Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 339

The bending moment at crownCusing the influence line

MCDl



P0:0127Cq



0:0468

2

C0:0246C

0:008

2



1:2

D6:15kNm:

Relative error is
6:26:15
6:175

100 %D0:8%:

This error is due to approximate calculation of the area of influence lines.
Shear force at crownCis obtained by projecting all forces, located to the left of
this section, on the vertical:

QCDRAPD27:36 30 D2:64kN:

Discussion. Influence lines for reactions of supports have the fundamental mean-
ing, since they allow easy calculations of reactions of statically indeterminate arch,
subjected to any fixed load. After that, calculating internal forces at any section of
the arch is performed as for statically determinate structure.

10.1.3 Statically Indeterminate Trusses.............................

In case of trusses the general procedure forconstruction of influence lines has the
fundamental features. Let us consider the statically indeterminate truss shown in
Fig.10.10a. The axial stiffness of all members equalsEA.It is required to construct
the influence line for reaction at the middle support. The structure under consid-
eration is externally statically indeterminate truss of the first degree of redundancy.
Let the primary unknownX 1 be a reaction of the middle support; the primary system
isshowninFig.10.10b.
Force method leads to the following expression for the primary unknown

X 1 D.ı1P=ı 11 /;

whereı1P is the displacement in the direction of 1-th primary unknown due to
traveling loadPD 1.

Principal concept Expression forX 1 may be modified. According to the reciprocal
displacements theorem,ı1PDıP1and expression forX 1 becomes

X 1 D

ı1P

ı 11

D

ıP1

ı 11

(10.15)

This equation contains the in-depth fundamental concept:instead of calculation
of displacementı1P in the direction of primary unknownX 1 due to the moving
loadPD 1 , we will calculate displacementsıP1at points of application of load
PD 1 due to unit primary unknownX 1 D 1 (Fig.10.10c). This fundamental idea
also will be used in the last section of this chapter for construction ofmodelsof
influence lines.