Advanced Methods of Structural Analysis

10.1 Construction of Influence Lines by the Force Method 331

Solution.The bending moment at the middle supportBusing corresponding influ-
ence line is

X 1 DMBDl

X

PyD10 . 10 0:048 20 0:096/D 24 kNm:

After that the initial statically indeterminate continuous beam may be considered as
asetoftwostatically determinatesimply supported beams subjected to given load
and moment at the supportB.Inotherwords,the statical indeterminacyhas been
disclosed.
Reaction at supportAis

RAD

10  8 C 20  4  24

10

D13:6kN:

Bending moment at specified points are

M 2 D13:6 2 D27:2kN m;

M 4 D13:6 6  10  4 D41:6kN m;

MBD13:6 10  10  8  20  4 D 24 kNm:

Discussion. Influence line forX 1 should be considered as a reference data for
analysis of beams subjected to different set of fixed loads. If we need to analyze
a structure once due to given set of loads, we can use any classical method or use in-
fluence line as a referred data. If we need toanalyze the same structure many times,
and each time the structure is subjected to different set of loads,then it is much more
convenient to construct influence line once and then use it as reference data for all
other sets of loadings. Thus, combination of two approaches, i.e., moving and fixed
load, is extremely effective for analysis of structures.

10.1.2 Hingeless Nonuniform Arches...............................

Let us apply the general procedure for analysis of symmetrical parabolic nonuni-
form arch with clamped ends shown in Fig.10.6a. The equation of the neutral line is

yD

4f

l^2

.lx/ x:

Assume that the cross-sectional moments of inertia varies by law

IXD

Ic

cos'x

;