Design of Steel Beam to Eurocode 3 – an overview

Table of Contents

This article presents an overview on the design of Structural Steel sections in bending to EN 1993 -1-1:2005 (Eurocode 3) and the UK National Annex to EN 1993 -1-1:2005. In this article EN 1993 -1-1:2005 is often referred to as the standard, while the UK National Annex to the standard is sometimes shortened as the UK NA

Beams are structural members that spans horizontally between two supports or restraints and are subjected to external load perpendicular to the axis along the member. Structural steel sections are often adopted as beams for different types of structures. Their relative advantages to other structural materials include high strength-to-weight ratio, and ease of erection.

In the following sections, we shall discuss the steps required in designing a steel beam for ultimate limit state and serviceability limit state.

Selection of appropriate Cross-Section

One of the first major decision of the engineer is to select the appropriate steel cross-section for beam to meet the specific project requirement. The criteria for selection often boil down to performance and economy.

Performance of the section has to be determined based on its cross-sectional resistance to internal forces, stability against lateral torsional buckling, and stiffness against vertical deformation (deflection). After meeting performance requirement, the adopted cross-section must also be cost-effective for optimal and economical design. The most common cross-sections are discussed below.

I-sections, and H-sections

These sections have one horizontal element each (flanges) on each end of the vertical element (web). Due to the distance between the flanges, these sections have high second moment area and flexural stiffness, so they are good for spanning long distance and supporting large loads. However, when the compression flanges are unrestrained, they are vulnerable to torsion which does not allow them to achieve their full strength before failure.

Although the I and H sections look similar, I-sections are by far often adopted as beams as they have higher depth of web which makes their flanges further apart than those of H-sections. This increases their bending stiffness about the major axis. The H-section is mostly adopted as columns because it is much better for resisting bending about both the major and minor axis. The I-sections and H-sections are respectively traditionally referred to as universal beams (UB) and universal columns (UC) in specification documents such as BS4 for standardized sections across the UK. Those with CE markings are referred to as UKB and UKC to distinguish them from the UB and UC which are without CE markings

Circular Hollow Sections (CHS), and Rectangular Hollow Sections (RHS) as Beams.

Hollow sections are best suited as beams where vulnerability to torsion is very significant. In such cases, the rectangular hollow section is often the desire section, the circular hollow section is rarely used as a beam because of its poor flexural stiffness.

Angle Sections, and Channel Sections

Angle and channel sections are economical and appropriate for light structures of small spans such as in roof purlins, side rails etc. They are also appropriate as top and bottom chords of truss or lattice girders spanning large spaces.

Pick a Trial Beam

Having concluded the desired section to be adopted as beam, then a trial cross-section is selected. The expected plastic modulus of the prospective beam can be used as a guide to pick a trial cross-section. The plastic modulus of a beam in bending is given as:

$
W_{p,ly}\,\,=\,\,\frac{M_{y,Ed}\,\,.\,\,\gamma _m}{f_y}
$

Where;

M_y,Ed is the design moment to be resisted

f_yis the yield strength of the beam

ϒ_m is the material factor of the beam.

Once a probable section modulus has been determined using the above equation, then a trial cross-section can be picked using SCI P363. It is a data sheet from which different standard steel sections and their properties can be taken. The cross-section to be selected should have a higher plastic section modulus than what is calculated.

Classification of Selected Beam Sections

The selected cross-section should then be classified based on its susceptibility to local buckling according to clause 5.5 of the standard as either Class 1, Class 2, Class 3, or Class 4. For more details on classification on steel section read; Classification of Structural Steel Cross-sections.

Verification of Resistance of Selected Beam Cross-Section

Having classified the beam, the resistance of the beam against various critical action effects (internal forces) such as bending moment, shear force, and buckling is verified

The verification of the beam cross-section should include the following:

moment resistance of the cross-section
shear resistance of the cross-section
lateral torsional buckling resistance
deflection
web bearing/buckling.

Shear Resistance of the Beam Cross-Section

The shear resistance of the cross-section should be verified using:

$$
\,\,\frac{V_{Ed}}{V_{cRd}}\,\,\leqslant \,\,1
$$

where V_c,Rdis the design plastic resistance (V_pl,Rd)

$
\,\,V_{c,Rd}\,\,=\,\,V_{pl,Rd}\,\,=\,\,\frac{A_v\,\,f_y/\sqrt{3}}{\gamma _{m0}}
$

where;

A_v is shear area which is defined in clause 6.2.6(3) for different sections

For rolled I and H sections: A_v = A – 2bt_f + (t_w + 2r)t_f ≥ ηh_wt_w

For welded I and H sections: A_v = η∑h_wt_w

For rolled channel sections: A_v = A – 2bt_f + (t_w + r) t_f

For rolled T sections: A_v = A – bt_f + (t_w + r) t_f/2

For welded T sections: A_v =t_w (h – t_f/2)

Rectangular hollow sections: A_v = Ah/ (b + h) (for load parallel to depth)

= Ab/ (b + h) (for load parallel to width)

For circular hollow section or tube of uniform section A_v = 2A/π

A is the cross-sectional area;

b is the overall breadth;

h is the overall depth;

h_w is the depth of the web;

r is the root radius;

t_f is the flange thickness;

t_w is the web thickness (If the web thickness in not constant, tw should be taken as the minimum thickness.).

η is a coefficient to allow for steel hardening. It can be conservatively taken as 1.

Υ_m0 is partial safety factor for resistance of cross-section

Shear Buckling

For most rolled sections, shear buckling is not a problem. In very rare cases where shear buckling is an issue, then resistance should be verified according to EN 1993-1-5.

It will not be necessary to verify the web of a beam for shear buckling provided that

$
\,\,\frac{h_w\,\,}{t_w}\,\,\leqslant \,\,72\frac{\varepsilon}{\eta}\,\,\left( for\,\,webs\,\,without\,\,intermediate\,\,stiffeners \right)
$

Moment Resistance of the Beam Cross-Section

The design bending resistance of a section about a principal axis is largely based on the classification of the steel cross-section. Also, when there is co-existent high shear within the same section, the bending resistance of the section is reduced. However, for vast majority of scenarios (except for cases like in support of a cantilever etc.), the effect of shear is always negligible as the shear is always low.

For adequate moment resistance, the equation below should be satisfied.

$
\,\,\frac{M_{Ed}}{M_{cRd}}\,\,\leqslant \,\,1
$

where M_c,Rdis the design bending resistance

$
\,\,Mc,Rd\,\,=\,\,Mpl,Rd\,\,=\,\,\frac{W_{pl}.\,\,f_y\,\,}{\varUpsilon _{m0}}\,\,\,\,
$ (for class 1, and class 2 sections)

$
\,\,Mc,Rd\,\,=\,\,Mel,Rd\,\,=\,\,\frac{W_{el,min}.\,\,f_y\,\,}{\varUpsilon _{m0}}\,\,\,\,
$ (for class 3 sections)

$
\,\,Mc,Rd\,\,=\,\,\frac{W_{eff,min}.\,\,f_y\,\,}{\varUpsilon _{m0}}\,\,\,\,
$ (for class 4 sections)

Fastener holes for zones of the cross-section in compression need not be allowed for provided that they are filled with fasteners. The effect of fastener holes in tension flange can be ignored provided that for the tension flange the expression below is true:

$
\frac{A_{fnet}\,\,x\,\,0.9f_u}{\varUpsilon _{m2}}\,\,\geqslant \,\,\frac{A_f\,\,f_y}{\varUpsilon _{mo}}
$

A_fis the area of the tension flange

Combined Bending and Shear of the cross-section

When bending and high shear occurs together in a cross-section, the effect of shear should be allowed for by reducing the moment resistance of the cross-section. The effect of shear force should be neglected if shear is low (i.e.: the shear force is less than half the shear resistance of the section) except in few sections susceptible to shear buckling such as in plate girders.

ie: V_Ed≤ 0.5V_pl,Rd(Low Shear)

V_Ed≥ 0.5V_pl,Rd(High Shear)

This reduction in moment resistance for high shear scenario can be achieved by reducing the plate thickness of the relevant part of the cross section or by reducing the yield strength of the cross-section. It is more straight-forward to reduce the yield strength as shown below:

(1-ρ)f_y (reduced yield strength)

$
\rho \,\,=\,\,\left[ \frac{VEd}{VcRd}-1 \right] ^2
$

According to clause 6.2.8(5) of the standard, the reduced design plastic resistance moment allowing for the shear force may be obtained for I-sections with equal flanges and bending about the major axis as follows:

$
M_{y,v,Rd}\,\,=\,\,\left[ \frac{Wply\,\,-\,\,\frac{\rho A_w^2}{4t_w}}{\varUpsilon _{m0}} \right] \,\,\leqslant \,\,M_{y,c,Rd}
$

Where;

M_y,c,Rd is the moment resistance without shear consideration

A_w = f_wt_w

Lateral torsional Buckling

Resistance of the beam to lateral torsional buckling needs to be checked only when the beam is an open section (I-section, H-section, angle-section etc.) and its compression flange is unrestrained. For further details on lateral torsional buckling, read “Lateral torsional buckling of steel beams”

Deflection

Deflection is an important serviceability state that has to be verified in steel beams. Excessive deflection of members can affect the aesthetic of the structure, cause damage to brittle finishes, cause problem to fixtures. EN 1993-1-1 does not specify limit for vertical deflection, however the UK National Annex to the standard in clause NA.2.23 specifies some limit in Table NA.2 which is reproduced below

Limit for vertical deformation of steel beams (Table NA.1; NA+A1:2014 to BS EN 1993-1-1:2005+A1:2014)

According to the UK National Annex, vertical deflections should be verified under the characteristic load combination due to variable loads and should not include permanent loads. Expressions to estimate deflections of beams with different support conditions and loading configurations can be found in standard structural design textbooks.

References

EN 1993-1-1:2005: Eurocode 3, Design of Steel structures – Part 1:1: General rules for buildings.

UK National Annex to Eurocode 3: Design of steel structures Part 1-1: General rules and rules for buildings (NA+A1:2014 to BS EN 1993-1-1:2005+A1:2014)