
MECHANICS  THEORY



Column Buckling

Basic Pinned Column
before and after Buckling 

In addition to understanding stress and material failure due to tension and
compression, buckling needs to be investigated.
While buckling can occur in plates and shells, column buckling is most common
and easiest understood. If an object buckles, it does not necessarily mean it
has failed, but a buckled structure will experience significant lateral
deflection. Furthermore, a buckled column structure will not sustain any significant
additional load.
Column buckling is not an exact science, and many issues can cause premature
buckling, such as misaligned loads, material imperfections, and varying structural
dimensions. However, for even perfect systems, buckling will be significantly
lower than the crushing or compression limit of the material. 



LoadDeflection Diagram for
Column Buckling 

When any long slender object is loaded in the axial direction, it will buckle
when the load reaches a particular load. This load, called the critical load,
P_{cr}, is when the object will deflect significantly in the
lateral direction (perpendicular to the load). After buckling starts, the structure
will not be able to sustain any additional loads.
For an ideal column, buckling will not occur until P_{cr} has been reached.
Thus the column will have no lateral deflection up to P_{cr}, and then
all at one time, the member will deflect. This is plotted in the diagram at the
left. The beam's internal resistance to bending keeps the beam from buckling.
However, at some point, the potential moment generated by the axial load will
be greater than the internal resisting moment, and the column will buckle.
For columns, it is generally assumed that if buckling occurs, the structure
has failed. Thus, a column structure should be designed not to buckle. 





Euler's Formula for Simply Supported Column (pinnedpinned)

Simply Supported (pinnedpinned)
Column in Buckled Mode
FreeBody Diagram of
Lower Column Section 

The theoretical buckling load, P_{cr}, for a basic column can be determined
and the resulting equation(s) are called Euler's formula. The simplest column
to develop the buckling equations is when both ends are simply supported by
a pin joint (also called a pinnedpinned column). This means that it can not
deflect at the joint, but it can rotate as shown in the diagram at the left.
If the column is loaded and starts to deflect laterally, then the moment in
the column must equal the moment cased by the load P at each end. This condition
can be better understood by taking a section cut at any point along the column
and constructing a freebody diagram. Summing the moments at any point along
the lower column section gives,
ΣM = 0
M + Pv = 0
The moment M, is drawn in the positive direction, even though it may be actually
act in the opposite direction. The distance v is the lateral deflection at the
cut location, x, from the bottom pin joint. 



Column Rotated  Similar to Beam


If the column is rotated 90^{o}, it becomes a simply supported beam.
As such, the beam bending equations are still valid and can be used. Recall from beam deflections, the moment is related to the deflection as
Combining the two previous equations gives
This is a standard differential equation which has a general solution of
where C_{1} and C_{2} are unknown constants that depend on the
boundary conditions of the differential equation. In this particular case, both
ends of the beam or column are pinned so that v = 0 when x = 0 and x = L. The
first condition, v_{x=0} = 0 gives 





0 = C_{1} (0) + C_{2} (1) => C_{2}
= 0
The second condition, v_{x=L} = 0 gives
Since C_{1} cannot be zero (would be trivial solution, v = 0), the sin
function must be equal to zero which requires,




First Three Column Buckling Modes 

The lowest load is when n equals 1, and is referred to as the critical load
P_{cr}. Euler's formula for a pinnedpinned column is
Other buckling modes (i.e. n > 1) do exist, but are less common since it takes
a larger load to produce this configuration. The first three buckling modes are
shown in the diagram at the left. 



Simply Supported Column
Possible Buckling Directions 

Buckling Direction


Simply supported columns (pinned at each end) will buckle around the axis with
the lowest moment of inertia. For example, for a rectangular cross sectional
column, as shown at the left, the column will buckle around the zaxis since
I_{z} will be less than I_{y}. If it is an unsymmetric cross
section, then it may not be obvious and may even buckle at a particular angle
between the y and zaxis. In
this eBook, only symmetric cross section members are considered.
When calculating buckling, check both directions to find the lowest moment
of inertia. Then use that for the critical buckling load calculations. 



