## EC3 solver

### General

The overall approach has been to comply exactly with EC3 where possible, using the highest tier solution where possible.

No attempt has been made to comply with specific National Annex’s. Instead, some general options have been included. Refer Figure “Eurocode parameters”.

All sections supported by BCDsteel are analysed to EC3 except for Class 4 round hollow sections.

### Resistance of cross-sections

All section types supported by BCDsteel are analysed for Class 2 to 4. Class 1 is simply treated as Class 2. The exception is Class 4 round hollow sections which are presently not supported.

For bending about a principal axis, there are 2 values of cross-section resistance MRd corresponding to compression first on 1 side of the neutral axis and then on the other. For a section that is symmetrical about an axis, the 2 values are identical. For a section that is unsymmetrical about an axis and for which the 2 capacities are not identical, BCDsteel uses the lower value. This is conservative. However, it does penalise T-sections in particular. Other sections that may be affected include unequal flange I-sections, channels and angles.

The cross-section resistance clauses are evaluated using the forces and section properties at each of 9 equi-spaced sections along each member. Two extra sections are used for all calculations except clauses 6.2.9, 6.2.10, 6.3.3. These are the sections at which MyEd, MzEd are a maximum.

**6.2.9 Bending and axial force**

For Class 2 sections, BCDsteel calculates an “exact” solution to (6.31) for all section types supported and for any combination of N My Mz without needing to use the approximate equations in 6.2.9.1.

For round hollow sections and solid round sections, BCDsteel uses an iterative solver to calculate the plastic section resistance for any combination of N My Mz.

For all other section types, BCDsteel generates 3D plastic yield surfaces. These are complex shapes as shown in the figures below. A plastic yield surface represents a fully plastic stress distribution for any combination of N My Mz. BCDsteel interpolates to the surface to determine the Utilisation Ratio. For a Class 2 section, a point on the surface has Utilisation Ratio = 1.0, and any point outside the surface has Utilisation Ratio > 1.0.

Class 4 cross-sections are analysed using 6.2.9.3(1). The stresses are calculated using effective section properties and the forces and moments acting simultaneously as per footnote to 4.3(3) and 4.3(4) of EN 1993-1-5. This analysis takes into account the shift eN of the centroid as per Figure 4.1 of EN 1993-1-5.

### Buckling resistance of members

**6.3.1 Compression buckling resistance**

Uniform members subject to axial compression (with or without bending) are analysed as though the member was subject to compression only, (ignoring any bending that may be present), using the maximum compression along the member, and using the Class for that section property when subject to compression only.

For non-uniform members, the members are analysed using the “minimum” section properties: The members are analysed as an equivalent uniform member using the section properties and corresponding Class at each of 9 equi-spaced sections along the member. The Utilisation Ratio reported is the maximum value from all 9 analyses.

Torsional and flexural torsional buckling is calculated for all section types including hollow sections.

**6.3.2 Bending buckling resistance**

For uniform members subject to bending about the major axis, (6.54) is calculated for each segment assuming the member was subject to bending about the major axis only, (ignoring axial force and bending about the minor axis), using the Class for that section property when subject to bending about the major axis only. For a given load combination, the Utilisation Ratio is the maximum value of (6.54) for all segments.

For non-uniform members, (6.54) is calculated for each segment assuming an equivalent uniform segment using the section properties at 9 equi-spaced sections along the segment. The Utilisation Ratio for a segment is the maximum value of (6.54) calculated for all sections along the segment. For a given load combination, the Utilisation Ratio for the member is the maximum Utilisation Ratio for all segments.

BCDsteel calculates the elastic critical moment for lateral-torsional buckling Mcr in accordance with Reference (1).

**6.3.3 Bending and axial compression**

Equations (6.61) and (6.62) are analysed using Annex B. Annex A is not available at this time.

BCDsteel analyses both uniform and non-uniform members using either of the following 2 methods:

- Using the “critical” section properties: The members are analysed as an equivalent uniform member using the section properties and Class at the critical section. The critical section is defined here as the cross-section with the largest Utilisation Ratio from clauses 6.2.3 to 6.2.10.
- Using the “minimum” section properties: The members are analysed as an equivalent uniform member using the section properties and corresponding Class at each of 9 equi-spaced sections along the member. The Utilisation Ratio reported is the maximum value from all 9 analyses.

An option is included to de-rate certain Class 1 and 2 sections as Class 3. This is a requirement specified in some National Annex’s. This option is not applicable for (i) doubly symmetric I-sections and (ii) all rectangular hollow sections (including IBoxed and Stiffened Box sections).

**Annex B**

The interaction factors Kij and the equivalent uniform moment factors Cm in Tables B.1, B.2, B.3 take into account intermediate restraints in both the major and minor axes directions.

Table B.3 makes a distinction between uniform loading and concentrated loading. BCDsteel determines whether it is uniform loading or concentrated loading from the bending moment diagrams and interpolates between them when necessary. No user input is required.

Table B.3 makes a distinction braced and sway members. BCDsteel determines whether it is braced or sway from the effective length factors for axial compression. Again, no user input is required.

**Reference**

(1) N.S. Trahair, M.A. Bradford, D.A. Nethercot, and L. Gardner. The Behaviour and Design of Steel Structures to EC3 Fourth edition