### Buckling utilization differences

The Eurocode EN 1993-1-1 offers basically
two methods for the buckling verification of members:

(1) based on buckling reduction factors
(buckling curves) and

(2) based on equivalent geometrical
imperfections.

This part reviews how these two methods
relate to each other in terms of the final member utilization. For the sake of simplicity,
we consider only members subjected to pure compression or pure bending,
undergoing

**or***flexural-buckling***. For case (1) the chapters 6.3.1 and 6.3.2 are used while for case (2) the imperfections are considered to take the shape of a buckling mode and the basic chapter is the 5.3.2 (11).***lateral-torsional buckling*

*It is an obvious expectation that these two standard procedures should yield the same utilization for the same problem. However, this is by far not the case in general.*

Let’s see the following simple example of a
simply supported, compressed column with a Class 2 cross-section (plastic
resistance calculation allowed). The column is 6 meters high and has an IPE300
cross-section made of S235 steel. The standard amplitude for the buckling mode
based imperfection is calculated by Eq. 5.9-5.11, that is equal to

**. The next figure shows the model, the buckling mode shape – which is a classic flexural buckling about the weak axis – and the second order weak axis bending moment distribution.***v*_{0}= 13.4 mm
Ä Important to note that the
verification based on equivalent geometrical imperfection should be calculated
from the results of a second order analysis using the linear cross section
check defined by Eq. 6.2 (or Eq. 6.1 for elastic cases).

*Figure 1. The compressed column, its flexural buckling mode shape, and the second order bending moment distribution due to the imperfection*

The most utilized cross-section will be the middle one, where the

where

is the amplification factor depending on the elastic critical load

Figure 2. shows the relationship between the applied compressive force and the second order bending moment and utilization of the middle cross-section where

It gives identical result to the one calculated by the equivalent imperfection method, which means that the two methods, i.e. Eqs. 5.9-5.11 and Eqs. 6.46-6.49 are consistent in terms of buckling resistance value. However, the buckling utilization of the column can be calculated by the common

In Figure 3. the two utilization values are illustrated. It can be observed that the two utilizations are equal at the buckling resistance load level only, when the applied compressive force is lower the reduction factor method gives higher utilization; when the applied compressive force is higher the imperfection method gives higher utilization.

**second order bending moment**value depends naturally on the level of compressive force according to the well-known**amplification relationship**:*N.c,Ed*is the applied compressive force,*v.0*is the amplitude of the equivalent geometrical imperfection and*N.cr = 347,6 kN*(Euler load). The utilization of this critical cross-section can be calculated according to Eq. (6.2):*N.Rd = 1264,6 kN*and*M.z.Rd = 29,28 kNm*It can be clearly seen that the**utilization is nonlinearly**depends on the compressive force level due to the nonlinearity of the second order bending moment. The utilization corresponding to the 100% value gives the buckling resistance of the column:*Figure 2. The second order bending and the utilization*

The buckling resistance can also be calculated by the reduction factor method using Eqs. 6.46-6.49:

**linear relationship**of Eq. 6.46:

In Figure 3. the two utilization values are illustrated. It can be observed that the two utilizations are equal at the buckling resistance load level only, when the applied compressive force is lower the reduction factor method gives higher utilization; when the applied compressive force is higher the imperfection method gives higher utilization.

*Figure 3. The utilization from the reduction factor and the imperfection methods*

The
difference is clearly because of the nonlinearity of the imperfection method. For
instance, if the applied compression force is

*N.c.Ed = 250 kN*the utilization values show considerable difference (see Figure 3.):
It can generally be stated that the

**while the reduction factor method shows accurate utilization at the buckling resistance only. For the adequacy of the member, however, both methods are correct.***imperfection method gives the “true” (nonlinear) utilization*