Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 333

This arch is statically indeterminate to the third degree. Let us accept the primary
system presented in Fig.10.6b, so the primary unknowns are the bending moment
X 1 , the normal forceX 2 , and shearX 3 at the crownCof the arch.
Canonical equations of the force method are

ı 11 X 1 Cı 12 X 2 Cı 13 X 3 C1PD0;

ı 21 X 1 Cı 22 X 2 Cı 23 X 3 C2PD0; (10.12)

ı 31 X 1 Cı 32 X 2 Cı 33 X 3 C3PD0:

The unit coefficients can be calculated by formula

ıikD

Zl

0

MiMk

EIx

ds:

SinceX 1 andX 2 are symmetrical unknowns, the unit bending moment diagrams
M 1 andM 2 are symmetrical, whileM 3 diagram is antisymmetrical. Is obvious
that all displacements computed by multiplying symmetrical diagram by antisym-
metrical ones equal to zero. Therefore,ı 13 Dı 31 D 0 ,ı 23 Dı 32 D 0 ,andthe
canonical equations (10.12) fall into two independent systems

8

<

:

ı 11 X 1 Cı 12 X 2 C1PD 0

ı 21 X 1 Cı 22 X 2 C2PD^0

and ı 33 X 3 C3PD0: (10.13)

Note that it is possible to find a special type of primary system in order thatall
secondary coefficients will be equal to zero. Corresponding primary system is called
a rational one. We will not discuss this question.
Coefficients and free terms of canonical equations will be calculated taking into
account only bending moments, which arise in the arch. The expression for bending
moments in the left part of the primary system for unit and loaded states (the force
PD 1 is located within the left part of the arch) are presented in Table10.2.

Table 10.2 Bending moments due to unit primary unknowns and

given unit load

M 1 M 2 M 3 MP^0 D 1

Bending moment

expression

1 1 .fy/ 1



l

2

x



MP^0 D 1 .ax/