Advanced Methods of Structural Analysis

62 3 Multispan Beams and Trusses

only. These horizontal reactionsHADHBDHare calledthrust. Often such types
of structures are called thrusted structures. Some examples of three-hinged trusses
are presented in Fig.3.20. Supports may be located at the same or different levels.
Truss (a) contains two pinned supports and the thrust is taken by these supports. A
modification of a three-hinged truss is presented in Fig.3.20b. This structure con-
tains pinned and rolled supports. Instead of one “lost” constraint at supportB,we
have introduced an additional member that connects both trusses. This member is
called a tie; the thrust is taken by the tie.

HA

C

f

h

Tie

A B

b

d

H H

RA RB

a

A f B

C

d

h

HA HB

RA RB

Fig. 3.20 Three-hinged trusses

The structure (a) is geometrically unchangeable. Indeed, two rigid discs,ACand
BCare connected to the ground by two hinges,AandB, and lineABdoes not pass
through the intermediate hingeC. Similarly, the kinematical analysis can be carried
out for truss (b).
All three-hinged trusses shown in Fig.3.20are statically determinate structures.
Indeed, the structures in Fig.3.20a have four unknown reactions, i.e., two vertical
reactions,RA,RBand two horizontal reactions,HA,HB. For scheme (b) we have
three unknown reactions (RA,RB, and horizontal reactionHA/as well as internal
forceH(thrust) in the tie. For their determination, three equilibrium equations can
be formulated considering the whole system. Since the bending moment at hingeC
is zero, this provides an additional equation of equilibrium. It means that the sum
of the moments of all external forces located on the right or on the left part of the
structure with respect to hinge C is zero, i.e.,

X

left

MCD 0 or

X

right

MCD0:

These four equations of equilibrium determine all four unknowns.
Three-hinged symmetrical truss is shown in Fig.3.21;thespanlD6d. We need
to construct the influence lines for the reactions and for the internal force in indicated
memberU 3 - 5. It is obvious that the influence lines for vertical reactionsRAandRB
are the same as for a simply supported beam.