This article presents an overview on the design of column to EN 1993-1-1:2005. However, since there are not much details in Eurocode 3 for columns in simple frames, the superseded BS 5950-1:2000, and other non-contradictory complementay Information (NCCI) are consulted.
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Columns are compression members and are mostly used to transfer loads from superstructure to foundation or to transfer members. However, columns in building or frames in general are often subjected to combined moment and axial load unlike struct that are idealized as being subjected to pure compression. This moment in columns can either be nominal or principal depending on the frame type. Consequently, frames can be categorized based on the connection between the beams and columns as:
- Rigid Frames
- Simple Frames
Rigid Frames
Rigid frames have rigid connection between beams and columns which makes the column to resist substantial moment from the beam together with shear. The stability of this type of frames against lateral load is dependent on the rigidity of the beam-column connection. These types of frames are very advantageous when adopted for building construction that are sensitive to vibration and deflection or building where a large uninterrupted space is desired.
Design of Columns in Rigid Frames
Columns in rigid frames have to be designed for both cross-sectional resistance and buckling. The methods for verification of these limit states are discussed in subsequent sections.
Compression resistance
The expression below has to be satisfied
$
\frac{N_{ED}}{N_{cRd}}\,\,\leqslant \,\,1
$
i.e: $
N_{cRd}\,\,=\frac{A\,\,f_y}{\gamma _{m}}\,\,
$ for class 1, 2, & 3 sections
$
N_{cRd}\,\,=\frac{A_eff\,\,f_y}{\gamma _{m}}\,\,
$ for class 4 sections
Combined Bending and Axial resistance
For combined compression and bending resistance, the expression below should be satisfied
MEd ≤ MN,Rd (for class 1 and 2 sections. Guidance for class 3 & 4 should be check in cl 6.2.9.2 & cl 6.2.9.3 of the standard.)
where MN,Rd is the design plastic moment resistance reduced due to the axial force NEd
MN,Rd is considered as allowance has to be made for the effects of axial load on the plastic moment resistance of cross-sections. These allowances need not be made if these conditions are satisfied for various cross-sections as enumerated below
Open Cross-sections (I-sections, H-sections etc.)
Allowance need not be made for effect of axial force on plastic moment of resistance of open doubly-symmetric sections if the expressions below are satisfied.
About y-axis
NEd ≤ 0.25Npl,Rd
$
N_{Ed}\,\,\leqslant \,\,\frac{\text{0.5}h_w\,\,t_w\,\,f_y}{\gamma _{mo}}
$
If they are not satisfied, bending resistance is:
$
M_{N,y,Rd}\,\,=M_{Pl,y,Rd}\,\,\frac{\text{1}-\,\,n}{\text{1}-\,\,0.5a}
$
Where;
$
n\,\,=\frac{N_{Ed}}{N_{cRd}}
$
$
a\,\,=\frac{A\,\,-\,\,2b_{tf}}{A}\,\,\leqslant \,\,0.5
$
About z-axis
Allowance need not be made for effect of axial force on plastic moment of resistance of open doubly-symmetric sections about z-axis if the expression below is satisfied.
$
N_{Ed}\,\,\leqslant \,\,\frac{\\,\,h_w\,\,t_w\,\,f_y}{\gamma _{mo}}
$
If not
For n ≤ a
$
M_{N,Z,Rd}\,\,=M_{Pl,Z,Rd}\,\,=\text{1}-\,\,\left( \frac{n\,\,-\,\,a}{\text{1}-\,\,a} \right) ^2
$
For Closed Cross-sections (RHS, CHS, etc)
For a rectangular solid section without fastener holes MN,Rd should be taken as:
$
M_{N,Z,Rd}\,\,=M_{Pl,Z,Rd}\,\,=\left[ \text{1}-\,\,\left( \frac{N_{Ed}}{N_{pl,Rd}} \right) ^2 \right] \,\,\left( for\,\,solids \right)
$
for rectangular structural hollow sections of uniform thickness and for welded box sections with equal flanges and equal webs without fastener holes, MN,Rd should be taken as:
$
M_{N,y,Rd}\,\,=M_{Pl,y,Rd}\,\,\left( \frac{\text{1}-\,\,n}{\text{1}-\,\,0.5a_w} \right) \,\,but\,\,M_{N,y,Rd}\,\,\leqslant \,\,M_{ply,Rd}
$
$
M_{N,z,Rd}\,\,=M_{Pl,z,Rd}\,\,\left( \frac{\text{1}-\,\,n}{\text{1}-\,\,0.5a_f} \right) \,\,but\,\,M_{N,z,Rd}\,\,\leqslant \,\,M_{plz,Rd}
$
Biaxial Bending
To verify a cross-section for combined uni-axial/bi-axial bending and axial compression, a simplified elastic approach can be adopted for class 1, class 2, & class 3 cross-sections by using the expression below:
$
\frac{N_{Ed}}{N_{Rd}}\,\,+\,\,\frac{M_{y,Ed}}{M_{y,Rd}}\,\,+\,\,\frac{M_{z,Ed}}{M_{z,Rd}}\,\,\leqslant \,\,1
$
Alternatively for class 1 and class 2 cross-sections, biaxial bending can be verified with a less conservative approach allowing for plastic behaviour of cross-sections using the expression below:
$
\,\,\left[ \frac{M_{y,Ed}}{M_{Ny,Ed}} \right] ^{\alpha}\,\,+\,\,\left[ \frac{M_{z,Ed}}{M_{N,Z,Rd}} \right] ^{\,\,\beta}\leqslant \,\,1
$
α and β are constant and can both be taken as 1. Otherwise, they can be taken as:
for I and H sections
α = 2, β = 5n but β ≥ 1
for rectangular hollow sections
α = 2, β = 2
for rectangular hollow sections
α = β = 1.66/(1 – 1.13n²) but α = β ≤ 6
where,
n = NEd /Npl,Rd
Buckling Resistance of Members subjected to Combined axial compression and Bending
Members subjected to combined axial compression and bending must satisfied the two below expressions for stability.
$\frac{N_{E d}}{\frac{X_y N_{R k}}{Y_{m 1}}}+K_{y y} \frac{M_{y, E d}+\Delta M_{y, E d}}{\chi L T \frac{M_{y, R k}}{Y_{m 1}}}+K_{y z} \frac{M_{z, B d}+\Delta M_{z, E d}}{\frac{M_{z, R k}}{Y_{m 1}}} \leq 1$
$\frac{N_{E d}}{\frac{\chi_z N_{R k}}{Y_{m 1}}}+K_{z y} \frac{M_{y, E d}+\Delta M_{y, E d}}{M_{L T} \frac{M_{y, R k}}{Y_{m 1}}}+K_{z z} \frac{M_{z, E d}+\Delta M_{z, E d}}{\frac{M_{z, R k}}{Y_{m 1}}} \leq 1$
Kyy, Kyz, Kzy and Kzz are interaction factors to account for non-linear effects. The values of these factors can be determined from Annex A or B of EN 1993-1-1. However, the UK National Annex to EN 1993-1-1 limits the application of Method 1 (given in Annex A) to doubly symmetric sections, while Method 2 (given in Annex B) is applicable to all cases.
$\frac{\chi_z N_{R k}}{\Upsilon_{m 1}}=N_{b, z, R d}, \frac{\chi_y N_{R k}}{\Upsilon_{m 1}}=N_{b, y, R d}, \chi_{L T} \frac{M_{y, R k}}{\Upsilon_{m 1}}=M_{b, y, R d}, \frac{M_{z, R k}}{\Upsilon_{m 1}}=M_{c, z, R d}$
For class 1, class 2, & class 3 cross-sections ΔMy,Ed = 0, ΔMz,Ed = 0
If these are substituted into the two expressions, then they become:
$\frac{N_{E d}}{N_{b, y, R d}}+K_{y y} \frac{M_{y, R d}}{M_{b, y, R d}}+K_{y z} \frac{M_{z, R d}}{M_{c b, z, R d}}=\leq 1$
$\frac{N_{E d}}{N_{b, z, R d}}+K_{z y} \frac{M_{y, R d}}{M_{b, y, R d}}+K_{z z} \frac{M_{z, R d}}{M_{c b, z, R d}} \leq 1$
Click here to study a Worked Example on the design of steel column subjected to combined biaxial bending and axial compression
Columns in Simple Construction
Simple frames have simple connections between beams and columns as the connection is assumed pinned. The necessary flexibility of the joint between beams and columns may result in some non-elastic deformation of the materials, other than the bolts. These types of frames have independent bracing system to resist lateral loads and stabilize the whole frame. The columns in such frames are only meant to resist nominal moment due to eccentricity of the connections.
Design of Columns in Simple frames
EN 1993-1-1 does not give specific guidance on design of columns in simple frames. The guidance presented in this article is according to Access Steel guidance in a Non-Contradictory Complimentary Information (NCCI) document NCCI:SN048b-EN-GB, and also BS 5950-1:200.
According to NCCI:SN048b-EN-GB, the simplified interaction equation for combined bending and axial compression for column in simple construction is:
$\frac{N_{E d}}{N_{\min b, R d}}+\frac{M_{y, R d}}{M_{b, y, R d}}+1.5 \frac{M_{z, R d}}{M_{c b, z, R d}}\leq 1$
where:
Nminb Rd is the lesser of $\chi_y \frac{A\,\,f_y}{\gamma _m}$ and $\chi_z \frac{A\,\,f_z}{\gamma _m}$
$
M_{b,Rd}\,\,=\,\,M_{pl,Rd}\,\,=\,\,\chi _{LT}\,\,\frac{W_{ply}\,\,.\,\,f_y}{\varUpsilon _m}
$
$
\,\,Mc,z,Rd\,\,= \,\,\frac{W_{plz}.\,\,f_y\,\,}{\varUpsilon _{m1}}\,\,\,\,
$
To apply this equation, the following conditions must be satisfied:
i) The column is hot-rolled I or H section, or rectangular hollow section,
(ii) the cross-section is Class 1, Class 2 or Class 3 under compression,
(iii) the bending moment diagrams about each axis are linear,
(iv) the column is restrained laterally in both the y and z directions at each floor but is unrestrained between floors,
(v) the design moments have been derived assuming ‘simple construction’ in which the beam vertical reactions are assumed to act at nominal eccentricity (a distance of 100 mm from the web/flange face of the column)
(vi) The end-moment ratio ψ is restricted as indicated in the Tables below
Nominal Moment in column in Simple Construction
It is imperative to be able to determine how much nominal moment a column in simple frame will have to resist. According to clause 4.7.7 of BS 5950-1, the nominal moments applied to the column by simple beams or other simply-supported members should be calculated from the eccentricity of their reactions, taken as follows:
1) For a beam supported on the cap plate, the reaction should be taken as acting at the face of the column, or edge of packing if used, towards the span of the beam.
2) For a roof truss supported on the cap plate, the eccentricity may be neglected provided that simple connections are used that do not develop significant moments adversely affecting the structure.
3) In all other cases the reaction should be taken as acting 100 mm from the face of the steel column, or at the centre of the length of stiff bearing, whichever gives the greater eccentricity
References
EN 1993-1-1:2005: Eurocode 3, Design of Steel structures – Part 1:1: General rules for buildings.
NCCI: Verification of Column in Simple Construction – a simplified interaction criterion (SN048b-EN-GB)
BS 5950-1-2000: Structural use of steelwork in building — Part 1: Code of practice for design — Rolled and welded sections
Design of Structural Elements to Eurocode 2 by Mc Kenzie
