Page 165 - Marine Structural Design

Part I1

Ultimate Strength

Chapter 8 Buckling/Collapse of Columns and Beam-Columns

8.1 Buckling Behavior and Ultimate Strength of Columns
This Chapter does not intend to repeat the equations and concept that may be found from
exiting books on buckling and ultimate strength, e.g. (Timoshenko, 1961 and Galambos, 2000).
Instead, some unique formulation and practical engineering applications will be addressed.
8.1.1 Buckling Behavior
For a column subjected to an axial force, the deflection produced by the axial force will be
substantially amplified by the initial imperfections.

P I I P

Y

Figure 8.1 Coordinate System and Displacements of a Beam-Column
with Sinusoidal Imperfections
Let's consider the case in which the initial shape of the axis of the column is given by the
following equation, see Figure 8.1 :
.m
w, = w,,, wn-
1
Initially, the axis of the beam-column has the form of a sine curve with a maximum value of
worn in the middle. If this column is under the action of a longitudinal compressive force P,
an additional deflection w, will be produced and the final form of the deflection curve is:
w=w,+w, (8.2)
The bending moment at any point along the column axis is:
M = P(wo + w,) (8.3)