Advanced Methods of Structural Analysis

508 13 Stability of Elastic Systems

l 2

P

k

l 1

EI

Fig. P13.10

Ans. sinnl 1 sinnl 2 Dnlsinnl



l 1 l 2

l^2



P

kl



;nD

r
P
EI
For problems 13.11 through 13.15, it is recommended to apply the Displacement
method. All stability functions 1 ; 2 ; :::are presented in Table A25.

13.11.Design diagram of the continuous beam is presented in Fig. P13.11. Derive
the equation for critical load in term of parameter ̨. Consider a special case for
̨D0:5.

EI P

l 1 =aL l 2 =(1−a)L

L

Fig. P13.11

Ans.

4

̨

' 2. ̨ 0 /C

3

1  ̨

' 1 ..1 ̨/  0 /D0;  1 Dl 1

r

P

EI

D ̨L

r

P

EI

D ̨ 0 ;

 2 D.1 ̨/ L

r

P

EI

D 0 .1 ̨/

13.12.Two-span beam of spansl 1 andl 2 Dˇl 1 is subjected to axial forcesPand
̨P(Fig. P13.12). The flexural rigidity for the left and right spans areEIandkEI.
Derive the equation for critical load in term of parameters ̨; ˇandk. Consider the
special cases: a) ̨D3; ˇD 1 ,kD 4 Iand b) ̨D0; kD1; ˇD 1. Explain
obtained results.

P

EI

l 1 l 2 =bl 1

aP

kEI

Fig. P13.12

Ans.' 1 . 1 /C

k

ˇ

' 1 . 2 / D 0 , 1 D l 1

r

P

EI

; 2 D l 2

r

PC ̨P

kEI

D

 1 ˇ

r

1 C ̨

k