Advanced Methods of Structural Analysis

9.2 Canonical Equations of the Mixed Method 317

Thus, the character of each equation ismixedbecause it is the displacement (or
reaction) caused by primary unknowns of the force methodXandthe displacement
methodZsimultaneously. Moreover, a meaning of each equation in (9.1)ismixed
because they representdisplacementalong of eliminated constraintAand reaction
of introduced constraint 2.

9.2.2 Calculation of Coefficients and Free Terms

The general properties of elastic structures and reciprocal theorems provide the fol-
lowing properties and relationships between coefficients of canonical equation of
the mixed method

ınn>0; rkk>0; ınmDımn;rnmDrmn;rnmDımn: (9.2)

Bending moment diagrams due to the primary unknownsX 1 D 1 andZ 2 D 1
are presented in Fig.9.2c, d. Construction of bending moment diagram caused by
primary unknownX 1 D 1 is obvious. For construction of bending moment dia-
gram caused by primary unknownZ 2 D 1 we need to rotate introduced constraint
2 through angleZ 2 D 1. In this case, members of the right part only will sustain
deformation, while left part will move as absolutely rigid body. Corresponding de-
formable position is shown by dotted line. Unit displacements.ı 11 ;ı 12 /and unit
reactions.r 22 ;r 210 /are shown in the corresponding diagrams.
Ordinates of bending moment diagram for clamped–clamped beam due to the
unit rotation of joint 2 are

M 23 D

4 EI 2  3

l 2  3

D 4

2 EI

8

D1:0EI;M 21 D

4 EI 2  1

l 2  1

D 4

1 EI

8

D0:5EI:

Having the unit bending moment diagrams due toX 1 D 1 andZ 2 D 1 we can
calculate the coefficients and free terms of canonical equations of the mixed method.
Multiplication of bending moment diagramM 1 by itself leads to coefficient of
the force method

ı 11 D

XZ M 1 M 1

EI

dsD

1

2 EI



12  12

2



2

3

 12 C

1

1 EI

 12  4  12 D

864

EI

.m=kN/

Equilibrium condition of induced constraint 2 from bendingmoment diagramM 1
leads to
r 210 D 12 m

The units of coefficients of canonical equations of the mixed method can be defined
as for the force and displacement methods. In general, forrik^0 , the unit ofreaction
which corresponds to the indexi, should be divided by unit of factor, which corre-
sponds to the indexk. In our case we get