Advanced Methods of Structural Analysis

86 4 Three-Hinged Arches

diagrams satisfy to Schwedler’s differential relationships. In particularly, if at any
point a shear changes sign, then a slope of the bending moment diagram equals zero,
i.e., at this point the bending moment has local extreme (for example, points 2, 7,
etc.).
It can be seen that the bending moments which arise in cross sections of the arch
are much less than in a reference beam.

4.3 Influence Lines for Reactions and Internal Forces....................

Equations (4.6), (4.8), and (4.10) can be used for deriving of equations for influ-
ence lines.

Vertical reactions The equations for influence lines for vertical reactions of the arch
are derived from (4.6). Therefore the equations for influence lines become

IL.RA/DIL



R^0 A



I IL.RB/DIL



R^0 B



: (4.11)

Thus, influence lines for vertical reactions of the arch do not differ from influence
lines for reactions of the reference simply supported beam.

Thrust The equation of influence lines for thrust is derived from (4.8). Since for
given arch a risefis afixednumber, then the equation for influence lines become

IL.H /D

1

f

IL



MC^0



: (4.12)

Thus, influence line for trustHmay be obtained from the influence line for bending
moment at sectionCof the reference beam, if all ordinates of the latter will be
divided by parameterf.

Internal forces The equations for influence lines for internal forces at any section
kmay be derived from (4.10). Since for given sectionk, the parametersyk,sin'k,
and cos'karefixednumbers, then the equations for influence lines become

IL.Mk/DIL



Mk^0



ykIL.H / ;

IL.Qk/Dcos'kIL



Q^0 k



sin'kIL.H / ;

IL.Nk/Dsin'kIL



Qk^0



cos'kIL.H / : (4.13)

In order to construct the influence line for bending moment at sectionk, it is neces-
sary to sum two graphs: one of them is influence line for bending moment at section
kfor reference beam and secondis influence line for thrustHwith all ordinates of
which have been multiplied by a constant factor.yk/.
Equation of influence lines for shear also has two terms. The first term presents
influence line for shear at sectionkin the reference beam all the ordinates of which