Evaluating the slenderness ratio in columns.

Following on from our previous article on buckling failure modes, we’re going to dive deeper into the types of buckling in columns.

Column Buckling.

Column buckling, or column stability, can be expressed as the failure of load-bearing capacity cause by the effect of pressure.  It’s manifested as bending, or buckling, of the column.  You can see an example of it in the image below:

Types of Buckling in Columns

There are generally three types of buckling in columns:

• Flexural buckling.
• Torsional buckling.
• Flexural-torsional buckling.

You can also see an image showing these types of buckling in this image:

Flexural buckling

Flexural buckling is the buckling mode, which governs the design of a member in pure compression. For buckling calculation of the column, it is necessary to check the slenderness of the column.

Depending on the support conditions, material, and geometric characteristics of the column, it is possible to calculate the slenderness of the column.

Torsional buckling and flexural-torsional buckling

Torsional buckling is simply a twisting of the entire cross section about its shear centre.

Flexural-torsional buckling is a compression member instability involving a combination of member bending and twisting as well as any local buckling of slender elements. in this behavioural sense, it resembles lateral-torsional buckling of unbraced beams.

How can we calculate the length of the column?

The length of the column is calculated using the following expression:

Le=K L

Where:

• K: effective length coefficient depending on the support conditions
• L: length of the column (m).

Column slenderness is defined by the following expression:

λ=Leimin 

Where are:

• L_e – effective length of the column (m),
• i_min – minimum radius of gyration, imin=IA (m),
• I – moment of inertia (m⁴),
• A – cross-section area (m²).

Let’s look at an example.  Take a 3m long column, with on end pinned and the other fixed.  Imagine it has a cross-section area of a = 3.125×10^-3m² and moment of inertia i = 4.7526×10^-6m⁴.

We can calculate the column slenderness as:
Le=K·L=0.7·3.0m=2.1m

imin=IA=4.7526×10-60.003125=0.01233 m

λ=Leimin=2.1m0.01233m=170.3

Non-dimensional slenderness

Since the buckling occurs under compression of the column, material characteristics affect the buckling critical force. Due to that, EC3 defined non-dimensional slenderness by the following expression:

=1 

Where are:

• λ – column slenderness,
• 1=π xEfy,
• E – Young modulus of elasticity (Pa),
• f_y – steel yield strength (Pa).

The table below shows steel grade mechanical properties:

Let’s look at an example.  Imagine a 3m long column with one end pinned and the other fixed.  It has a cross-section area of a = 3.125×10^-3 m² and moment of inertia I=4.7526×10^-6 m⁴.

We can calculate the column non-dimensional slenderness for steel grade S235 and E=210 x 10⁹ Pa by:

Le=K x L=0.7×3.0m=2.1m

imin=IA=4.7526×10-60.003125=0.01233 m

λ=Leimin=2.1m0.01233m=170.3

1=π x Efy=3.14 x 210000 x 106 Pa235 x 106 Pa=29.9

=1=170.329.9=5.69

Keep an eye out for future courses on different types of buckling and various calculations around buckling in columns

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