Page 175 - Marine Structural Design

Chapter 8 Buckling/CoNapse of Columns and Beam-columns 151

e e
N=bea, =bh-a, =Np- (8.65)
h h
Combination of Eqs. (8.64) and (8.65) gives

(8.66)

The above equation is the interaction formula for a rectangular cross-section under combined
axial load and bending.
Tubular Members
For tubular members, a fully plastic yielding condition for the cross-section may be obtained
as:
- aP
M
cos(-.-)
=
(8.67)
MP
where,
Pp = crYA
M, = 2lrRta,

where R is radius of the cross-section.

8.4 Examples

8.4.1 Example 8.1: Elastic Buckling of Columns with Alternative Boundary Conditions
Problem:
Derive elastic buckling strength equation based on the basic differential equation for initially
straight columns:
d4w d2w
-+k2-- -0 (8.68)
dr4 dx2
Solution:
The general solution of Eq. (8.68) is:
w = Asin kx i Bcoskx -I- Cx + D (8.69)
(1) Columns with Hinged Ends
The deflection and bending moments are zero at both ends:
d2w
w=-=O at x=O and x=l (8.70)
dx2
Applying the boundary conditions to the general solution, we may get:

B=C=D=O sinkl=O (8.71)
Hence,