8.6 Analysis of Symmetrical Structures 303
fact: the degree of statical (or kinematical) indeterminacy for a half-frame is less
than for an entire frame. Equivalent half-frames should be analyzed by classi-
cal methods. It will be shown below that for analysis of the half-frame, different
methods (Force and Displacement methods) should be applied for symmetrical and
antisymmetrical components. Therefore, the method under consideration is called
the combined method.
8.6.2 Concept of Half-Structure....................................
In case of symmetrical structure subjected to symmetrical and antisymmetrical
loads, the elastic curve at the point on the axis of symmetry (AS) has specific
properties. Mainly these properties allow an entire frame to replace by an equiva-
lent half-frame, separately for symmetrical and antisymmetrical loading. In case of
nonsymmetrical change of temperature or settlements of supports, these exposures
could also be replaced by symmetricaland antisymmetrical components.
At any section of a member the followinginternal forcesarise:symmetrical
unknowns, such as bending momentM and axial forceNandantisymmetrical
unknown shear forceQ(Fig.8.15).
Fig. 8.15 Symmetrical and
antisymmetrical internal
forces
MMQQNNAt the point of the axis of symmetry (AS) the followingdisplacementsarise:
the verticalv, horizontalh, and angular'. Considering of these displace-
ments at the AS (pointA) allow us to construct an equivalent half-frame. De-
pending on loading (symmetrical or antisymmetrical), the different displacements
will be at the pointAand, as result, the different support conditions for equivalent
half-frame.
Let us consider the arbitrary symmetrical frames (Table8.6). Elastic curves in
the neighborhood of the pointAon the axis of symmetry are shown by dotted line.
Assume that number of spans is odd. In the case of symmetrical load, the horizontal
and angular displacements at the pointAare zero, while the vertical displacement
occurs. Therefore, an equivalent half-frame must contain a support at pointA,which
would model corresponding displacements, i.e., which would allow the vertical dis-
placement and does not allow the horizontal and angular displacements. Only the
slide support corresponds to these types of displacements.