234 7 The Force Method
abcFig. 7.12 Trusses. Types of statically indeterminacy
The truss (7.12a) contains one redundantsupportmember and does not contain
the redundant members in the web. This structure presents the first degree ofexter-
nallystatically indeterminate truss. In cases (b) and (c), the diagonal members do not
have a point of intersection, i.e., these members are not connected (nor hinged, nor
fixed) with each other. Case (7.12b) presents theinternallystatically indeterminate
truss to six degrees of redundancy. Case (7.12c) presents theinternallystatically
indeterminate truss to six degrees andexternallyto the first degree.
The analysis of statically indeterminate trusses may be effectively performed by
the force method in canonical form. The primary system is obtained by elimination
of redundant constraints. As in the general case (7.4) of canonical equations,Xnare
primary unknowns;ıikare unit displacements of the primary system in the direction
ofith primary unknown due to unit primary unknownXkD 1 ;iPare displace-
ments of the primary system in the direction ofith primary unknownXkD 1 Iip
due to acting load.
For computation of coefficient and free terms of canonical equations, we will use
the second term of the Maxwell–Mohr integral (6.11).
ıikDXnNNiNNkl
EA;iPDXnNNiNP^0 l
EA; (7.7)wherelis length ofnth member of the truss;NNi,NNkare axial forces innth member
due to unit primary unknownsXi D1; Xk D 1 ;andNP^0 is axial force innth
member of the primary system due to acting load.
Summation procedure in (7.7) should be performed on all members of the truss
(subscriptnis omitted).
Solution of (7.4) is the primary unknownsXi.iD1;:::;n/. Internal forces in
the members of the truss may be constructed by the formula
NDNN 1 X 1 CNN 2 X 2 CCNP^0 : (7.8)