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001656

Dipl.-Ing. (FH) René Flori

2020-09-11

Design of K-Type Joint Using CHS Sections According to EN 1993-1-8

Closed circular cross-sections are ideal for welded truss structures. The architecture of such constructions is popular when designing transparent roofs. This article shows the special features of the connection design using hollow sections.

1 Spatial Truss Girder

General

Slender truss structures made of closed cross-sections are popular in architecture. Due to computer-aided production of cuts and connection geometries, it is possible also to have complex spatial nodes. This technical article deals with the design of a K‑node. The node definition and the design, in particular, are described in detail.

Roof Structure, Truss Girder, Curved, Made of CHS

2 Curved Spatial Trusses Made of Welded CHS

Details of Model

Material: S355
Chord cross-section: RO 108x6.3 | DIN 2448, DIN 2458
Strut cross-section: RO 60.3x4 | DIN 2448, DIN 2458
Dimensions: see the graphic

3 Truss Dimensions

Assigning Node to Joint Type

The connection type for a hollow section node is not only defined by the geometry, but also by the orientation of the axial forces in the struts. In our example, there is a tensile force in Member 35 (Strut 1) and a compression force in Member 36 (Strut 2) on Node 28 to be designed. Given this distribution of internal forces and moments, the joint type is a K‑node. If there was compression or tension in both struts, the connection type would be a Y‑node.

4 Distribution of Internal Forces for Axial Force

Checking Validity Limits

Compliance with the validity limits is crucial for any design. The ratio of the diameters of struts and chords is important. If it is not in the range of 0.2 ≤ d_i / d_o ≤ 1.0, it is impossible to perform the design. The diameter ratio d_i / d_o is also referred to as β. The European standard EN 1993-1-8 [1] specifies in Table 7.1 the validity limits for struts and chord members, as well as a limitation of the struts' overlap. If you want to have a gap between the struts, a minimum dimension of g ≥ t₁ + t₂ must be observed, too. t is the respective wall thickness of the struts. Structural components subjected to compression also need to be classified into cross-section classes 1 or 2. A corresponding check is carried out according to EN 1993-1-1 [2], Chapter 5.5.

5 Displaying Window 1.3 Geometry in RF-/HSS

Design

In our example, the connection meets the validity limits according to Table 7.1. Therefore, it is sufficient according to EN 1993-1-8, Section 7.4.1 (2), to analyze the chord member for flange failure and punching shear.

Flange Failure of Chord Member due to Axial Force According to EN 1993-1-8, Table 7.2, Row 3.2

Determination of Diameter-Wall Ratio γ

γ = \frac{d_{0}}{2 \cdot t_{0}}

γ	Ratio of the chord member's width or diameter to the double of its wall thickness
d₀	Total diameter of the chord member
t₀	Wall thickness of the chord cross-section

Determination of Diameter-Wall Thickness Ratio γ

γ = 8.57

Determination of Factor k_g

k_{g} = γ^{0.2} \cdot (1 + \frac{0.024 \cdot γ^{1.2}}{1 + e^{(0.5 \cdot g / t_{0} - 1.33)}})

k_g	Factor for nodal connections with a gap g
γ	Ratio of the chord member's width or diameter to the double of its wall thickness
e	Euler's number
g	Gap width between the struts of a K or N joint
t₀	Wall thickness of the chord cross-section

Factor k_g According to EN 1993-1-8, Table 7.2

k_g = 1.72

Determination of Chord Tension Coefficient k_p

k_{p} = 1 + 0.3 \cdot n_{p} \cdot (1 - n_{p})

n_{p} = \frac{f_{p}}{f_{y}}

f_{p} = \frac{N_{p}}{A_{0}} - \frac{|M_{0}|}{W_{0}}

k_p	Chord prestress coefficient
n_p	Ratio
f_p	Value of acting compressive stress in the chord member without stresses due to components of strut forces at the connection parallel to the chord
f_y	Yield strength
N_p	Starting axial compressive force in the chord
A₀	Cross-sectional area of the chord member
M₀	Secondary moment from eccentricity
W₀	Elastic section modulus of the chord cross-section

Chord Prestress Coefficient k_p EN 1993-1-8

f_p is the chord tension from the axial force N_p and the additional moment from eccentricity. Since there is compression and tension in the chors, it is assumed that N_p = 0. Furthermore, the eccentricity of the joint is so small that an additional moment from any eccentric connection of the struts does not have to be considered. The auxiliary coefficient is thus zero. The sign convention for compression and tension forces in RFEM and RSTAB differs from those of the European standard EN 1993-1-8. Therefore, the formula is adjusted to k_p.
k_p = 1.0

Determination of Allowable Limit Internal Force N_Rd

N_{1, Rd} = \frac{k_{g} \cdot k_{p} \cdot f_{y 0} \cdot {t_{0}}^{2}}{\sin (θ_{1})} \cdot (1.8 + 10.2 \cdot \frac{d_{1}}{d_{0}}) / γ_{M 5}

N_{2, Rd} = \frac{\sin (θ_{1})}{\sin (θ_{2})} \cdot N_{1, Rd}

N_1,Rd	Design value of the axial force resistance of the connection for Strut 1
k_g	Factor for nodal connections with a gap g
k_p	Chord prestress coefficient
f_y0	Yield strength of material of the chord member
t₀	Wall thickness of the chord cross-section
θ₁	Enclosed angle between Strut 1 and the chord member
d₁	Total diameter of Strut 1
d₀	Total diameter of the chord member
y_M5	Partial safety factor
N_2,Rd	Design value of the axial force resistance of the connection for Strut 2
θ₂	Enclosed angle between Strut 2 and the chord member

Flange Failure According to EN 1993-1-8, Table 7.2, Row 3.2

N_1,Rd = N_2,Rd = 257.36 kN
N_1,Ed / N_1,Rd = 197.56 / 257.36 = 0.77 < 1.0
N_2,Ed / N_2,Rd = 186.89 / 257.36 = 0.73 < 1.0

Punching of Chord Member due to Axial Force According to EN 1993-1-8, Table 7.2, Row 4

Determination of Allowable Limit Internal Force N_Rd

N_{i, Rd} = \frac{f_{y 0}}{\sqrt{3}} \cdot t_{0} \cdot π \cdot d_{i} \cdot \frac{1 + \sin (θ_{i})}{2 \cdot \sin^{2} (θ_{i})} / γ_{M 5}

N_i,Rd	Design value of the axial force resistance of the connection for a structural component i
f_y0	Yield strength of material of the chord member
t₀	Wall thickness of the chord cross-section
π	Circle number
d_i	Total diameter for CHS components i
θ_i	Enclosed angle between the strut i and the chord member
γ_M5	Partial safety factor

Punching Chord Member due to Axial Force According to EN 1993-1-8, Table 7.2, Row 4

N_1,Rd = N_2,Rd = 417.58 kN
N_1,Ed / N_1,Rd = 197.56 / 417.58 = 0.47 < 1.0
N_2,Ed / N_2,Rd = 186.89 / 417.58 = 0.45 < 1.0

Flange Failure of Chord Member due to Moment M_op According to EN 1993-1-8, Table 7.5, Row 2

This design is only relevant for 3D structures where moments may also occur from the truss plane.

Determination of Allowable Limit Internal Force M_op,Rd

M_{op, i, Rd} = \frac{f_{y 0} \cdot {t_{0}}^{2} \cdot d_{i}}{\sin (θ_{i})} \cdot \frac{2.7}{1 - 0.81 \cdot β} \cdot k_{p} / γ_{M 5}

M_op,i,Rd	Design value of the moment resistance of the connection for bending out of plane of the structural system for the structural component i
f_y0	Yield strength of material of the chord member
t₀	Wall thickness of the chord cross-section
d_i	Total diameter for CHS components i
θ_i	Enclosed angle between the strut i and the chord member
β	Ratio of mean diameters or mean widths of the strut and the chord
k_p	Chord prestress coefficient
y_M5	Partial safety factor

Flange Failure of Chord Member due to Moment Mₒₚ According to EN 1993-1-8, Table 7.5, Row 2

M_op,1,Rd = M_op,2,Rd = 5.92 kNm
M_op,1,Ed / M_op,1,Rd = 0.08 / 5.92 = 0.01 < 1.0
M_op,2,Ed / M_op,2,Rd = 0.01 / 5.92 = 0.00 < 1.0

Flange Failure of Chord Member due to Moment M_ip According to EN 1993-1-8, Table 7.5, Row 1

Determination of Allowable Limit Internal Force M_ip,Rd

M_{ip, i, Rd} = 4.85 \cdot \frac{f_{y 0} \cdot {t_{0}}^{2} \cdot d_{i}}{\sin (θ_{i})} \cdot \sqrt{γ} \cdot β \cdot k_{p} / γ_{M 5}

M_ip,i,Rd	Design value of the moment resistance of the connection for bending in plane of the structural system for the structural component i
f_y0	Yield strength of material of the chord member
t₀	Wall thickness of the chord cross-section
d_i	Total diameter for CHS components i
θ_i	Enclosed angle between the strut i and the chord member
γ	Ratio of the chord member's width or diameter to the double of its wall thickness
β	Ratio of mean diameters or mean widths of the strut and the chord
k_p	Chord prestress coefficient
γ_M5	Partial safety factor

Flange Failure of Chord Member due to Moment Mᵢₚ According to EN 1993-1-8, Table 7.5, Row 1

M_ip,1,Rd = M_ip,2,Rd = 9.53 kNm
M_ip,1,Ed / M_ip,1,Rd = 0.37 / 9.53 = 0.04 < 1.0
M_ip,2,Ed / M_ip,2,Rd = 0.14 / 9.53 = 0.01 < 1.0

Punching of Chord Member due to Moment M_op According to EN 1993-1-8, Table 7.5, Row 3.2

This design is only relevant for 3D structures where moments may also occur from the truss plane.

Determination of Allowable Limit Internal Force M_op,Rd

M_{op, i, Rd} = \frac{f_{y 0} \cdot t_{0} \cdot {d_{i}}^{2}}{\sqrt{3}} \cdot \frac{3 + \sin (θ_{i})}{4 \cdot \sin^{2} (θ_{i})} / γ_{M 5}

M_op,i,Rd	Design value of the moment resistance of the connection for bending out of plane of the structural system for the structural component i
f_y0	Yield strength of material of the chord member
t₀	Wall thickness of the chord cross-section
d_i	Total diameter for CHS components i
θ_i	Enclosed angle between the strut i and the chord member
y_M5	Partial safety factor

Punching of Chord Member due to Moment M_op According to EN 1993-1-8, Table 7.5, Row 3.2

M_op,1,Rd = M_op,2,Rd = 8.70 kNm
M_op,1,Ed / M_op,1,Rd = 0.08 / 8.70 = 0.01 < 1.0
M_op,2,Ed / M_op,2,Rd = 0.01 / 8.70 = 0.00 < 1.0

Punching of Chord Member due to Moment M_ip According to EN 1993-1-8, Table 7.5, Row 3.1

Determination of Allowable Limit Internal Force M_ip,Rd

M_{ip, i, Rd} = \frac{f_{y 0} \cdot t_{0} \cdot {d_{i}}^{2}}{\sqrt{3}} \cdot \frac{1 + 3 \cdot \sin (θ_{i})}{4 \cdot \sin^{2} (θ_{i})} / γ_{M 5}

M_ip,i,Rd	Design value of the moment resistance of the connection for bending in plane of the structural system for the structural component i
f_y0	Yield strength of material of the chord member
t₀	Wall thickness of the chord cross-section
d_i	Total diameter for CHS components i
θ_i	Enclosed angle between the strut i and the chord member
y_M5	Partial safety factor

Punching of Chord Member due to Moment M_ip According to EN 1993-1-8, Table 7.5, Row 3.1

M_ip,1,Rd = M_ip,2,Rd = 7.33 kNm
M_ip,1,Ed / M_ip,1,Rd = 0.37 / 7.33 = 0.05 < 1.0
M_ip,2,Ed / M_ip,2,Rd = 0.14 / 7.33 = 0.02 < 1.0

Interaction Conditions According to EN 1993-1-8, Chapter 7.4.2, Equation 7.3

In this design step, the struts are designed for the shared loading from axial force and bending. Currently, only bending perpendicular to the truss plane is considered here.

\frac{N_{i, Ed}}{N_{i, Rd}} + \frac{|M_{op, i, Ed}|}{M_{op, i, Rd}} \leq 1.0

\frac{197.56}{257.36} + \frac{|- 0.08|}{5.92} = 0.78 < 1.0 (S 1)

\frac{- 186.89}{257.36} + \frac{|- 0.01|}{5.92} = 0.72 < 1.0 (S 2)

N_i,Ed	Design value of the acting axial force for the structural component i
N_i,Rd	Design value of the axial force resistance of the connection for the structural component i
M_op,i,Ed	Design value of the acting moment out of plane of the structural system for the structural component i
M_op,i,Rd	Design value of the moment resistance of the connection for bending out of plane of the structural system for the structural component i
S1	Strut 1
S2	Strut 2

Interaction Conditions According to EN 1993-1-8, Section 7.4.2, Equation 7.3

Conclusion

The technical article shows that the design for a K-node is not trivial. With the RF-/HSS add-on module, Dlubal offers a tool for designing all nodal types defined in the European standard, for CHS as well as SHS and RHS sections.

Knowledge Base

KB 001656 | Design of K-Type Joint Using CHS-Sections According to EN 1993-1-8

Author

Mr. Flori is the customer support team leader and provides technical support for customers of Dlubal Software.

Links

References

European Committee for Standardization. (2009). Eurocode 3: Design of Steel Structures – Part 1-8: Design of Joints; EN 1993‑1‑8:2005 + AC:2009. Berlin: Beuth Verlag GmbH.
European Committee for Standardization (CEN). (2010). Eurocode 3: Design of Steel Structures – Part 1‑1: General Rules and Rules for Buildings, EN 1993‑1‑1:2010‑12. Berlin: Beuth Verlag GmbH.

Downloads

Model of Technical Article | RFEM 5 File

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Design of K-Type Joint Using CHS Sections According to EN 1993-1-8

General

Details of Model

Assigning Node to Joint Type

Checking Validity Limits

Design

Flange Failure of Chord Member due to Axial Force According to EN 1993-1-8, Table 7.2, Row 3.2

Punching of Chord Member due to Axial Force According to EN 1993-1-8, Table 7.2, Row 4

Flange Failure of Chord Member due to Moment Mop According to EN 1993-1-8, Table 7.5, Row 2

Flange Failure of Chord Member due to Moment Mip According to EN 1993-1-8, Table 7.5, Row 1

Punching of Chord Member due to Moment Mop According to EN 1993-1-8, Table 7.5, Row 3.2

Punching of Chord Member due to Moment Mip According to EN 1993-1-8, Table 7.5, Row 3.1

Interaction Conditions According to EN 1993-1-8, Chapter 7.4.2, Equation 7.3

Conclusion

Flange Failure of Chord Member due to Moment M_op According to EN 1993-1-8, Table 7.5, Row 2

Flange Failure of Chord Member due to Moment M_ip According to EN 1993-1-8, Table 7.5, Row 1

Punching of Chord Member due to Moment M_op According to EN 1993-1-8, Table 7.5, Row 3.2

Punching of Chord Member due to Moment M_ip According to EN 1993-1-8, Table 7.5, Row 3.1