Chapter 9
Mixed Method
This chapter is devoted to the analysis of special types of structures. They are the
frames with different flexibility ranges of their separate parts. These structures have
specific properties which allow simplify their analysis.
9.1 Fundamental Idea of the Mixed Method..............................
In engineering practice the statically indeterminate frames with specified features
may be found: one part of a structure contains a small number of reactions and
a large number of the rigid joints, while another part contains a large number of
reactions and a small number of the rigid joints. Two examples of this type of struc-
tures are presented in Fig.9.1a, b. Part 1 of a structure (a) contains only one support
constraint (one vertical reaction at pointA) and three rigid joints, while the part 2
contains nine unknown reactions and two rigid joints. We can say that parts 1 and 2
are “soft” and “rigid”, respectively. Theframe (b) may be considered also as a struc-
ture with different ranges of flexibility of their separate parts 1 and 2. The part 1 is
soft while the part 2 is rigid one.
For analysis of these types of structures the mixed method should be applied.
In this method, some unknowns represent unknowns of the force method and some
represent unknowns of the displacement method. The mixed method was introduced
and developed by Prof. A. Gvozdev in 1927.
9.1.1 Mixed Indeterminacy and Primary Unknowns
It is convenient to apply the force method to the “soft” part of the structure and the
displacement method to the “rigid” part of the structure. The primary system of the
mixed method is obtained from a given structure byeliminatingthe redundant con-
straints in the “soft” part of a structure andintroducingadditional constraints at rigid
joints in the “rigid” part of a structure (Fig.9.1c, d). Thus, the primary unknowns
of the mixed method areforces and displacements simultaneously. For scheme (a)
the primary unknowns are theforceX 1 anddisplacementsZ 2 andZ 3. For scheme
I.A. Karnovsky and O. Lebed,Advanced Methods of Structural Analysis,
DOI 10.1007/978-1-4419-1047-99,cSpringer Science+Business Media, LLC 2010
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