# Determining the compression resistance of a steel column using BS5950

The compression resistance of a column depends on its cross sectional area and effective length, but also on its section classification - slender or non-slender. When using BS5950-1:2000 the classification is determined by reference to Tables 11 or 12. For I and channel sections the flanges and web are classified separately and the section classification is taken as the worse of these.

The flange classification limits are fixed for different section types: e.g. for an I section the section will be slender if the flange slenderness (half the section width/flange thickness) exceeds 15e where e=√ (275/py) so the flange is either slender or it is not.

With webs the situation is more complicated: most web classification limits depend on the axial load being applied. For an I section the web is slender if its depth between fillets/web thickness exceeds 120e/(1+2r2) (min 40e). r2 is Fc (the applied load)/Ag.py (cross sectional area x grade stress). If r2 is zero (e.g. when the section is used as a beam subject to bending only, no axial load) the limit will be 120e, if large 40e.

If the web slenderness is less than 40e (as is the case with all UC's) then the web will never be slender. If it were to be larger than 120e (no rolled I sections are so) it would always be slender. Where the web slenderness falls between these (as is the case with many UB sections especially if Grade S355) then the classification may change with the load.

Consider a 406x140x39 UB S355: the web slenderness is 360.4/6.4=56.3, 63.9e. Ag.py is 1,764kN. If the applied load is less than 772kN then r2 will be 0.438 or less and the web limit will be at least 120x0.88/(1+2*0.438) = 56.3 so the web is classed as non-slender, and the maximum axial capacity will (theoretically) be Ag.pc (=py if very short) 1,764kN. However once the load exceeds 772kN r2 will be more than 0.438 and the web limit drops below the actual figure, thus changing the web classification to slender. The compression resistance, now given by Aeff.pc, drops to 1,457kN (rounded to 1,460kN in the 'Blue Book' D.4) which is the true compression resistance.

In most cases the pattern above occurs: when the threshold for the section is passed the compression resistance drops, but it is still higher than the applied load. In a few cases the compression resistance drops to below the applied load:

Consider a 300x300x6.3 SHS S355: the d/t ratio is 281/6.3 = 44.6. For the member to be classed as non-slender the Table 12 limit of 120e/(1+2r2) must be greater or equal to this and by calculation the load which gives this value is 1,786kN. If the load is less than this r2 will be 0.683 or less giving a limiting web slenderness of 44.6 or more: as the actual figure is 44.6 the section will be classed as non-slender. If the load is greater r2 will be more than 0.683 giving a limiting web slenderness of less than 44.6 and the section will be classed as slender.

With an effective length of 8m the non-slender compression resistance (Ag.pc) is 2,061kN. However with this load applied the member has to be classed as slender so the compression resistance falls to 1,755kN (Aeff.pc). The applicable compression resistance is therefore the maximum load at which the member is still non-slender, i.e. 1,786kN (1,790kN rounded) as shown in the 'Blue Book' table D-12, though at first sight it might be hard to see how this figure was arrived at. This table may help:

Effective Length | Compression resistance kN when member is classed as non-slender (load<1,786kN) | Compression resistance kN when member is classed as slender (load>=1,786kN) | Effective Compression resistance kN |

7.0m | 2,222 | 1,846 | 1,846 |

8.0m | 2,061 | 1,755 | 1,786* |

9.0m | 1,862 | 1,637 | 1,786* |

10.0m | 1,646 | N/A | 1,646 |

Italics: member is slender with this load applied. * Keep load to the figure at which the member is still classed as non-slender

Where the member is subject to an axial load and bending, then the higher compression resistance can be used in the unity equation if the compression load is such that the member is non-slender ('Blue Book' D-115, 1,760kN). Considering the 406x240x39 UB again if an axial load of more than 772kN were to be applied the member would become slender and the lower compression resistance would need to be used. In addition the moment resistance would drop sharply, but this might be an appropriate section where the axial load is low but some degree of bending about the major axis. It will almost never be sensible to choose section that is classified as a slender for a column.

Finally, it should be pointed out that these step changes in compression resistance are a reflection of the BS's attempt to keep complexity within limits. A more refined analysis would show a transitional reduction in compression resistance as the load reached the level at which the member becomes slender, but we may perhaps be grateful to have been spared this!