78 4 Three-Hinged Arches
The each post which connects the beams of overarched structure with arch itself
has the hinges at the ends. It means that in the poles only axial force arises. Idealized
design diagram of the three-hinged arch without overarched members is shown in
Fig.4.1b.
Degrees of freedom of three-hinged arch according to Chebushev formula (1.1)
isW D 0 , so this structure is geometrically unchangeable. Indeed, two rigid discs
ACandBCare connected with the ground by two hingesAandBand lineABdoes
not pass through the intermediate hingeC.
This structure has four unknown reactions, i.e., two vertical reactionsRA,RB
and two horizontal reactionsHA,HB. For their determination, three equilibrium
equations can be formulated considering the structure in whole. Since bending
moment at the hingeCis zero, this provides additional equation for equilibrium of
the part of the system. It means that the sum of the moments of all external forces,
which are located on the right (or on the left) part of the structure with respect to
hingeCis zero X
leftMCD 0 orXrightMCD0: (4.1)These four equations of equilibrium determine all four reactions at the supports.
Therefore, three-hinged arch is a geometrically unchangeable and statically deter-
minate structure.
4.1.2 Peculiarities of the Arches...................................
The fundamental feature of arched structure is that horizontal reactions appear even
if the structure is subjected to vertical load only. These horizontal reactionsHAD
HBDHare called athrust; such types of structures are often called as thrusted
structures. One type of trusted structures (thrusted frame) was considered in Chap. 3.
It will be shown later that at any cross section of the arch the bending moments,
shear, and axial forces arise. However, the bending moments and shear forces are
considerably smaller than corresponding internal forces in a simply supported beam
covering the same span and subjected to the same load. This is the fundamental
property of the arch thanks to thrust. Thrusts in both supports are oriented toward
each other and reduce the bending moments that would arise in beams of the same
span and load. Therefore, the height ofthe cross section of the arch can be much
less than the height of a beam to resist the same loading. So the three-hinged arch is
more economical than simply supported beam, especially for large-span structures.
Both parts of the arch may be connected by a tie. In this case in order for the
structure to remain statically determinate, one of the supports of the arch should be
rolled (Fig.4.2a, b).
Prestressed tie allows controlling the internal forces in arch itself. Tie is an el-
ement connected by its ends to the arch by mean of hinges, therefore the tie is
subjected only to an axial internal force. So even if horizontal reactions of supports
equal zero, an extended force (thrust) arises in the tie.