3.5 Special Types of Trusses 69
supports, respectively. For this truss, we haveW D 2 14 24 4 D 0. Thus, the
truss in Fig.3.24is statically determinate.
In this truss two rigid discs are connected using members 1, 2, and the rolled
supportC. If the constraintCwas absent, then this structure would be geometrically
changeable. However, the connections of both discs to the ground using constraint
Clead to a geometrically unchangeable structure.
A peculiarity of this multispan truss is that even though this structure is stati-
cally determinate, its reactions cannot bedetermined using the three equilibrium
equations for the truss as a whole. Therefore, the Wichert truss requires a new
approach for its analysis: the replacement bar method, also called the Henneberg
method (1886). The main idea behind this method is that the entire structure is
transformed into a new structure. For this, the intermediate support is eliminated
and a new element is introduced in such a way that the new structure becomes ge-
ometrically unchangeable and can be easily analyzed. The equivalent condition of
both systems, new and original, allows us to determine the unknown reaction in the
eliminated constraint.
As an example, let us consider the symmetrical Wichert truss supported at points
A,B,andCand carrying a loadP, as shown in Fig.3.25a. Assume that angle
̨D 60 ı.
BPXCad
45 °CAL RKaXCPC1(^1) K
N 1
A B
b
Substituted bar
L R
Fig. 3.25 (a) Design diagram of a Wichert truss; (b) Substituted system
To analyze this system, let us replace supportCby an additional vertical member
and apply external forceXC, which is equal to the unknown reaction of supportC
(Fig.3.25b). The additional elementCKis called a substituted bar. According to the
superposition principle, the internal force in the substituted bar isFCCFP, i.e., it is
the sum of the internal forces due to unknown reactionXCof supportCand given
external loadP. Both systems (a) and (b) are equivalent if the internal force in the
substituted bar is zero, i.e.,FCCFPD 0. In expanded form, this equation may be
written as:
FXCCFPD0; (3.3)
whereFPis the internal force in the substituted bar due to external forceP;Fis the
internal force in the substituted bar due to unit forceXCD 1 ;andFXCrepresents
the internal force in the substituted bar due to unknown reactionXC.
Two conditions should be considered. They areP-loading andXC-loading. De-
sign diagram for both conditions present a truss simply supported at pointsAandB.