**Buckling of Slender Struts**

**Introduction**

A type of failure that is sometimes overlooked for a body subject to compressive loading, is that due to instability,
called buckling.
The longer and more slender the column is, the lower the safe compressive stress that it can stand.
The slenderness of a column is measured by the slenderness ratio, *L/k*, where *L*
is the length of the column and (lower case) *k* = *(I/A) ^{0.5}* the radius of
gyration of the cross sectional area about the centroidal axis. The minimum radius
of gyration is the one to be considered. This corresponds to the minimum value of

A link to a 'buckling load calculator' based on the thoeory below, is given at the bottom of this page.

**Eulers Formula**

Euler analysis applies to slender columns, the formula, for the critical axial concentric load that causes the column to be on the point of collapse for frictionless pinned ends (no bending moment at the ends) is given below.

** **

Click here for a derivation of the above equation.

For a particular column cross section and length, the load capacity
*F _{c}* depends only upon the modulus of elasticity

**Effective Length**

Eulers equation as written can be applied to a column with ends fixed in any
manner if the length is taken as that between sections of zero bending moment.
This length is called the effective length, *L _{e}* and is equal to

End Fixings | Theoretical K value | Practical K value |

pinned frictionless ends: | K=1 | K=1 |

fixed ends: | K=0.5 | K=0.65 |

fixed - pinned and guided: | K=0.7 | K=0.8 |

fixed - free: | K=2 | K=2.1 |

A typical factor of safety, or design factor, for Euler structural columns is
between 2 and 3.5, **but** this is based on the **critical load**, not on the yield
or ultimate strength of the material.

If the long column remains straight and the load concentric, the average stress in
the column at the point of collapse is *s _{c} = F_{c}/A* and it is local
buckling at some point where the stress is below the yield stress of the material that
leads to failure.

**Short and Intermediate Columns**

If *L _{e}/k* is below a certain value for a particular material, the Euler
formula gives a critical load which causes a stress greater than the yield stress of the
material. Collapse in these cases is probably due to a combination of buckling and plastic
action. For very short columns the yield stress (with appropriate design factor) can be used. For columns that are not short, but where the Euler formula gives stress above the yield stress, empirical methods of design are used.
One popular equation in use since the early 1900s is the Johnson formula which can be used
for columns with slenderness ratios below a transition slenderness ratio or column
constant,

The value of *L _{e}/k* that indicates the transition slenderness ratio is
given by: and when

As a very rough guide, for steel, the Euler buckling formula is only applicable for
columns with L_{e}/k exceeding about 100, depending upon the yield stress.

Link to Buckling Load Calculator and Interactive Graph (Netscape 4.5 or IE 4 or later).

Link to Buckling Load Calculator

Further reading: 'Design of Machine Elements', by V M Faires, Collier Macmillan, 1965, chapter 7.

It should be noted that sheets and plates may suffer buckling. Where a fabricated 'I' section beam has an insufficiently thick web, this can also suffer from buckling. Standard sections are sized so this is unlikely to be a problem.

David J Grieve, Amended, 1st March 2004, original dated 27th August 1999.