Page 167 - Marine Structural Design
P. 167
Chapter 8 BucklingKollapse of Columns and Beam-columns 143
I
From Eq. @.lo), the maximum bending moment at x = - may be obtained as follows:
2
(8.1 1)
I
The maximum stress in the cross section where x = - is:
2
', MMAX
OMM: =- (8.12)
A W
Eq. (8.12) may be rewritten as follows:
(8.13)
where,
W = Section modulus
A = Area of the cross section
C = Distance from the neutral axis to the extreme fiber
r = Radius of gyration of the cross section
W
S = Radius of the core: s = -
A
By taking the first term of the Fourier expansion
- + ......
-- -I+-+ (3
pE
P-P, PE
we obtain:
-- -1+- P (8.14)
PE
P-PE PE
Combining Eq. (8.14) and Eq. (8.13), the maximum stress is given by:
(8.15)
8.1.2 Perry-Robertson Formula
A simple method to derive the ultimate strength of a column is to equate oMAX Eq. (8.12) to
in
yield stress cry :
(8.16)
