Advanced Methods of Structural Analysis

8.1 Fundamental Idea of the Displacement Method 275

In case 8.3a, the primary unknownsZ 1 andZ 2 are angular displacements of in-

troduced rigid joints 1 and 2; the primary unknown,Z 3 , is the linear displacement

of introduced constraint 3. In case 8.3b,the primary unknowns are angular displace-

mentsZ 1 ;:::;Z 4 and linear displacementsZ 5 ,Z6:In case 8.3c, the primary un-

knowns are angular displacementsZ 1 ,Z 2 ,Z 3 and linear displacementsZ 4 andZ 5.

It can be seen that a primary system of the displacement method consists of a

number of single-span redundant beams. In most cases, they are fixed-fixed and

fixed-pinned uniform beams; fixed-guided beams and beams with elastic support as

well as nonuniform beams are also possible. The next sections show how to use

the primary system and tabulated data (TablesA.3–A.8) for the analysis of statically

indeterminate structures in canonical form.

8.1.3 Compatibility Equation. Concept of Unit Reaction

The fundamental idea of the displacement method presentation in canonical form is

explained below by considering the simplest frame in Fig.8.4a.

1

Z P

Entire

structure

1

R 1 P

P

r 11

1

Z 1 = 1

1

P

Primary

system

abc d

Fig. 8.4 (a,b) Design diagram and primary system; (c,d) The primary system is subjected to

given load P and unit rotation of introduced constrain 1

1.The elastic curve of the frame caused by given loadPis shown by the dashed
line; rigid joint 1 rotates in a clockwise direction by some angleZ.
2.The primary system is obtained by introducing additional constraint 1 (Fig.8.4b).
This additional constraint prevents angular displacement. Therefore, the reactive
moment due to external loadParises in the introduced constraint and deforma-
tion occurs in the horizontal element only.The elastic curve and reactive moment
R1Pat constraint 1 in the primary system are shown in Fig.8.4c. The difference
between the given structure and the primary one is obvious: in the primary system
the reactive moment arises at the additional constraint.
3.In order to eliminate the difference between the given and primary structures,
constraint 1 must be rotated by some angleZ 1. In this case, the reactive moment
arises in constraint 1. If the angle of rotation is a unit angle.Z 1 D1/, then the
reactive moment is called aunit reactionand is denoted asr 11. The term “unit”
means that this moment is caused by aunit displacementof the introduced con-
straint. The first numeral inr 11 is the label of the constraint where the reactive