Free vibrations response of a 3DOF model

Given a structure of RC frame consisting of 3 floors, represented by a 3DOF lumped

model of Fig. 1, let us found the free vibrations response using the stiffness method.

A. Initial Data

a. Geometrical characteristics:

b cm col ? 25 4

3

3 3 2 2 32552.083

12

25* 25

I I cm col col ? ? ? ? ?

h cm col ? 25 A 25* 25 625cm2 col ? ?

L cm col ? 400

b. Material characteristics:

2 270000

cm

daN

Econcrete ? ? ? 0.2 0.0025 ( ) 3 weight density

cm

daN

Wconcrete ?

Fig. 1. 3DOF model of a 3 stories RC frame

c. Evaluation of Mass matrix:

cm

s

m daN

2

3 / 2 0.3185

981

0.0025* 25* 25* 400 ? ?

cm

s

m daN

2

2 0.6371

981

0.0025* 25* 25* 400 ? ?

cm

s

m daN

2

1 0.6371

981

0.0025* 25* 25* 400 ? ?

The final Mass matrix is:

d. Computation of Stiffness Matrix:

The elastic force vector: F u el ? K

The stories shear force vector is: T ? K ? ?

The stories drift: ?1 ? ? ? i i u u

The lateral stiffness of the story:

daN cm

L

EI

L

EI

k

col

col

column col

col 1647.949 /

400

12 12 12 * 270000 *32552.083

3 3 3 ? ? ? ? ?

and the lateral elastic forces are: