Advanced Methods of Structural Analysis

Problems 549

Ans.!i^2 D

EI

i0:7854MR^3

;

 1 D1:4195;  2 D0:03395 ˆD



11

0:6591:517

14.10.Symmetrical frame with absolutely rigid cross bars of total massM and
2Mand massless columns is shown in Fig.P14.10. Calculate the frequencies of
free vibrations, and find the modal matrix and show the shapes of vibrations.

y 2

EI=∞, 2M

h 2 EI

h

EI=∞, M

EI

y 1

Fig. P14.10

Ans.! 1 D3:4641

r

EI

h^3 M

,! 2 D6:9282

r

EI

h^3 M

,ˆD



11

0:5 1

14.11.Symmetrical frame with absolutely rigid cross bars of total massM and
2Mand massless columns is shown in Fig.P14.11. Find the frequencies of free
vibrations.

y 2

EI=∞, 2M

h EI

h

EI=∞, M

EI

y 1

EI

Fig. P14.11

Ans.! 12 D7:7628

EI

h^3 M

,! 22 D41;7366

EI

h^3 M

,ˆD



11

0:67650:7390

14.12.Derive the frequency equation, calculate the frequencies of vibration and
find the mode shape of vibrations for beam (lengthl, flexural stiffnessEI, mass per
unit lengthm) with following classical boundary conditions: (a) Fixed-fixed beam;
(b) Clamped-pinned beam; (c) Guided-clamped beam

!iD

^2 i

l^2

r

EI

m

: