Advanced Methods of Structural Analysis

504 13 Stability of Elastic Systems

M

b M

d

MP l

a

c

P

P

Mtr

F

D

Fl

F= 1

1 ·l

ab

Fig. 13.27(a) Graph multiplication method for clamped-free beam-column; (b) Calculation of
the deflectionat the free end

According to graph multiplication method, the general formula for displace-
ment is

D

l

6 EI

Œ2ab  1 ./C2cd  2 ./C.adCbc/ 3 ./;DnlDl

r

P

EI

 1 ./D

tan



̨./I 2 ./D ̨./C

tan

12

ˇ^2 ./I 3 ./D

tan



ˇ./

For beam in Fig.13.27b for vertical displacement at the free end we get:

D

l

6 EI

Œ2F l 1 l 1 . /C 2  0  0  2 . /C.0 1 lC 0 Fl/  3 . /

D

Fl^3

3 EI

 1 . /

This result had been obtained previously by the initial parameter method.

Problems.......................................................................

13.1.Absolutely rigid column is loaded by two forcesP 1 DPandP 2 D ̨Pas
shown in Fig. P13.1; ̨is any positive number. Determine a critical force.

k 1

k 2

P 1 =P

P 2 =αP l 1

l 2

Fig. P13.1