Advanced Methods of Structural Analysis

8.2 Canonical Equations of Displacement Method 281

displacementZ 1 at support 1. The primary system is obtained from a given structure
by introducing constraint 1 at middle support 1 (Fig.8.6a); this constraint prevents
angular displacement at support 1.

q

P

1

Z 1

q=2kN/m

l 1 =8 m

P=12 kN

l 2 =10 m

ul 2 =6 m ul 2 =4 m

A^1 B

EI

a

r 11

0.375EI 0.3EI

r 11 = 0.675EI

1 1

Z 1 =1

l 1

3 EI

= 0.375EI

l 2

3 EI= 0.3EI

0.12EI

M 1

ul 2 =4 m

Extended fibers Elastic curve

bc

de

R 1 P=−4.16 (kNm)

R 1 P

16

1

20.16

q P

M 10 A M 10 B

Mk^0

k Mp^0

f

M 1

q P

k

Mk

MP

Extended fibers

Fig. 8.6 (a) Design diagram of the beam and primary system. (b) Bending moment diagram
caused by unit angular displacement; (c) Calculation ofr 11 .(d) Bending moment diagram of a
primary system caused by a given load; (e) Calculation of free term R1P.(f) Final bending mo-
ment diagram

The canonical equation of the displacement method is

r 11 Z 1 CR1PD0: (a)

To calculate unit reactionr 11 , we need to rotate the introduced constraint clockwise
by angleZ 1 D 1. The corresponding elastic curve, the location of the extended