## Design properties for rectangular hollow steel sections (RHS) and square hollow steel sections (SHS) according to EN1993-1-1

### Differences between Rectangular Hollow Sections (RHS) and Square Hollow Sections (SHS)

In general the procedure for calculating the properties of Square Hollow Sections (SHS) is the same as for Rectangular Hollow Sections (RHS).
The equations for Rectangular Hollow Sections (RHS) may also be used for Square Hollow Sections (SHS) by substituting *h* = *b*.
The rest of the text covers the more general case of Rectangular Hollow Sections (RHS).

### Definition of the cross-section

For typical Rectangular Hollow Sections (RHS) the geometric properties of the cross-section are defined in the following standards:

The geometric properties that define the cross-section are: height *h*, width *b* and wall thickness *t*.
In addition the rounding of the corners is defined by the outer rounding radius *r*_{o} and the inner rounding radius *r*_{i}.
The notation is defined in EN1993-1-1 §1.7 which is reproduced in the figure above.

The outer rounding radius *r*_{o} and the inner rounding radius *r*_{i} are specified as nominal values to be used in calculations in the standards and they are specified in the design standards EN 10210-2 §A.3 for hot finished structural hollow sections and EN 10219-2 §B.3 for cold formed structural hollow sections:

The actual rounding radii may vary within the tolerances allowed by the aforementioned standards and the sectional properties still remain valid.

### Geometric properties

The basic geometric properties of the cross-section are calculated by using the fundamental relations of mechanics.
Due to symmetry the centroid of the cross-section (center of mass) as well as the shear center are located at the center.
The geometric quantities include the total area of the cross section *A* and the second moments of the area *I*_{y} and *I*_{z} about the major axis y-y and about the minor axis z-z respectively.
They are calculated by adding the contribution of the external boundary and then subtracting the contribution of the internal void where the rounding of the corners is properly taken into account:

*A* = 2*t*⋅(*b* + *h* - 2*t*) - (4 - π)⋅(*r*_{o}^{2} - *r*_{i}^{2})

*I*_{y} = *b*⋅*h*^{3} / 12 - (*b* - 2⋅*t*)⋅(*h* - 2*t*)^{3} / 12 - 4(*I*_{g} + *A*_{g}⋅*h*_{g,y}^{2}) + 4(*I*_{ξξ} + *A*_{ξξ}⋅*h*_{ξ,y}^{2})

*I*_{z} = *h*⋅*b*^{3} / 12 - (*h* - 2⋅*t*)⋅(*b* - 2*t*)^{3} / 12 - 4(*I*_{g} + *A*_{g}⋅*h*_{g,z}^{2}) + 4(*I*_{ξξ} + *A*_{ξξ}⋅*h*_{ξ,z}^{2})

where the following auxiliary quantities are defined in EN 10210-2 §A.3 and EN 10219-2 §B.3:

*A*_{g} = (1 - π / 4)⋅*r*_{o}^{2}

*A*_{ξξ} = (1 - π / 4)⋅*r*_{i}^{2}

*h*_{g,y} = *h* / 2 - *r*_{o} ⋅ (10 - 3π) / (12 - 3π) and *h*_{g,z} = *b* / 2 - *r*_{o} ⋅ (10 - 3π) / (12 - 3π)

*h*_{ξ,y} = (*h* - 2*t*) / 2 - *r*_{i} ⋅ (10 - 3π) / (12 - 3π) and *h*_{ξ,z} = (*b* - 2*t*) / 2 - *r*_{i} ⋅ (10 - 3π) / (12 - 3π)

*I*_{g} = [1/3 - π/16 - 1 / (3⋅(12 - 3π))]⋅*r*_{o}^{4}

*I*_{ξξ} = [1/3 - π/16 - 1 / (3⋅(12 - 3π))]⋅*r*_{i}^{4}

### Shear area

The shear areas *A*_{v,z} and *A*_{v,y} for the case of rectangular hollow sections are specified in EN1993-1-1 §6.2.6(3) as:

__Load parallel to depth:__ *A*_{v,z} = *A*⋅*h* / (*b* + *h*)

__Load parallel to width:__ *A*_{v,y} = *A*⋅*b* / (*b* + *h*)

### 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.

*W*_{pl,y} = *b*⋅*h*^{2} / 4 - (*b* - 2*t*)⋅(*h* - 2*t*)^{2} / 4 - 4⋅*A*_{g}⋅*h*_{g,y} + 4⋅*A*_{ξξ}⋅*h*_{ξ,y}

*W*_{pl,z} = *h*⋅*b*^{2} / 4 - (*h* - 2*t*)⋅(*b* - 2*t*)^{2} / 4 - 4⋅*A*_{g}⋅*h*_{g,z} + 4⋅*A*_{ξξ}⋅*h*_{ξ,z}

### Torsional properties

The torsional constant *I*_{T} and the torsional modulus *W*_{T} are defined in EN 10210-2 §A.3 and EN 10219-2 §B.3.
The specified values are good approximations for the case of thin walled rectangular hollow sections with rounded corners.
In the following equations the quantity *h* in EN 10210 and EN 10219 is denoted as *h*_{0}:

*I*_{T} = *t*^{3}⋅*h*_{0} / 3 + 2⋅*K*⋅*A*_{h}

*W*_{T} = *I*_{T} / (*t* + *K* / *t*)

where the following auxiliary quantities are defined in EN 10210-2 §A.3 and EN 10219-2 §B.3:

*R*_{c} = (*r*_{o} + *r*_{i}) / 2

*A*_{h} = (*b* - *t*)⋅(*h* - *t*) - *R*_{c}^{2}⋅(4 - π)

*h*_{0} = 2⋅[(*b* - *t*) + (*h* - *t*)] - 2⋅*R*_{c}⋅(4 - π)

*K* = 2⋅*A*_{h}⋅*t* / *h*_{0}

### 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 torsional moment resistance

The design torsional moment resistance of the cross-section for the case of St. Venant torsion is specified in EN1993-1-1 §6.2.7.
The shear stress due to St. Venant torsion is derived from the theory of elasticity as specified above.
For elastic verification the yield criterion in EN1993-1-1 §6.2.1(5) is applied, i.e. the shear stress *τ* is limited the steel yield stress in pure shear *f*_{y} / √3:

*T*_{Rd} = *W*_{T} ⋅ ( *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 (web parallel to depth, web parallel to width) 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 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.

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 in pure compression |
Web in pure bending |

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

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

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

For the classification of the web parallel to depth: *c* = *h* - 2*t* - 2*r*_{i}

For the classification of the web parallel to width: *c* = *b* - 2*t* - 2*r*_{i}

### Buckling curves

The appropriate buckling curve for rectangular hollow sections and square hollow sections is specified in EN1993-1-1 Table 6.2 depending on the steel yield stress *f*_{y}, and whether the section is hot finished or cold formed as described in the following table:

Buckling curve of rectangular hollow sections and square hollow sections according to EN1993-1-1 Table 6.2
Type |
Steel Class |
Buckling Curve |

Hot finished |
S235, S275, S355, S420 |
a |

Hot finished |
S460 |
a0 |

Cold formed |
S235, S275, S355, S420 |
c |

Cold formed |
S460 |
c |