Members subject to combined
bending and compression
David Brown of the SCI reviews the options and available resources that can be used to
simplify the design checks and determine the required resistance data.
Expressions 6.61 and 6.62
These two expressions are well-known in the Eurocode steel design world.
They bring together a number of intermediate calculations in a final crescendo
of complexity, not helped by an unfamiliar presentation of familiar terms. In
fact, the expressions are conceptually similar to the “more exact” approaches
found in BS 5950, containing an axial term, a major axis moment term and a
minor axis moment term. The denominators in the three terms are the flexural
buckling resistance, the lateral torsional buckling resistance and the minor axis
cross sectional resistances respectively. The second two terms are modified by
factors that allow for the interaction between the different modes of buckling.
If Class 4 sections are excluded the ΔM terms due to a shift in the neutral
axis can be removed, and if the denominators are presented in more familiar
terms, the two expressions become:
+ k+ k 1 (6.61)
Technical Digest 2018
+ kzy 1 (6.62)
The main ratios are each
. Purists should note that the
denominator in the final term is really
, but this is equal to the cross
sectional resistance Mc,z,Rd since γM1 = γM0 = 1.0
The first task in using these expressions is to determine the member
Member resistances from the Blue Book
The calculation of member resistances always starts from section classification.
The easy way to classify a section under combined bending and axial load is to
use the “n” limit given in the axial force and bending tables of the Blue Book.
An extract from the tables is
shown in Figure 1.
The Class 2 limit is the axial load
ratio (compared to Npl,Rd) when a
member changes from Class 2 to
Class 3. The Class 3 limit is the axial
load ratio when a section becomes
Class 4 (and the designer may prefer
to choose a different section!).
The limitations are so defined
because, as shown in Table 1, the
different Classes demand different
properties to be used in the calculation of member resistance.
For the resistance calculations, it does not matter if the member is Class
1 or 2; both use the same member properties. Thus all that is needed is to
know that the member is “at least Class 2”, and hence why a Class 1 limit is not
For the beam data shown in Figure 1, the member becomes Class 3 when
the axial load exceeds 0.263 × 3620 = 952 kN. The member becomes Class 4
when the axial load exceeds 0.839 × 3620 = 3037 kN.
These limits are simply a rearrangement of the conditions found in Table
5.2 of BS EN 1993-1-1.
Flexural buckling resistances can be obtained directly from the axial force
and bending tables for the appropriate buckling length. There can be an
advantage in taking resistances from the axial force and bending tables, as
the resistances are limited to Class 3. In the pure compression tables, under
uniform compression, the section may become Class 4 and the resistance
Lateral torsional buckling resistances are best taken from the resistance
table for bending alone. This is because the tables dedicated to bending
alone allow designers to select a resistance appropriate to the shape of
bending moment diagram, based on the C1 value. The bending resistances in
the axial force and bending tables are for a value of C1 = 1.0, so can be very
There is however an immediate problem if the section is Class 3. The axial
force and bending tables provide a LTB resistance for Class 3 sections, but for C1
= 1.0. All UB in bending alone are Class 1, so the bending tables do not cover
Class 3 sections. If a section becomes Class 3 due to the axial compression,
but has a non-uniform bending moment diagram, use of the values in the
axial force and bending tables will be conservative. For a precise value, manual
calculations would require the calculation of the LTB resistance using the
The interaction factors
The interaction factors are given in both Annex A and Annex B of BS EN 1993-
1-1. Annex B is recommended, because it is simpler, and because the Annex A
method is to be relegated when the revised Eurocode is published.
A typical term from Annex B is shown in Figure 2, (below).
The C factor deals with the shape of the bending moment diagram, and is
taken from Table B3 of the Standard.
Cmy (1 + 0.8 )
Nkyy = C(1 + (y – 0.2) ) Ed
my NyRk M1
Figure 2: Typical interaction factor
Again, the presentation of these terms is not very attractive. In particular,
the term χyNRk/γM1 is unhelpful, as it is simply the flexural resistance (in this
case in the major axis), or Nb,y,Rd . The expressions might more helpfully be
presented in the form in Figure 3.
y( Nkyy = C1 + ( – 0.2) ) C(1 + 0.8 Ed
) my Nmy Nb,y,Rd
Figure 3: Typical interaction factor, revised presentation
Figure 1:”n” limit from the Blue Book
Class Axial resistance Bending resistance
1 Ag Wpl
2 Ag Wpl
3 Ag Wel
4 Aeff Weff
Table 1: Member class and resistance calculations