Advanced Methods of Structural Analysis

314 9 Mixed Method

1 2

A

a

c

A

X 1

Z 2 Z 3

b

d X 1 X 1

Z X^2

3 Z 4

1

2

Fig. 9.1 (a, b)Design diagrams of frames with different ranges of flexibility of their separate
parts;(c, d)primary systems of the mixed method

(b) the primary unknowns of the force method are labeled asX 1 ;X 2 ;X 3 , while
the primary unknowns of thedisplacement method areZ 3 ;Z 4. A numbering of the
unknowns is through sequence.
For frames with different ranges of flexibility of their separate parts the number
of unknowns of the mixed method is less than a number of unknowns of the force
or displacement methods. This is an advantage of the mixed method. The mixed
method may also be presented in canonical form.
LetusconsideraframeshowninFig.9.2a; the flexural rigidity of vertical and
horizontal elements areEIand 2EI, respectively. For given structure the number
of unknowns by the force method equals 4, and the number of unknowns by the
displacement method also equals 4 (angles of rotation of rigid joints 2, 4, 5 and
linear displacement of the cross bar 4-5).
Both parts of the frame have different ranges of flexibility, mainly, the right part
is “rigid” and left part is “soft.” Indeed, the right part 1-2-3 has six constraints of
supports at the points 1 and 3, while the left part 2-4-5-Aof the frame has only
one constraint supportA. Therefore, for the left and right parts of the frame it is
convenient to apply the force and displacement methods, respectively.

9.1.2 Primary System...............................................

The primary system of the mixed method is obtained from the given structure by
eliminatingthe left vertical constraintAandintroducingthe additional constraint at
rigid joint 2, simultaneously (Fig.9.2b). Primary unknowns areX 1 andZ 2 ,where
X 1 is unknown of the force method andZ 2 is unknown of the displacement method.