Advanced Methods of Structural Analysis

536 14 Dynamics of Elastic Systems

abm

1 =m,

EI=•

EI

EI

EI

h

h

h

1

2

3

m 2 =2m,

EI=•

m 3 =2m,

EI=•

1

2

3

M 1

r 31

r 21

r 11

Z 1 =1

6 ih

6 ih

6 ih

1

2

3

r 31

r 21

12 ih^2 r^11

6 ih^12 ih^2

c

Fig. 14.13 (a) Design diagram of the frame; (b) Primary system; (c) Bending moment diagram
caused by unit displacement of the constraint 1 and calculation of unit reactionsr 11 ;r 21 ,andr 31

Similarly, considering the secondand third unit displacements, we get

r 12 D (^24) hi 2 ;r 22 D (^48) hi 2 ;r 32 D (^24) hi 2 ;
r 13 D0; r 23 D (^24) hi 2 ;r 33 D (^48) hi 2
Letr 0 D (^24) hi 2. Equations (14.9) becomes

r 0 m!^2

A 1 r 0 A 2 C0:A 3 D0;
r 0 A 1 C 2

r 0 m!^2

A 2 r 0 A 3 D0;
0:A 1 r 0 A 2 C 2

r 0 m!^2

A 3 D0:
(a)
The frequency equation is
DD
2
4
r 0 m!^2 r 0 0
r 0 2r 0 2m!^2 r 0
0 r 0 2r 0 2m!^2
3
(^5) D 0