9 Axial force and bending tables
Generally, members subject to axial compression forces and bending should be verified for cross section resistance (EN 1993-1-1, 6.2.9) and for member buckling resistance (EN 1993-1-1, 6.3.3.(4)).
The relevant parameters required to evaluate the interaction equations given in the above clauses are presented in tabular form.
The tables are applicable to members subject to combined tension and bending and to members subject to combined compression and bending. However, the values in the tables are conservative for tension, as the more onerous compression section classification limits have been used for calculating resistances to axial force.
Tables are given for cross-section resistances and for member buckling resistances, as explained below.
9.1 Cross-section resistance
EN 1993-1-1 For Class 1, 2 and 3 cross sections the following conservative approximation may be used:
Eq (6.2) |
For CHS, SHS and RHS, values of N_{pl,Rd}, M_{c,y,Rd} and M_{c,z,Rd} are given in the tables. The section classification depends on the axial force applied and therefore M_{c,Rd} values are given for different values of N_{Ed} / N_{pl,Rd}. The values of these resistances have been calculated as described in the following sections.
No M_{c,y,Rd} and M_{c,z,Rd} values are given for combined axial force and bending on Class 4 sections. Refer to EN 1993-1-1, 6.2.9.3(2).
The symbol * denotes that the section is Class 4 when fully stressed under axial compression only.
For elliptical hollow sections, values of N_{pl,Rd}, M_{c,y,Rd} and M_{c,z,Rd} are given in the tables.
If an elliptical hollow section is Class 4, the design compression and bending resistances are calculated based on the area and section modulus appropriate for a Class 3 section, using a reduced design strength such that the section is Class 3. The reduced design strength is calculated using the expression given in Section 6.2.
(a) Design resistance for axial force
where:
A | is the gross area of the cross-section |
γ_{M0} | is the partial factor for cross-sections |
It should be noted that N_{pl,Rd} for a Class 4 elliptical hollow section is based on the gross area and a reduced design strength, as described above.
(b) Design resistances for bending about the principal axes of the cross section
The design resistances for bending about the major and minor axes, M_{c,Rd} (for CHS and SHS), M_{c,y,Rd} and M_{c,z,Rd} (for RHS and elliptical hollow sections) have been calculated as in Section 7 using W_{pl,y}, W_{el,y} and W_{pl,z} and W_{el,z} as appropriate. No values are given for the reduced resistance in the presence of high shear.
M_{c,Rd} values for Class 4 cross sections are not given in the tables. For checks of Class 4 cross sections, see EN 1993-1-1, 6.2.9.3(2).
If an elliptical hollow section is Class 4, the design resistances are calculated based on the section modulus appropriate for a Class 3 section and a reduced design strength such that the section is Class 3. When subject to combined axial compression and bending, elliptical hollow sections are classified for axial compression, which may be more onerous than major axis bending.
The symbol ~ indicates that the section is Class 4 for a given value of N_{Ed} / N_{pl,Rd}.
The symbol $ indicates that the section is Class 4 and that it would be overloaded due to compressive axial force alone i.e. the section is Class 4 and N_{Ed} > A_{eff} f_{y} for the given value of N_{Ed} / N_{pl,Rd}.
The Class 2 and Class 3 limits for the section subject to bending and compression are the maximum values of N_{Ed} / N_{pl,Rd} up to which the section is either Class 2 or Class 3, respectively. The Class 2 limit is given in bold type.
Limits are not given for elliptical hollow sections; the classification depends purely on the geometry of the section.
As an alternative to Equation (6.2) for Class 1 and 2 CHS, SHS and RHS sections a less conservative criterion may be used as follows:
M_{Ed} ≤ M_{N,Rd} | Eq (6.31) |
Values of M_{N,Rd} are given in the tables.
(c) Design moment resistance reduced due to axial force
The reduced design resistances for bending have been calculated as follows:
(i) Circular hollow sections
No information is available in EN 1993-1-1 or EN 1993-1-6 for the resistance of cross sections of circular hollow sections under combined axial force and bending. The values of M_{N,Rd} have been calculated using the methodology adopted in Steelwork Design Guide to BS 5950-1: 2000, Volume 1 - Section properties member capacities (SCI publication P202)^{[8]}. For Class 1 and 2 sections, the expression is as follows:
where:
n | = N_{Ed} / N_{pl,Rd} |
(ii) Rectangular and square hollow sections
For bending about the y-y axis:
M_{N,y,Rd} = | but ≤ M_{pl,y,Rd} | Eq (6.39) |
where:
a_{w} | = but ≤ 0.5 |
M_{pl,y,Rd} | is the plastic resistance moment about the y-y axis |
n | is the ratio N_{Ed} / N_{pl,Rd} |
Because the values of M_{N,y,Rd} are only valid for Class 1 and Class 2 sections, no values are shown when N_{Ed} / N_{pl,Rd} exceeds the limit for a Class 2 section (shown as the symbol '–' in the tables).
For square hollow sections, the reduced moment resistance is the same for both axes, and is displayed as M_{N,Rd}.
For bending about the z-z axis:
M_{N,z,Rd} = | but ≤ M_{pl,z,Rd} | Eq (6.40) |
where:
A_{f} | = but ≤ 0.5 |
M_{pl,z,Rd} | is the plastic resistance moment about the z-z axis |
n | is the ratio N_{Ed} / N_{pl,Rd} |
Because the values of M_{N,z,Rd} are only valid for Class 1 and Class 2 sections, no values are shown when N_{Ed} / N_{pl,Rd} exceeds the limit for a Class 2 section (shown as the symbol '–' in tables).
9.2Member buckling resistance
Interaction equations (6.61) and (6.62) from EN 1993-1-1, 6.3.3(4) must be satisfied for members subject to combined axial compression and bending. To check these interaction equations the following parameters given in (a), (b) and (c) are needed:
These are given in the tables as N_{b,y,Rd} and N_{b,z,Rd} for rectangular hollow sections and elliptical hollow sections, and as N_{b,Rd} for circular and square hollow sections. They are the compression resistances for buckling about the y-y and the z-z axis respectively. For CHS, SHS and RHS, the adjacent n limit (i.e. N_{Ed} / N_{pl,Rd}) ensures that the section is not Class 4 and has been calculated as in Section 0. For Class 4 elliptical hollow sections, the design compression is calculated based on the gross area, using a reduced design strength such that the section is Class 3.
When subject to combined axial compression and bending, elliptical hollow sections are classified for axial compression, which may be more onerous than major axis bending.
This term is given in the table as M_{b,Rd} for rectangular and elliptical hollow sections. For rectangular hollow sections, values are given for two N_{Ed} / N_{pl,Rd} limits. The higher limit ensures the section is Class 3 and the lower limit (in bold) ensures the section is Class 2, M_{b,Rd} is calculated accordingly. The reduction factor χ_{LT} is calculated for the 'rolled sections' case using the following buckling curves, with _{LT,0} = 0.4 (Section 0, as recommended in Reference 6) and β = 0.75.
Table 9.1
Cross-section | Limits | Buckling curve |
Hot-finished hollow section | h / b ≤ 2.0 | b |
2.0 < h / b ≤ 3.1 | c | |
Cold-formed hollow section | h / b ≤ 2.0 | c |
2.0 < h / b ≤ 3.1 | d |
For circular and square hollow sections, M_{b,Rd} is not applicable and is therefore not given in the tables.
M_{z,Rk} may be based on the elastic or plastic modulus, depending on the section classification, as shown in Table 6.7 of EN 1993-1-1. Values of f_{y} W_{el,z} are given (for Class 4 elliptical hollow sections, a reduced value of f_{y} is used with the elastic modulus). Values of f_{y} W_{pl,z} are given as M_{c,z,Rd} (M_{c,Rd} for CHS) in the tables for cross section resistance check.
In addition, values of f_{y} W_{el,z} and N_{pl,Rd} are given in the tables for completeness.
The UK and German National Annexes permit the use of either Annex A or Annex B of EN 1993-1-1 to calculate the interaction factors k for circular, square and rectangular hollow sections. The French National Annex permits the use of Annex A only and the Dutch National Annex permits the use of Annex B only.
For CHS, SHS and RHS, the symbol * denotes that the cross section is Class 4 under axial compression only (due to the web becoming Class 4). Under combined axial compression and bending, the class of the cross-section depends on the axial force, expressed in terms of n (= N_{Ed} / N_{pl,Rd}). Values of n are given, up to which the cross-sections would be Class 2 or Class 3.
The limits in normal and bold type are the maximum values up to which the section is either Class 3 or Class 2, respectively. The tabulated resistances are only valid up to the given N_{Ed} / N_{pl,Rd} limit.
For certain rectangular hollow sections, webs and flanges are Class 4. In these cases section resistances provided are based on effective section properties. These sections are indicated by the ♦ symbol.