shape of influence line is often referred as a model of influence line. The

model of influence lines may be constructed by kinematical method using M ̈uller–

Breslau principle, which is considered below.

Let us considerntimes statically indeterminate continuous beam. It is required

to construct an influence line for any reaction (or internal force)Xat any section of

a beam. Primary system of the force method is obtained by eliminating constraint,

which corresponds to required forceXand replacing this constrain by forceX.

Primary system presents.n1/times statically indeterminate structure. Canonical

equation for specified unknownX 1 in case of unit loadPis presented in the form

ı 11 X 1 Cı1PD 0. Influence line for primary unknownXbecomes

IL.X 1 /D

1

ı 11

IL.ı1P/;

whereı 11 is the displacement in the direction of primary unknownX 1 caused by

unit primary unknownX 1 D 1 Iı1Ppresents displacement in the direction of pri-

mary unknown caused by moving unit loadP.

According to reciprocaldisplacements theorem,ı1PDıP1,whereıP1presents

displacement in the direction of moving loadPcaused by unit primary unknown

X 1. Therefore, the influence line for primary unknown may be constructed by

formula

IL.X 1 /D

1

ı 11

IL.ıP1/: (10.24)

The ordinates of influence line for any function X (reaction, bending moment, etc.)

are proportional to ordinates of the elastic curve due to unit force X, which replaces

the eliminated constraint where the force X arises(M ̈uller–Breslau principle). This

principle with elastic loads method was effectively applied previously for analytical

construction of influence lines for statically indeterminate truss. Now we illustrate

the M ̈uller–Breslau principle for two types of problems. They are analytical compu-

tation of ordinates of influence lines and construction of models of influence lines.

Both of these problems are referred as kinematical method. Our consideration of

this method will be limited only the continuous beams.

In order to construct the model of influence line for a certain factorX(reaction,

bending moment, shear force) by a kinematical method, the following steps must be

performed: