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2020-09-11

Design of a 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.

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

Curved Spatial Trusses Made of Welded CHS

Details of the 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 graphic

Truss Dimensions

Assigning the Node to the 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 no. 35 (strut 1) and a compression force in member no. 36 (strut 2) on node no. 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.

Distribution of Internal Forces for Axial Force

Checking the 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.

Displaying Window 1.3 Geometry in RF-/HSS

Design Process

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

Flange failure of a 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
Ψ_2,i	Beiwert für quasi-ständige Werte der veränderlichen Einwirkungen i
t₀	Wall thickness of 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 struts of a K or N joint
t₀	Wall thickness of chord cross-section

Factor k𝚐 according to EN 1993-1-8, Table 7.2

k_g = 1.72

Determination of the 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 chord member without stresses due to components of strut forces at connection parallel to chord
f_y	Yield strength
N_p	Starting axial compressive force in chord
A₀	Cross-sectional area of chord member
M₀	Secondary moment from eccentricity
W₀	Elastic section modulus of chord cross-section

Chord prestress coefficient kₚ EN 1993-1-8

f_p is the chord tension from the axial force N_p and the additional moment from eccentricity. Since there are compression and tension in the chord, 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. Thus, the auxiliary factor f_p is zero. The sign rule for compression and tension forces in RFEM and RSTAB differs from those of the European standard EN 1993-1-8. Therefore, the formula for k_p has been adjusted.

k_p = 1.0

Determination of the 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 axial force resistance of 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 a chord member
t₀	Wall thickness of chord cross-section
θ₁	Enclosed angle between strut 1 and chord member
d₁	Total diameter of strut 1
d₀	Total diameter of chord member
y_M5	Partial safety factor
N_2,Rd	Design value of axial force resistance of connection for strut 2
θ₂	Enclosed angle between strut 2 and 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 the chord member due to axial force according to EN 1993-1-8, Table 7.2, Row 4:

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

Punching the 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 a 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 the 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 moment resistance of connection for bending out of plane of structural system for structural component i
f_y0	Yield strength of material of a chord member
t₀	Wall thickness of chord cross-section
d_i	Total diameter for CHS components i
θ_i	Enclosed angle between strut i and chord member
β	Ratio of mean diameters or mean widths of strut and 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 a chord member due to moment M_ip according to EN 1993-1-8, Table 7.5, Row 1:

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

Punching the chord member due to moment Mₒₚ 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 the chord member due to moment M_ip according to EN 1993-1-8, Table 7.5, Row 3.1:

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

Punching the chord member due to moment Mᵢₚ 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 (strut 1)

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

N_i,Ed	Design value of acting axial force for structural component i
N_i,Rd	Design value of axial force resistance of connection for structural component i
M_op,i,Ed	Design value of acting moment out of plane of structural system for structural component i
M_op,i,Rd	Design value of moment resistance of connection for bending out of plane of structural system for structural component i

Interaction conditions according to EN 1993-1-8 Chapter 7.4.2 Equation 7.3

Summary

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.

Links

References

EN 1993-1-8: Bemessung und Konstruktion von Stahlbauten – Teil 1-8: Bemessung von Anschlüssen. Beuth Verlag GmbH, Berlin, 2010
Eurocode 3: Bemessung und Konstruktion von Stahlbauten − Teil 1-1: Allgemeine Bemessungsregeln und Regeln für den Hochbau. Beuth Verlag GmbH, Berlin, 2010

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