Axially loaded steel members are subjected to external loads along the longitudinal axis of the member. These loads either induces pure tension or compression in these members, as opposed to members in frames that are subjected to combine bending and compression. Axially loaded members are mostly found in pin-jointed structures such as lattice girders, bracing systems, pin-jointed trusses etc. Varieties of steel cross-sections such as angles, rectangular hollow sections (RHS), circular hollow sections (CHS), flat plate, ropes, cables etc. are employed as axially loaded members.
Verification of Axially loaded Tension Members
For tension members the expression below must be satisfied
$
\frac{N_{ED}}{N_{tRd}}\,\,\leqslant \,\,1
$
NEd is the design tensile force
NtRd is the tensile resistance. It should be taken as the design plastic resistance of the gross cross-section when the section is without holes.
i.e: $
N_{plRd}\,\,=\frac{A\,\,f_y}{\gamma _{mo}}\,\,
$
For sections with holes, the design tension resistance should be taken as smaller of the plastic resistance and the design ultimate resistance of the net cross-section at holes and fasteners.
i.e: NtRd = Ncr,T is the smaller of;
i) $
N_{tRd}\,\,=\frac{A\,\,f_y}{\gamma _{mo}}\,\,
$
2) $
N_{u.Rd}\,\,=\frac{0.9A_net\,\,f_u}{\gamma _{m2}}\,\,
$
Anet is the area of the plate in tension after deducting bolt holes
Effects of Eccentric Connections
Due to eccentric connections in tension members, such as when an angle section is connected through one leg or a channel section is connected through one flange, there is a localized secondary bending effect induced in members. These effects of asymmetric connection of tension members are allowed for in clause 3.10.3 of EN 1993-1-8. However, EN 1993-1-8 only considers only single row of bolts in angle sections. More elaborate guidance concerning design of tension members with eccentric connections, covering different cross-sections and bolts arrangement can be found in BS 5950-1 and UK engineers may resort to it.
Verification of Axially loaded Compression Members
For compression members the expression below must be satisfied
$
\frac{N_{ED}}{N_{cRd}}\,\,\leqslant \,\,1
$
NEd is the design compressive force
NcRd is the compression resistance.
i.e: $
N_{cRd}\,\,=\frac{A\,\,f_y}{\gamma _{m0}}\,\,
$ for class 1, 2, & 3 sections
$
N_{cRd}\,\,=\frac{A_eff\,\,f_y}{\gamma _{m0}}\,\,
$ for class 4 sections
Stability of Compression Members
Members subjected to axial compression can buckle in three ways, namely:
- Flexural buckling
- Torsional buckling
- Torsional-flexural buckling
Flexural Buckling
Flexural buckling can occur in all compression members. This is also known as struct buckling. It is the commonest buckling mode.
Torsional Buckling, and Torsional-Flexural Buckling
For members with open cross-sections, the possibility that the resistance of the member to either torsional or torsional-flexural buckling could be less than its resistance to flexural buckling should be considered. It should, however, be noted that these two buckling modes are not always critical for doubly symmetric I and H sections. Torsional buckling is often critical for cruciform sections, while torsional-flexural buckling mode is often the critical mode for asymmetric cross-sections.
Buckling Resistance of Compression Members
For all compression members regardless of mode of buckling, buckling resistance should be verified using the expression below:
$
\frac{N_{ED}}{N_{bRd}}\,\,\leqslant \,\,1
$
NEd is the design axial load
NbRd is the buckling resistance of a compression member.
$
N_{bRd}\,\,=\chi \frac{A\,\,f_y}{\gamma _m1}
$ for class 1, 2, & 3 sections
$
N_{bRd}\,\,=\chi \frac{A_eff\,\,f_y}{\gamma _m1}
$ for class 4 sections
χ is the reduction factor
The value of the reduction factor can be read off from the buckling curve given in figure 6.4 of the standard having determined the slenderness. Table 6.2 of the standard enables designers to determine the appropriate buckling curve for flexural buckling and other buckling modes for compression members. Alternatively, the reduction factor can be evaluated using the expression below.
$
\,\,\chi \,\,=\,\,\frac{1}{\phi \,\,+\,\,\sqrt{\phi ^2\,\,-\,\,\lambda ^2}}\,\,\leqslant \,\,1
$
Φ = 0.5 (1 + α(λ – 0.2) + λ²)
α is an imperfection factor and it is given in Table 6.1 of the standard (It should be noted that table 6.1 can only be consulted after determining the appropriate buckling curve using table 6.2)
λ is the non-dimension slenderness. This depends on the buckling mode concerned. Expression to evaluate non-dimension slenderness for the three buckling modes are given in the following sections.
Slenderness due to flexural buckling
$
\,\,\lambda \,\,=\,\,\sqrt{\frac{A\,\,f_y}{N_{cr}}}\,\,=\,\,\frac{L_{cr}}{i}\,\,\frac{1}{\lambda _1}
$ for class 1, 2 & 3 cross-sections
Lcr is the buckling length in the plane being considered.
Ncr is the elastic critical force for the relevant buckling mode based on gross cross-section properties
i is the radius of gyration about the relevant axis.
$
\,\,\lambda _1\,\,=\,\,\varPi \,\,\sqrt{\frac{E}{f_y}}\,\,=\,\,93.9\varepsilon
$
Slenderness due to Torsional & Torsional-Flexural Buckling
$
\,\,\lambda \,\,=\,\,\sqrt{\frac{A\,\,f_y}{N_{cr}}}\,\,$ for class 1, 2 & 3 cross-sections
Where;
Ncr = Ncr,T (For torsional buckling)
Ncr = Ncr,TF (For torsional-flexural buckling)
$
\,\,N_{cr,T}\,\,=\,\,\frac{1}{i_o^2}\,\,\left( GI_{T\,\,}+\,\,\frac{\varPi ^2EI_w}{L_T^2} \right)
$
$
i_o^2\,\,=\,\,i_y^2\,\,+\,\,i_z^2\,\,+\,\,y_0^2\,\,+\,\,z_0^2
$
$
N_{cr,TF}\,\,=\frac{N_{cr,y}}{2\beta}\,\,\left( \text{1}+\,\,\frac{N_{cr,T}}{N_{cr,y}}\,\,-\,\,\sqrt{\left( \text{1}-\,\,\frac{N_{cr,T}}{N_{cr,y}} \right) ^2\,\,+\,\,\left( \frac{y_o}{i_o} \right)}^2 \right)
$
$
\beta \,\,=\,\,\text{1}-\,\,\left( \frac{y_o}{i_o} \right) ^2
$
Buckling Length (Lcr )
Guidance on buckling length is restricted to informative Annex BB of the standard which gives information on few structures such lattice structures and triangulated structure. Although Annex BB should be sufficient for members in pure axial compression, there are a few calculations and parameters involved. Engineers can alternatively adopt Table 22 of BS 5950-1 which present a more straightforward approach to determine effective lengths (Effective length is the equivalent of buckling length in BS 5950-1)
References
EN 1993-1-1:2005: Eurocode 3, Design of Steel structures – Part 1:1: General rules for buildings.
BS 5950-1:2000: Structural Use of Steelwork in building – part 1 Code of practice for design – rolled and welded sections
Handbook of Structural Steelwork: Eurocode Edition
