Page 169 - Marine Structural Design

Chapter 8 Buckling/Collapse of Columns and Beam-columns 145

The Johnson-Ostenfeld approach was recommended in the first edition of the book "Guide to
Stability Design Criteria for Metal Structures" in 1960 and has been adopted in many North
American structural design codes in which a moderate amount of imperfection has been
implicitly accounted for. The Johnson-Ostenfeld formula was actually an empirical equation
derived from column tests in the 1950s. It has since then been applied to many kinds of
structural components and loads, see Part 2 Chapters 10 and 11 of this book.

8.2 Buckling Behavior and Ultimate Strength of Beam-Columns
8.2.1 Beam-Column with Eccentric Load

Figure 8.3 Beam-Column Applied Eccentrical Load
Consider a beam-column with an eccentricitye, at each end, see Figure 8.3. The equilibrium
equation may be written as:

d2W
EI- + P(w+ e, ) = 0 (8.21)
ak2
The general solution of Eq. (8.21) is:
w= Asinkx+bcoskx-e, (8.22)

Using boundary conditions
I
w =o ut x=f-
2
d2w I
EI- =-Pel ut x=+-
dr2 2
the integral constant may be obtained and the solution of Eq. (8.21) is:
c 1
w=e, see-coskx-1 (8.23)

The maximum deflection at the middle of the beam-column is given by:
kl
wMM = e, sec- (8.24)
2
The maximum moment and stress at the middle of the beam-column are expressed as follows: