Rectangular hollow sections

This is a selected chapter from book Component-based finite element design of steel connections by prof. Wald et al. The chapter is focused on verification of joints of rectangular hollow sections.

Description

In this chapter, uniplanar welded rectangular, square, hollow sections T, X, and K-joints with gap predicted by CBFEM are verified. Square hollow sections (SHS) brace is welded directly onto an RHS chord without the use of reinforcing plates. The joints are loaded by an axial force. In CBFEM, the design resistance is limited by 5 % of strain or a force corresponding to 0,03b0 joint deformation and in FMM generally by plate out of plane deformation 0,03b0 where b0 is the depth of the RHS chord; see Lu et al. (1994).

Failure mode method

In the case of the axially loaded T, Y, X or K-joint with gap of the welded rectangular hollow sections, five failure modes can occur. These are chord face failure, chord plastification, chord side wall failure, chord web failure, chord shear failure, punching shear failure, and brace failure. In this study, chord face failure, brace failure, and punching shear failure are examined for T, Y and X-joint and chord face failure, chord shear failure, brace failure, and punching shear failure are examined for K-joint with gap; see Fig. 7.2.1. The welds designed according to EN 1993-1-8:2005 are not the weakest components in the joint.

Fig. 7.2.1 Examined failure modes: a) Chord face failure, b) Chord shear failure, c) Brace failure, and d) Punching shear failure

Chord face failure

The design resistance of an RHS chord face is determined by FMM model in section 9.5 of EN 1993‑1-8:2020. The method is also given in the ISO/FDIS 14346 and is described in detail in Wardenier et al. (2010). The design resistance of the axially loaded T, Y or X-joint of welded rectangular hollow sections is

\[ N_{i,Rd} = C_f \frac{f_{y0} t_0^2}{\sin{\theta_i}} \left ( \frac{2 \eta}{(1-\beta) \sin{\theta_i}} + \frac{4}{\sqrt{1-\beta}} \right ) Q_f / \gamma_{M5} \]

The design resistance of the axially loaded K-joint with gap of welded rectangular hollow sections is

\[ N_{i,Rd} = 8.9 C_f \beta \gamma^{0.5} \frac{f_{y,0} t_0^2}{\sin{\theta_i}} Q_f / \gamma_{M5} \]

where Cf is the material factor, fy0 is the yield stress of the chord, t0 is the wall thickness of the chord, η is the brace height to the chord width ratio, β is the brace width to the chord width ratio, qi is the included angle between the brace member i and the chord (i = 1, 2), Qf is the chord stress function, and γ is the chord slenderness ratio.

Brace failure

The design resistance of an RHS chord face can be determined using the method given by FMM model in section 9.5 of EN 1993-1-8:2020. The design resistance of the axially loaded T, Y or X-joint of welded rectangular hollow sections is

\[ N_{i,Rd} = C_f f_{yi} t_i (2 h_i - 4 t_i + 2 b_{eff} ) / \gamma_{M5} \]

The design resistance of the axially loaded K-joint with gap of welded rectangular hollow sections is

\[ N_{i,Rd} = C_f f_{yi} t_i (2 h_i - 4 t_i + b_i + b_{eff} ) / \gamma_{M5} \]

where Cf is the material factor, fyi is the yield stress of the brace member i (i = 1, 2), ti is the wall thickness of the brace member i, hi is the height of the brace member i, bi is the width of the brace member i, beff is the effective width of the brace member.

Punching shear

The design resistance of the axially loaded T, Y or X-joint of welded rectangular hollow sections is

\[ N_{i,Rd} = C_f \frac{f_{y0} t_0}{\sqrt{3}\sin{\theta_i}} \left( \frac{2h_i}{\sin{\theta_i}} + 2b_{e,p} \right ) / \gamma_{M5} \]

The design resistance of the axially loaded K-joint with gap of welded rectangular hollow sections is

\[ N_{i,Rd} = C_f \frac{f_{y0} t_0}{\sqrt{3}\sin{\theta_i}} \left( \frac{2h_i}{\sin{\theta_i}} + b_i+b_{e,p} \right ) / \gamma_{M5} \]

Where Cf is the material factor, fy0 is the yield stress of the chord, t0 is the wall thickness of the chord, qi is the included angle between the brace member i and the chord (i = 1, 2), hi is the height of the brace member i, bi is the width of the brace member i and be,p is the effective width for punching shear.

Chord shear failure

The design resistance of the axially loaded K-joint with gap of welded rectangular hollow sections is

\[ N_{i,Rd} = \frac{f_{y0}A_{V,0,gap}}{\sqrt{3}\sin{\theta_i}}/\gamma_{M5} \]

where fy0 is the yield stress of the chord, Av,0,gap is the effective area for chord shear failure, and qi is the included angle between the brace member i and the chord (i = 1, 2).

Range of validity

CBFEM was verified for typical T, Y X, and K-joints with gap of the welded rectangular hollow sections. Range of validity for these joints is defined in Table 9.2 of prEN 1993-1-8:2020; see Tab. 7.2.1. The same range of validity is applied to CBFEM model. Outside the range of validity of FMM, an experiment should be prepared for validation or verification performed for verification according to a validated research model.

Tab. 7.2.1 Range of validity for method of failure modes, Table 9.2 of EN 1993-1-8:2020

General\(0.2 \le \frac{d_i}{d_0} \le 1.0 \)\( \theta_i \ge 30^{\circ} \)\(\frac{e}{d_0} \le 0.25 \)

\(g \ge t_1+t_2 \)\(f_{yi} \le f_{y0} \)\( t_i \le t_0 \)
ChordCompressionClass 1 or 2 and \( d_0 / t_0 \le 50 \) (but for X joints: \( d_0/t_0 \le 40 \))

 Tension\(d_0 / t_0 \le 50 \) (but for X joints: \( d_0/t_0 \le 40 \))
CHS bracesCompressionClass 1 or 2 and \(b_i / t_i \le 35\) and \(\frac{h_i}{t_i} \le 35 \)

Tension\(b_i / t_i \le 35\) and \(\frac{h_i}{t_i} \le 35 \)


7.2.2 Uniplanar T and Y-SHS joint

An overview of the considered examples is given in Tab. 7.2.2. Selected cases cover a wide range of joint geometric ratios. Geometry of joints with dimensions is shown in Fig. 7.2.2. Selected joints failed according to the method based on FMM by the chord face failure or brace failure.

Tab. 7.2.2 Examples overview

ExampleChordBraceAngles
Material 
 SectionSectionθ1fyfuE
   [°][MPa][MPa][MPa]
1SHS200/6.3SHS70/8.090275430210
2SHS200/8.0SHS90/8.090275430210
3SHS200/10.0SHS140/12.590275430210
4SHS200/12.5SHS120/12.590275430210
5SHS200/16.0SHS50/6.390275430210
6SHS200/6.3SHS140/12.560275430210
7SHS200/8.0SHS80/8.060275430210
8SHS200/10.0SHS120/12.560275430210
9SHS200/12.5SHS90/8.060275430210
10SHS200/16.0SHS160/16.060275430210
11SHS200/6.3SHS80/8.030275430210
12SHS200/8.0SHS150/16.030275430210
13SHS200/10.0SHS100/10.030275430210
14SHS200/12.5SHS100/10.030275430210
15SHS200/16.0SHS90/8.030275430210

Fig. 7.2.2 Dimensions of T-joint

Verification of resistance

The results of FMM are compared with the results of CBFEM. The comparison is focused on the resistance and design failure mode. The results are presented in Tab. 7.2.3.

Tab. 7.2.3 Comparison of results of design resistances in tension/compression predicted by CBFEM and FMM

ExampleFMMMode of failureCBFEMMode of failureDiff.
 [kN] [kN] [%]
165,9Chord face failure63,0Chord face failure4
2123,7Chord face failure113,7Chord face failure8
3329,2Chord face failure354,0Chord face failure8
4400,7Chord face failure419,5Chord face failure5
5302,8Brace failure263,0Brace failure13
6160,0Chord face failure160,5Chord face failure0
7136,2Chord face failure125,3Chord face failure8
8310,8Chord face failure306,5Chord face failure1
9361,4Chord face failure327,0Chord face failure10
101478,0Chord face failure1373,7Chord face failure7
11170,9Chord face failure160,9Chord face failure6
12704,0Chord face failure638,5Chord face failure9
13531,1Chord face failure468,6Chord face failure12
14829,9Chord face failure731,9Chord face failure12
15721,6Brace failure647,8Brace failure10

The study shows a good agreement for the applied load cases. The results are summarized in a diagram comparing design resistances of CBFEM and FMM; see Fig. 7.2.3. The results show that the difference between the two calculation methods is in all cases less than 13 %.

Fig. 7.2.3 Verification of resistance determined by CBFEM to FMM for the uniplanar SHS T and Y-joint

Benchmark example

Inputs

Chord

  • Steel S275
  • Section SHS 200×200×6,3

Brace

  • Steel S275
  • Section SHS 70×70×8,0
  • Angle between the brace member and the chord 90°

Weld

  • Butt weld

Mesh size

  • 16 elements on the biggest web of rectangular hollow member

Loaded

  • By force to brace in compression/tension

Outputs

  • The design resistance in compression/tension is NRd = 63,0 kN
  • The design failure mode is chord face failure

Uniplanar X-SHS joint

An overview of the considered examples is given in the Tab. 7.2.4. Selected cases cover a wide range of joint geometric ratios. The selected joints failed according to the method based on FMM by the chord face failure or brace failure.

Tab. 7.2.4 Examples overview

ExampleChordBraceAngles
Material 
 SectionSectionθfyfuE
   [°][MPa][MPa][MPa]
1SHS200/6.3SHS140/12.590275430210
2SHS200/8.0SHS70/8.090275430210
3SHS200/10.0SHS120/12.590275430210
4SHS200/12.5SHS90/8.090275430210
5SHS200/16.0SHS160/16.090275430210
6SHS200/6.3SHS70/8.060275430210
7SHS200/8.0SHS80/8.060275430210
8SHS200/10.0SHS150/6.060275430210
9SHS200/12.5SHS140/12.560275430210
10SHS200/16.0SHS120/12.560275430210
11SHS200/6.3SHS80/8.030275430210
12SHS200/8.0SHS150/16.030275430210
13SHS200/10.0SHS100/10.030275430210
14SHS200/12.5SHS160/16.030275430210
15SHS200/16.0SHS90/8.030275430210

Verification of resistance

The results of the method based on failure modes (FMM) are compared with the results of CBFEM. The comparison is focused on the resistance and design failure mode; see Tab. 7.2.5.

Tab. 7.2.5 Comparison of results of prediction of resistance by CBFEM and FMM

ExampleFMMode of failureCBFEMMode of failureDiff.
 [kN] [kN] [%]
1130,6Chord face failure131,6Chord face failure1
2106,3Chord face failure92,1Chord face failure13
3256,4Chord face failure252,5Chord face failure2
4302,1Chord face failure281,4Chord face failure7
51192,9Chord face failure1177,9Chord face failure1
678,2Chord face failure68,4Chord face failure12
7136,2Chord face failure120,9Chord face failure11
8474,0Chord face failure479,1Chord face failure1
9629,7Chord face failure659,1Chord face failure5
10795,7Chord face failure761,1Chord face failure4
11170,9Chord face failure159,8Chord face failure7
12704,0Chord face failure646,5Chord face failure8
13531,1Chord face failure469,5Chord face failure12
141813,9Brace failure1670,8Brace failure8
15721,6Brace failure650,3Brace failure10

The study shows a good agreement for the applied load cases. The results are summarized in a diagram comparing design resistances of CBFEM and FMM; see Fig. 7.2.4. The results show that the difference between the two calculation methods is in all cases less than 13 %.

Fig. 7.2.4 Verification of resistance determined by CBFEM to FMM for the uniplanar SHS X-joint

Benchmark example

Inputs

Chord

  • Steel S275
  • Section SHS 200×200×6,3

Braces

  • Steel S275
  • Sections SHS 140×140×12,5
  • Angle between the brace members and the chord 90°

Welds

  • Butt welds

Mesh size

  • 16 elements on the biggest web of rectangular hollow member

Loaded

  • By force to brace in compression/tension

Outputs

  • The design resistance in compression/tension is NRd = 131,6 kN
  • The design failure mode is chord face failure

7.2.4 Uniplanar K-SHS joint

An overview of the considered examples is given in the Tab. 7.2.6. Selected cases cover a wide range of joint geometric ratios. The selected joints failed according to the method based on FMM by the chord face failure or brace failure.

Tab. 7.2.6 Examples overview

ExampleChordBracesAngles
Material 
 SectionSectionsθfyfuE
   [°][MPa][MPa][MPa]
1SHS200/8.0SHS160/16.045275430210
2SHS200/10.0SHS150/16.045275430210
3SHS200/10.0SHS160/16.045275430210
4SHS200/12.5SHS140/12.545275430210
5SHS200/12.5SHS150/16.045275430210
6SHS200/12.5SHS160/16.045275430210
7SHS200/16.0SHS50/6.345275430210
8SHS200/16.0SHS70/8.045275430210
10SHS200/16.0SHS80/8.045275430210

Verification

The results of CBFEM are compared with the results of FMM. The comparison is focused on the resistance and design failure mode. The results are presented in Tab. 7.2.7.

Tab. 7.2.7 Comparison of results of prediction of resistances by CBFEM and FMM

ExampleFMMode of failureCBFEMMode of failurediff
 [kN] [kN] [%]
1626.6Chord face failure609Chord face failure3
2800.1Chord face failure754Chord face failure6
3875.6Chord face failure883Chord face failure1
41070.8Chord face failure973Chord face failure9
51147.3Chord face failure1116Chord face failure3
61223.8Chord face failure1219.4Chord face failure0
7302.8Brace failure258Brace failure15
8545.6Brace failure463Brace failure15
15633.6Brace failure548.4Brace failure13

The study shows a good agreement for the applied load cases. The results are summarized in a diagram comparing design resistances of CBFEM and FMM; see Fig. 7.2.5. The results show that the difference between the two calculation methods is in all cases less than 15%.

Fig. 7.2.5 Verification of resistance determined by CBFEM to FMM for the uniplanar SHS K-joint

Benchmark example

Inputs

Chord

  • Steel S275
  • Section SHS 200×200×8,0

Braces

  • Steel S275
  • Sections SHS 160×160×16,0
  • Angle between the brace members and the chord 45°

Welds

  • Butt welds

Mesh size

  • 16 elements on the biggest web of rectangular hollow member

Loaded

  • By force to brace in compression/tension

Outputs

  • The design resistance in compression/tension is NRd = 609 kN
  • The design failure mode is chord face failure

Sample files

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