Tuesday, 31 March 2020

Buckling resistance of members: imperfection or reduction factor? - Part 1

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 flexural-buckling or lateral-torsional 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).

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 v0 = 13.4 mm. 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.
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 second order bending moment value depends naturally on the level of compressive force according to the well-known amplification relationship:
where N.c,Ed is the applied compressive force, v.0 is the amplitude of the equivalent geometrical imperfection and
is the amplification factor depending on the elastic critical load N.cr = 347,6 kN (Euler load). The utilization of this critical cross-section can be calculated according to Eq. (6.2):
Figure 2. shows the relationship between the applied compressive force and the second order bending moment and utilization of the middle cross-section where 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:
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 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 imperfection method gives the “true” (nonlinear) utilization while the reduction factor method shows accurate utilization at the buckling resistance only. For the adequacy of the member, however, both methods are correct.