›
www.thestructuralengineer.org
Note 1 Level 2
Technical
Technical Guidance Note
TheStructuralEngineer
January 2013
25
Designing a steel beam
Introduction
ICON
LEGEND
This Technical Guidance Note is the first of the Level 2 guides. Guides in
this next level build on what has been described previously in the Level 1
series. The topics covered at Level 2 are of a more complex nature as they
typically deal with the design of elements as opposed to core concepts
such as loading and stability. As such, the amount of prior knowledge the
reader is assumed to have is at the very least the contents of relevant Level
1 Technical Guidance Notes.
W Principles of steel
beam design
W Applied practice
W Worked example
The subject of this guide is the design of non-composite steel beams to BS
EN 1993-1-1 – Eurocode 3: Design of Steel Structures – Part 1-1: General
Rules for Buildings. It covers both restrained and unrestrained rolled steel
‘I’ and ‘H’ beam sections. It does not encompass the design of ‘T’ sections,
hollow sections, castellated beams, angles and welded sections.
The key to understanding the design of
steel beams is to determine whether or
not the beam is restrained against lateral
torsional buckling. This subject is covered
in the Level 1 (No. 16) Technical Guidance
Note: Lateral torsional buckling. If a beam is
restrained, all that needs to be checked is
the bending moment and shear resistance
of the beam as well as serviceability limits
against the applied load/actions. All
of which are based on the beam’s core
properties.
If the beam is unrestrained along any part
of its length however, then there is a risk
it will fail due to lateral torsional buckling.
To address this, Eurocode 3 establishes
a reduction factor that is applied to the
bending moment resistance of the beam.
Calculating this factor is the cornerstone
of unrestrained steel beam design within
Eurocode 3.
Steel material properties
The density of steel is 7850 kg/m3 and the
Young’s Modulus (E) is 210,000 N/mm2.
S275 and S355 are the two strength
TSE13_25-29.indd 25
W Web resources
Description
Principles of steel
beam design
Frequent references will be made on the
section variables throughout this guide. You
are advised therefore to examine Figure 1
for the definition and nomenclature of these
variables.
W Further reading
Variable
b
Width of flange
h
Depth of beam
z-z
Minor axis
y-y
Major axis
d
Depth of web
tw
Thickness of web
tf
Thickness of flange
r
Radius of root fillet between web and flange
Wpl,y
Plastic section modulus about the y-y axis
Wel,y
Elastic section modulus about the y-y axis
iz
Radius of gyration about the z-z axis
A
Cross sectional area of the beam
Iyy
Second moment of area about the y-y axis
N
Figure 1
Beam section notation used in Eurocode 3
grades of steel most commonly used in
the construction industry within the UK.
S275’s nominal yield strength is 275 N/mm2
and Grade S355’s nominal yield strength
is 355 N/mm2. The actual yield strength is
dependent on the maximum thickness of
an element within a steel section, as the
thicker the element the lower the yield
strength. Table 1 defines what the yield
strength should be for the most common
ranges of thicknesses found in open rolled
steel sections. These figures are based on
the values given in BS EN 10025 Hot Rolled
Products of Structural Steels, which is the
product standard for rolled steel sections of
various sub-grades.
19/12/2012 10:13
›
Note 1 Level 2
26
Technical
Technical Guidance Note
TheStructuralEngineer
January 2013
Table 1: Yield strength fy vs. element thickness
Grade
Yield strength fy for
element thickness
< 16mm (N/mm2)
Yield strength fy for element
thickness > 16mm
and < 40mm (N/mm2)
Yield strength fy for element
thickness > 40mm
and < 63mm (N/mm2)
S275
275
265
255
S355
355
345
335
Classification of beam sections
Clause 5.5.2 in BS EN 1993-1-1 groups steel
beams into four classifications. These
classifications are based on a steel beam
section’s resistance to suffering from a local
failure due to buckling:
of the elements must lie. These limits are
further modified based on the yield strength
of the material; this is defined via
coefficient ε thus:
235 N/mm
fy
f =
Class 1/‘Plastic’ These sections can form
a plastic hinge when a bending moment is
applied to them without suffering from local
buckling failure.
Class 2/‘Compact’ These sections cannot
properly develop a plastic hinge as their
ability to rotate is limited before local
buckling failure occurs.
Class 3/‘Semi-Compact’ These are
sections that can withstand elastic stresses,
specifically at the extreme fibres of the
section, but cannot form a plastic hinge.
This has the effect of negating their plastic
bending moment capacity.
Class 4/‘Slender’ Sections that will fail
due to local buckling prior to the point of
yield stress. Their plastic bending capacity
therefore is non-existent.
When determining the classification of a
section, typically two parts of a rolled steel
beam section are considered. For a simply
supported beam, these are the edge of
the top flange and the web, both of which
are subjected to compression stress due
to bending. Figure 2 indicates where these
elements lie within a rolled steel beam section.
Table 5.2 in Clause 5.5 of BS EN 1993-1-1
defines the limits within which the geometry
E
Figure 2
Elements
of a rolled steel
beam that
determine its
classification
TSE13_25-29.indd 26
2
Where:
ε is the coefficient for section classification
fy is the yield strength of the steel, based on
element thickness
Table 2 indicates the limiting values for
various classes of section for both of the
elements identified in Figure 2. If any of the
ratios go beyond those stated in Table 2 the
section is considered to be in the Class 4
category.
Table 2: Limiting values of geometries
for section classes 1-3
Class
Web
Flange
1
c/t ≤ 72ε
c/t ≤ 9ε
2
c/t ≤ 83ε
c/t ≤ 10ε
3
c/t ≤ 124ε
c/t ≤ 14ε
Shear capacity
Typically the component of the beam that
takes the majority of the applied shear
force is its web. There are instances where
stiffeners are installed in order to support
high shear loads, but this is very much the
exception rather than the rule.
"Guides in this
next level build
on what has
been described
previously in the
Level 1 series"
19/12/2012 10:13
www.thestructuralengineer.org
27
All steel beams must satisfy the following
expression:
V Ed
# 1
V c, Rd
Where:
VEd is the applied shear force
Vc,Rd is the design shear resistance
In the case of Class 1 and 2 rolled steel
beams, the design shear resistance is
designated as Vpl,RD and is defined in Clause
6.2.6, equation 6.18 of BS EN 1993-1-1 as:
A v (f y /
V pl,Rd =
3)
c M0
Lateral torsional buckling
fy
M c,Rd = W y
c M0
Where:
Wy is the major axis section modulus of the
beam based on its classification:
Wy = Wpl,y (Plastic section modulus)
for Class 1 or 2
Wy = Wel,y (Elastic section modulus)
for Class 3
Wy = Weff,y (Minimum effective section
modulus) for Class 4
fy is the yield strength of the steel, based on
element thickness
γM0 is the partial factor for the resistance of
cross-sections, which in the UK is set at 1.0
Where:
Av is the cross section area of the part of the
beam that is resisting shear. For ‘I’ and ‘H’
sections this can conservatively be taken to
be htw, which is the cross sectional area of the
web and the thickness of the flange (Figure 3).
For all other classes of beam sections, you are
referred to Clause 6.2.6 (4) of BS EN 1993-1-1
for determining their design shear resistance.
The bending moment resistance should be
reduced if the applied shear force is more
than half of the plastic shear resistance
of the beam. Where it exceeds this value,
Clause 6.2.8 of BS EN 1993-1-1 applies. This
places a modification factor against the yield
strength, thus:
Modified yield strength = (1-ρ)fy
Where:
t =
` 2V
- 1j
2
Ed
V pl,Rd
This modified yield strength is then inserted
into the calculation that determines bending
moment resistance.
For Class 4 sections you are required to
follow the guidance given in BS EN 1993-15 – Eurocode 3: Design of Steel Structures
– Part 1-5: Plated Structural Elements. Class
4 sections are not found in rolled ‘I’ and
‘H’ elements and are therefore beyond the
scope of this guide.
N
Figure 3
Approximate extent of web
resisting shear in a steel beam
Clause 6.2.6 of BS EN 1993-1-1 has a more
accurate equation that takes into account
the radii of the root fillet to the web-toflange interface of the ‘H’ and ‘I’ sections.
These can be used if you are finding it
difficult to satisfy the shear resistance
requirements.
Bending moment resistance
of steel beams
BS EN 1993-1-1 defines the bending moment
resistance of restrained steel beams (Mc,Rd)
in clause 6.2.5(2) as:
TSE13_25-29.indd 27
In the case of an unrestrained portion of
a beam, a factor is applied to the bending
moment resistance (Mb,Rd) that takes into
account the risk of lateral torsional buckling.
This is described Clause 6.3.2.1 (3) of BS EN
1993-1-1, in equation 6.55 as:
M b, Rd = | LT W y
fy
c M1
Where:
Wy is the major axis section modulus of the
beam based on its classification and is the
same for restrained beams
fy is the yield strength of the steel, based on
element thickness
γM1 is the partial factor for resistance of
members subject to instability, which in
the UK is set at 1.0
χLT is the reduction factor that takes into
account lateral torsional buckling
Clause 6.3.2.3 of BS EN 1993-1-1 describes
how the value of χLT is related to the
slenderness of the beam. This is related
to the distance between restraints to the
element of the beam that is subject to
compression. For simply supported beams it
is its upper-most flange. This is known as the
non-dimensional slenderness ( m LT ) and is
defined thus:
m LT =
W y fy
M cr
Where:
Wy and fy are as per previous definitions
Mcr is the elastic critical moment for lateral
torsional buckling, which is based on the
slenderness of the beam
Mcr is not defined within Eurocode 3, which
offers no guidance in calculating its value.
There are however, many direct methods
for calculating slenderness, the most simple
of which is described in this guide.
Slenderness
For ‘H’ and ‘I’ sections it is possible to
use simplified methods to calculate the
relative slenderness of the beam. The most
conservative method is defined in Table 1.1
of NCCI: Determination of non-dimensional
slenderness of I and H sections SN002aEN-EU. It is based on applying the following
equations that vary depending on the grade
of steel being used:
For S275 Grade steel:
:L D
m LT =
iz
96
For S355 Grade steel:
:L D
m LT =
iz
85
Where:
L is the distance between restraints to the
compression flange of the beam
iz is the radius of gyration about the minor
axis of the beam
While valid, this method is very conservative
as it ignores the bending moment diagram
of the beam and can therefore result in
oversized members. There is however,
a more accurate yet complex method
that does take into account the bending
moment diagram and is described in
NCCI: Determination of non-dimensional
slenderness of I and H sections SN002aEN-EU. It is strongly recommended that
you examine this method in order to design
more efficiently sized steel beams.
19/12/2012 10:13
›
Note 1 Level 2
28
Technical
Technical Guidance Note
TheStructuralEngineer
January 2013
There are further methods in determining
Mcr described in the Non-Contradictory
Complimentary Information website for
structural steelwork.
Once the non-dimensional slenderness is
established, the value of χLT is determined
using equation 6.57 of Clause 6.3.2.3 of BS
EN 1993-1-1 thus:
Serviceability
The vertical deflection limits for steel beams
can be found in the Clause NA.2.23 of the UK
National Annexe to BS EN 1993-1-1. Table 3
is based on these stated limits, which are for
the deflection due to unfactored imposed
loads/variable actions only.
Table 3: Vertical deflection to steel beams
1
| LT =
U LT +
2
2
U LT - b m LT
U LT = 0.5 61 + a LT ( m LT - m LT, 0) + b m LT@
2
Where:
The values of β and m LT,0 are defined as
0.4 and 0.75 respectively, as described in
Clause NA.2.17 of NA to BS EN 1993-1-1.
Beam type/structure
Deflection limit
Cantilever
Length/180
Beams supporting
brittle finishes
Span/360
All other conditions
Span/200
Purlins and
cladding rails
To suit cladding
system
Eurocode 0.
αLT is the imperfection factor and is found
in Table 6.3 of BS EN 1993-1-1, which reads
against the steel beam’s buckling curve.
The buckling curves are labelled ‘a’ to ‘d’
and can be found in Clause NA.2.17 of NA
to BS EN 1993-1-1. The buckling curve is
dependent upon the h/b ratio of the beam
section.
"Class 4 sections
are not found in
rolled ‘I’ and ‘H’
elements and are
beyond the scope
of this guide"
In addition to deflection, it is prudent to
check the vibration of the beam i.e. its
dynamic response. Technical Guidance Note
No. 11, Level 1 explains how to do this.
Applied practice
The applicable codes of practice for
designing steel beams are as follows:
BS EN 1993-1-1 Eurocode 3: Design of
Steel Structures – Part 1-1: General Rules for
Buildings
BS EN 1993-1-1 UK National Annex to
Eurocode 3: Design of Steel Structures –
Part 1-1: General Rules for Buildings
Worked example
A simply supported, unrestrained steel beam spanning 8m is supporting another steel
beam in the middle of its span. The ultimate load from this beam is 500 kN, while the
serviceability load due to variable actions is 200 kN. The floor structure consists of one
way spanning precast concrete planks with a screed and tiled finish. These planks span
parallel to the steel beam and hence do not provide lateral restraint. Determine what size
of beam is required to support this load, assuming the steel grade to be S355.
TSE13_25-29.indd 28
19/12/2012 10:13
www.thestructuralengineer.org
29
Glossary and
further reading
Reduction factor – A variable applied to
the bending moment resistance of a beam
due to the fact that it is unrestrained and
hence subject to lateral torsional buckling.
Rolled steel section – A steel element
that is cast to a pre-set size and not built up
from separate plate elements.
Section classification – A categorisation
of steel elements that is based on the
element’s ability to develop a plastic hinge
when placed under load.
Further Reading
The Institution of Structural Engineers
(2010) Manual for the Design of Steelwork
Building Structures to Eurocode 3 London:
The Institution of Structural Engineers
Steel Construction Institute (2009) Steel
Building Design. Worked Examples for
Students (P387) Ascot, Berkshire: SCI
The Institution of Structural Engineers
(2012) ‘Principles of design’ The Structural
Engineer Vol. 90 (1) pp. 40-41
The Institution of Structural Engineers
(2012) ‘Derivation of dead loads’ The
Structural Engineer Vol. 90 (1) pp. 43-45
The Institution of Structural Engineers
(2012) ‘Derivation of imposed loads’ The
Structural Engineer Vol. 90 (2) pp. 46-48
The Institution of Structural Engineers
(2012) ‘Derivation of wind load’ The
Structural Engineer Vol. 90 (2) pp. 49-52
The Institution of Structural Engineers
(2012) ‘Derivation of snow load’ The
Structural Engineer Vol. 90 (3) pp. 22-24
The Institution of Structural Engineers
(2012) ‘Lateral torsional buckling’ The
Structural Engineer Vol. 90 (16) pp. 28-30
Eurocode 0.
Web
resources
Tata Steel Interactive ‘Blue Book’:
www.tatasteelconstruction.com/en/design_
guidance/the_blue_book/
Non-Contradictory Complimentary
Information website for structural steelwork:
www.steel-ncci.co.uk/
TSE13_25-29.indd 29
19/12/2012 10:13