## Design properties for flanged steel profiles (IPE, HEA, HEB, HEM) according to EN1993-1-1

### Definition of the cross-section

For typical steel profiles (IPE, HEA, HEB, HEM, UB, UC, UBP) the geometric properties of the cross-section are defined in the following standard: EN 10365 - Hot rolled steel channels, I and H sections - Dimensions and masses.

The European Standard EN 10365 has superseded the following older standards:

Geometry of IPE profiles: Euronorm 19 – 57, DIN 1025/5

Geometry of HEA profiles: Euronorm 53 – 62, DIN 1025/3

Geometry of HEB profiles: Euronorm 53 – 62, DIN 1025/2

Geometry of HEM profiles: Euronorm 53 – 62, DIN 1025/4

Geometry of UB, UC ,UBP profiles: BS 4-1

The geometric properties that fully define the cross-section are: total height *h*, flange width *b*, web thickness *t*_{w}, flange thickness *t*_{f}, and root radius *r*.
The notation is defined in EN1993-1-1 §1.7 which is reproduced in the figure above.

The root radius *r* is not specified in the latest standard EN 10365 because it varies with individual manufacturers.
In this calculations the root radius *r* is considered as specified in the older standards.

### Geometric properties

The basic geometric properties of the cross-section are calculated by using the fundamental relations of mechanics.
The geometric quantities include the total area of the cross section *A* and the second moments of the area about the major axis *I*_{y} and about the minor axis *I*_{z}, where the orientation of the major axis of bending y-y and the minor axis of bending z-z is specified in EN1993-1-1 §1.7 which is reproduced in the figure above.
The root fillets are taken into account in the calculated geometric properties.
Due to symmetry the centroid of the cross-section (center of mass) as well as the shear center are located in the middle of the height and width.

### Shear area

For shear load parallel to the web the shear area *A*_{v,z}, for the case of rolled I and H sections, is specified in EN1993-1-1 §6.2.6(3) as:

*A*_{v,z} = max( *A* - 2*b**t*_{f} + (*t*_{w} + 2*r*)*t*_{f} , *η**t*_{w}*h*_{w} )

where according to EN1993-1-1 §5.1 the value of the coefficient *η* is assumed equal to *η* = 1.2 for steel grades up to and including S460 and *h*_{w} = *h* - 2*t*_{f} is the height of the web.

For shear load parallel to the flanges the corresponding shear area *A*_{v,y} is not specified in EN1993-1-1 for the case of rolled I and H sections.
In the provided tables the shear area *A*_{v,y} is assumed equal to the sum of areas of the flanges only, which is a reasonable conservative assumption:

*A*_{v,y} = 2*b**t*_{f}

### Elastic section modulus

The elastic section modulii *W*_{el,y} and *W*_{el,z} about the major axis y-y and the minor axis z-z respectively are calculated by dividing the second moment of the area *I*_{y} and *I*_{z} with the corresponding distance from the centroid to the most distant edge:

*W*_{el,y} = *I*_{y} / (*h* / 2)

*W*_{el,z} = *I*_{z} / (*b* / 2)

### Plastic section modulus

The plastic section modulii *W*_{pl,y} and *W*_{pl,z} about the major axis y-y and the minor axis z-z respectively correspond to the maximum plastic bending moment when the axial force of the cross-section is zero and the stress profile is fully plastic.
Due to symmetry when the full plastic bending stress profile is reached with zero axial force the section is divided into two parts separated by the axis of symmetry.
The plastic section modulus corresponds to the sum of first moments of the area of the two halves about the major axis y-y and the minor axis z-z respectively.

### Torsional and warping properties

For open thin-walled cross-sections the torsional constant *I*_{T}, torsional modulus *W*_{T}, warping constant *I*_{w}, and warping modulus *W*_{w} may be calculated according to the procedure described in EN1993-1-3 Annex C.
For stock profiles IPE, HEA, HEB, HEM the values presented in the tables for the torsional and warping properties are accurate results obtained from finite element analysis of the cross-section and they are reproduced from Table 1 of the following scientific paper: M.Kraus & R. Kindmann, 'St. Venants Torsion Constant of Hot Rolled Steel Profiles and Position of the Shear Centre'.
The presented values take into account the actual thickness of the cross-section elements and the presence of the root fillets.
For stock profiles UB, UC, UBP torsional constant *I*_{T} and warping constant *I*_{w} are results provided by the manufacturer 'British Steel.
For the calculation of the warping modulus W_{w} the maximum warping ordinate max*ω* is calculated based on the assumption of thin-walled cross-section in accordance with EN1993-1-3 Annex C.

### Design cross-section resistance

The design resistance of the cross-section for axial force, shear force, and bending moment are calculated in accordance with EN1993-1-1 §6.2.
They correspond to the gross cross-section resistance reduced by the steel partial material safety factor for cross-section resistance *γ*_{M0} that is specified in EN1993-1-1 §6.1 for buildings, or the relevant parts of EN1993 for other type of structures, and the National Annex.

The aforementioned design resistances do not take into account a) flexural buckling, b) lateral torsional buckling, c) interaction effects of axial force, shear force, bending moment, and d) interaction effects of biaxial bending.
Therefore the presented cross-section resistances are indicative values applicable for special cases.
In general the overall element resistance is smaller and must be verified according to the relevant clauses of EN1993-1-1 Section 6.

#### Design axial force resistance

The design plastic resistance of the cross-section in uniform tension is specified in EN1993-1-1 §6.2.3(2).
The design plastic resistance of the cross-section in uniform compression for cross-section class 1, 2, 3 is specified in EN1993-1-1 §6.2.4(2).
The aforementioned axial force resistances correspond to the gross cross-sectional area *A* and the steel yield stress *f*_{y}:

*N*_{pl,Rd} = *A*⋅*f*_{y} / *γ*_{M0}

#### Design shear force resistance

The design plastic shear resistance of the cross-section is specified in EN1993-1-1 §6.2.6(2).
It corresponds to the relevant shear area *A*_{v,z} or *A*_{v,y}, for shear force along the axis z-z and y-y respectively, multiplied by the steel yield stress in pure shear *f*_{y} / √3 corresponding to the yield criterion in EN1993-1-1 §6.2.1(5)::

*V*_{pl,Rd,z} = *A*_{v,z} ⋅ ( *f*_{y} / √3 ) / *γ*_{M0}

*V*_{pl,Rd,y} = *A*_{v,y} ⋅ ( *f*_{y} / √3 ) / *γ*_{M0}

#### Design elastic bending moment resistance

The design elastic bending moment resistance of the cross-section is specified in EN1993-1-1 §6.2.5(2).
It corresponds to the relevant elastic section modulus *W*_{el,y} or *W*_{el,z}, for bending about the major axis y-y or about the minor axis z-z respectively, multiplied by the steel yield stress *f*_{y}:

*M*_{el,Rd,y} = *W*_{el,y} ⋅ *f*_{y} / *γ*_{M0}

*M*_{el,Rd,z} = *W*_{el,z} ⋅ *f*_{y} / *γ*_{M0}

The elastic bending moment resistance is applicable for class 3 cross-sections.
For class 4 cross-sections the effective cross-section properties must be defined that take into account the reduced effective widths of the compression parts of the cross-section as specified in EN1993-1-1 §6.2.2.5.

#### Design plastic bending moment resistance

The design plastic bending moment resistance of the cross-section is specified in EN1993-1-1 §6.2.5(2).
It corresponds to the relevant plastic section modulus *W*_{pl,y} or *W*_{pl,z}, for bending about the major axis y-y or about the minor axis z-z respectively, multiplied by the steel yield stress *f*_{y}:

*M*_{pl,Rd,y} = *W*_{pl,y} ⋅ *f*_{y} / *γ*_{M0}

*M*_{pl,Rd,z} = *W*_{pl,z} ⋅ *f*_{y} / *γ*_{M0}

The plastic bending moment resistance is applicable for class 1 or 2 cross-sections.

### Cross-section class

The classification of cross-sections is specified in EN1993-1-1 §5.5.
The role of the classification is to identify the extent to which the resistance and rotation capacity of the cross-section are limited by local buckling of its parts.

Four section classes are identified:

__Class 1:__ Plastic bending moment resistance develops and plastic hinge develops with rotation capacity adequate for plastic analysis.

__Class 2:__ Plastic bending moment resistance develops but the rotation capacity is limited by local buckling.

__Class 3:__ Elastic bending moment resistance develops but local buckling prevents the development of plastic resistance.

__Class 4:__ Elastic bending moment resistance cannot develop because local buckling occurs before the yield stress is reached at the extreme fiber. Effective widths are used to account for the effects of local buckling of compression parts.

The classification of the cross-section parts (flanges and web) is specified in EN1993-1-1 Table 5.2.
The class of the compression part depends on its width *c* to thickness *t* ratio, adjusted by the factor *ε* that takes into account the value of the steel yield stress *f*_{y}:

*ε* = (235 MPa / *f*_{y})^{0.5}

In general the class of the compression part is more unfavorable when it is subjected to uniform compression, as compared to pure bending.
Indicative classification of the flanges and webs of the steel profiles is presented for the characteristic cases of pure uniform compression and pure bending moment.
In general the class may have an intermediate value if the stress profile of the compression part occurs from a combination of compressive axial force and bending moment.
The classification of the total cross-section is determined by the class of its most unfavorable compression part, web or flange.

The examined width to thickness limits *c* / *t* for cross-section classification according to EN1993-1-1 Table 5.2 are presented below:

Width to thickness limits for cross-section classification according to EN1993-1-1 Table 5.2
Class |
Web |
Outstand Flanges |

Web in pure compression |
Web in pure bending |
Flanges in pure compression due to axial force or bending moment |

Class 1 |
*c* / *t* ≤ 33*ε* |
*c* / *t* ≤ 72*ε* |
*c* / *t* ≤ 9*ε* |

Class 2 |
*c* / *t* ≤ 38*ε* |
*c* / *t* ≤ 83*ε* |
*c* / *t* ≤ 10*ε* |

Class 3 |
*c* / *t* ≤ 42*ε* |
*c* / *t* ≤ 124*ε* |
*c* / *t* ≤ 14*ε* |

For the classification of the webs *t* = *t*_{w} and *c* = *h* - 2*t*_{f} - 2*r*.

For the classification of the outstand flanges *t* = *t*_{f} and *c* = *b* / 2 - *t*_{w} /2 - *r*.

### Buckling curves

The appropriate buckling curve for rolled flanged sections is specified in EN1993-1-1 Table 6.2 depending on the aspect ratio *h*/*b*, the flange thickness *t*_{f}, the steel yield stress *f*_{y}, and the orientation of bending axis.