What is slenderness ratio & why is it important in design?

Slenderness ratio is a geometrical parameter, defined for a compression member (column). It is the ratio of effective length and lateral dimension of the compression member. It is also a measure of the structural vulnerability to the failure of the structure. Generally we design the columns to resist the axial compression load. Sometimes a combination of bi-axial/ uni-axial moment acting on it.

Why it is needed?

Columns are made to support compressive loads, this compressive load may cause buckling in the column which may lead to failure well before the compressive strength of the column. These compressive loads are connected with buckling phenomenon by Euler’s elastic critical load formula. Euler’s define the critical load that a column can sustain before failure by buckling phenomenon.

Where do civil engineers use slenderness ratio in actual construction practice?

  • As a civil engineer, our main aim is to find the load which column can take before buckling failure, and this load depends on the slenderness ratio.
  • The slenderness ratio also defines the failure mode of the column based on the effective length and the radius of gyration.

What is slenderness ratio?

Slenderness ratio is the ratio of effective length of the column to the minimum radius of gyration.

Basis of slenderness ratio formula :

From the Euler formula, the slenderness ratio is inversely proportional to the radius of gyration. This means the tendency of buckling failure is maximum when r is minimum.

$$S \alpha \frac{1}{r}$$

Slenderness ratio is also proportional to K lu, where K is the effective length factor and lu is the unsupported length of the column.

$$S \alpha K \times l_u $$

$$ S \alpha l_{eff} $$

Combining bth equations, we get :

$$Slenderness\, ratio = \frac{l_{eff}}{r_{min}}$$

Where: l_{eff} is the effective length, and r_{min} is the minimum radius of gyration. 

  • Column will buckle in the weak side first if we use minimum radius of gyration. 
  • Columns with low slenderness ratio, fails as a result of failure of materials. i.e., this structure fails when concrete or steel reaches its ultimate strength and not due to buckling. On the other hand columns with high slenderness ration, the structure fails mainly due to buckling along with large lateral deflection.
Column will buckle in the direction where inertia is minimum hence we choose minimum inertia

Bending of column will depends on the radius of gyration. Column will bend in the direction where radius of gyration is minimum.  

$$
r_{\min }=\sqrt{\frac{I_{\min }}{A}}
$$

On the basis of slenderness ratio, we classify the columns as :

  1. Short columns – The value of slenderness ratio is less than or equal to 12.
  2. Long or slender columns – – The value of slenderness ratio is greater than 12.

Effective length (l_eff):

Euler’s column formula is for pin end column hence load carrying capacity changes with change in support condition. Incorporate this change in support condition by the effective length as shown in below figure.

Value of effective length based on the support condition for a column

As you can see in the figure that we are choosing the length of the column which will behave like pinned end condition column. 

What code says? 

In practical situation some more complexities comes in picture. There we can not defined ideal support condition as we have defined in the fig above. Hence code defined some rational number based on the experiments and concrete theories, which we need to follow.

Code uses a term Unsupported length. According to Clause 25.1.3 of IS 456:2000 it is the clear height of the column. Figure below shows the unsupported length of column in different cases.

(a) in case of flat slab, (b) in case of different beams

Why it is important ?

Slenderness ratio decide the failure mechanism of column weather column will fail in crushing failure or in buckling failure.

Check this post to read a brief about what is a column and what are the different types of column

The design of RC columns section is done according to the codal provisions given in IS: 456-2000. By directly using the charts available in SP 16, one can design the columns section efficiently.

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