Advanced Methods of Structural Analysis

7.3 Analysis of Statically Indeterminate Structures 225

q

Primary

system

X 1

1

2 3 ′^3 ′′

4

a

a

EI

EI

q

Design

diagram

a b

Unit

state

X 1 =1

M 1

a

d 11 a

Loaded

state

M^0 P

qa^2

2

qa^2

8

q

Δ 1 P a/2

cd

15

32 qa

2

qa^2

32

MP

7 qa^2

64

q

Loaded

state

qa^2

2

qa^2

q^8

a/2

+ =

M 1 ⋅X 1

X 1

M^0 P

e

P= 1

M

1·a

f

X 1 X 1 =1

M 1

1

1

qa^2 /8

q

M^0 P

g

Fig. 7.9 (a,b) Design diagram of a frame and primary system. (c,d) Unit and loaded states and
corresponding bending moment diagrams. (e) Construction of bending moment diagram using the
superposition principle. (f) Unit state for calculation of horizontal displacement. (g)Newversion
of primary system and corresponding bending moment diagrams

The unit displacementı 11 is obtained by “multiplying” theMN 1 graph by itself,
i.e.,

ı 11 D

MN 1 MN 1

EI

D

1

EI



1

2

aa

2

3

aC

1

EI

aaaD

4

3

a^3

EI

m

kN



: