Advanced Methods of Structural Analysis

8.2 Canonical Equations of Displacement Method 279

momentr 12 and unit reactive forcer 22 arise in constraints 1 and 2, respectively.

The specified ordinates for the vertical member at the bottom and at point 1 are

equal to 6 EI 1 =h^2.

Unit reactionsr 11 ,r 12 represent the reactivemomentsin constraint 1 for both

states and unit reactionsr 21 ,r 22 represent the reactiveforcesin constraint 2 for both

states. Unit reactionsrii, located on the main diagonal of the canonical equations,

are called the main reactions (r 11 ,r 22 ); other unit reactions are called secondary

ones (r 21 ,r 12 ).

To calculate all the unit reactions, we need to consider the free-body diagrams for

joint 1 and for crossbar 1-2 for each state.The free-body diagram for joint 1 in state 1

is shown in Fig.8.5e. The direction of moments 4 EI 1 =hand 3 EI 2 =lcorresponds to

the location of the extended fibers in the vicinity of joint 1 (Fig.8.5e); the extended

fibers are shown by dashed lines. Positive unit reactive momentr 11 is shown by the

direction of unit displacementZ 1.

The equilibrium condition for joint 1 is†MD 0 , therefore we get

r 11 D

4 EI 1

h

C

3 EI 2

l

:

To calculate unit reactionr 21 , we need to consider the free-body diagram for the

crossbar. The shear force infinitely close to joint 1 is found by considering the equi-

librium of the vertical element through the following steps. Moments at the end

points of the vertical element shown in Fig.8.5f are taken from the bending moment

diagram in Fig.8.5c. The moments 4 EI 1 =hat the top and 2 EI 1 =hat the support

are equilibrated by two equal forces, 6 EI 1 =h^2. The upper force is transmitted to

the crossbar. After that, unit reactionr 21 is found by considering the equilibrium

equation of the crossbar, (†FxD 0 ), so

r 21 D

1

h



4 EI 1

h

C

2 EI 1

h



D

6 EI 1

h^2

:

It is obvious that this reaction can also be taken directly from TableA.4,row1.

Similarly, equilibrium equations†MD 0 for the free-body diagrams for joint 1

in state 2 (Fig.8.5g) and equilibrium†FxD 0 for the crossbar in state 2 (Fig.8.5h)

lead to the following unit reactions:

r 12 D

6 EI 1

h^2

;r 22 D

12 EI 1

h^2

:

8.2.3 Properties of Unit Reactions.................................

1.Main reactions are strictly positive (rii>0). Secondary reactionsrikmay be
positive, negative, or zero.