## Design properties for circular hollow steel sections (CHS) according to EN1993-1-1

### Definition of the cross-section

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

The geometric properties that fully define the cross-section are: external diameter *D* and wall thickness *t*.
The notation is defined in EN1993-1-1 §1.7 which is reproduced in the figure above.

### 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*, which is constant for any axis of bending due to symmetry.
They are calculated by adding the contribution of the external boundary and then subtracting the contribution of the internal void.

*A* = π⋅*D*^{2} / 4 - π⋅(*D*-2⋅*t*)^{2} / 4

*I* = π⋅*D*^{4} / 64 - π⋅(*D*-2⋅*t*)^{4} / 64

### Shear area

The shear area *A*_{v} for the case of circular hollow sections is specified in EN1993-1-1 §6.2.6(3) as:

*A*_{v} = 2⋅*A* / π

Due to symmetry the shear area is constant for shear load along any axis.

### Elastic section modulus

The elastic section modulus *W*_{el} is calculated by dividing the second moment of the area *I* with the corresponding distance from the centroid to the most distant edge:

*W*_{el} = *I* / (*D* / 2)

### Plastic section modulus

The plastic section modulus *W*_{pl} corresponds 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 a bending axis passing through the centroid.

The plastic section modulus *W*_{pl} is calculated by adding the contribution of the external boundary and then subtracting the contribution of the internal void.
The centroid of a semi-circular segment is located 2*D* / 3π from the center of the arc.
The following result is obtained:

*W*_{pl} = 2⋅(π⋅*D*^{2} / 4 / 2)⋅(2⋅*D* / 3⋅π) - 2⋅(π⋅(*D*-2⋅*t*)^{2} / 4 / 2)⋅(2⋅(*D*-2⋅*t*) / 3⋅π) ⇒

*W*_{pl} = [*D*^{3} - (*D*-2⋅*t*)^{3}] / 6

### Torsional properties

For circular cross-sections and tubular cross sections the torsional constant *I*_{T} is equal to the polar moment of inertia of the area *I*_{P} that is given by the following relation:

*I*_{T} = *I*_{P} = 2⋅*I*

For circular cross-sections and tubular cross sections the maximum shear stress due to St. Venant torsion *τ* is related to the maximum torque *T* by the following relation:

*T* = *τ* ⋅ *I*_{P} / (*D* / 2)

Therefore the torsional modulus *W*_{T} is obtained as:

*W*_{T} = *T* / *τ* = 2⋅*W*_{el}

### Design cross-section resistance

The design resistance of the cross-section in 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 applicable 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) local shell 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} 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} = *A*_{v} ⋅ ( *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} multiplied by the steel yield stress *f*_{y}:

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

The elastic bending moment resistance is applicable for class 3 cross-sections.
For the case of circular hollow sections that cannot be classified as Class 3 their strength and stability verifications should be based on EN 1993-1-6: Strength and Stability of Shell Structures.

#### 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} multiplied by the steel yield stress *f*_{y}:

*M*_{pl,Rd} = *W*_{pl} ⋅ *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 circular hollow sections (tubular sections) is specified in EN1993-1-1 Table 5.2.
The class of the cross-section in bending and/or compression depends on the ratio of its diameter *D* to the wall thickness *t*, 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}

The diameter to wall thickness limits *D* / *t* for cross-section classification according to EN1993-1-1 Table 5.2 are presented below:

Classification of tubular sections according to EN1993-1-1 Table 5.2
Ratio *D* / *t* |
Section Class |

*D* / *t* ≤ 50*ε*^{2} |
Class 1 |

*D* / *t* ≤ 70*ε*^{2} |
Class 2 |

*D* / *t* ≤ 90*ε*^{2} |
Class 3 |

*D* / *t* > 90*ε*^{2} |
See EN1993-1-6 |

For the case of circular hollow sections (tubular sections) that cannot be classified as Class 3 their strength and stability verifications should be based on EN 1993-1-6: Strength and Stability of Shell Structures.

### Buckling curves

The appropriate buckling curve for circular 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 circular 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 |