Advanced Methods of Structural Analysis

338 10 Influence Lines Method

Internal forces may be defined using the corresponding influence lines by formula
SDPyCq ̋,whereyis the ordinate of influence line under concentrated force,
̋is area of influence line within acting distributed load. The area of curvilinear
influence line may be calculated approximately by replacing curvilinear segments
between two neighboring ordinates by straight lines (Fig.10.9).

Fig. 10.9 Approximation for
calculation of area of
curvilinear influence line

yn yn+1 yn+2 ym–1 ym

hh...... h

If a horizontal distanceh, which separates these ordinates, remains constant, then
the area bounded by two ordinatesynandymwill be given by formula

̋mn Dh

y

n

2

CynC 1 CynC 2 C:::Cym 1 C

ym

2



: (a)

Ordinates of influence lines in Fig.10.6c–f are presented over0:05lD1:2m. Now
we can calculate all reactions at supportAdue to fixed loadPandq.

Thrust

HD

l

f



P0:1320Cq



0:2344

2

C0:2295C

0:2160

2



1:2

D20:20kN

Vertical reaction

RAD



P0:844Cq



0:5

2

C0:425C

0:352

2



1:2

D27:36kN

Moment at support

MADl



P0:0528Cq



0:0312

2

C0:0418C

0:048

2



1:2

D33:32kNm:

Obtained values of reactions at supportA(as well as the influence lines for
primary unknownsXi) allow for calculating all internal forces at any section of
the arch. For this it is necessary to eliminate all constraints at the left end of the arch
and replace them by the reactive forces justfounded, i.e., to consider the given arch
as statically determinate one, which is clamped atBonly and is subjected to given
load and reactions at supportA. For example, bending moment at crownC,bydef-
inition, equals

MCDRA

l

2

HfCMAP



l

2

0:25l



D27:36 12 20:20 6 33:3230 .126/D6:2kNm:

Here we again use the fixed and moving load approaches in parallel way.