Joint strength reduction factor of internally ring-stiffened tubular joints at elevated temperatures

Abstract

This article presents the results of a numerical parametric study on the ultimate strength of internally crown- and saddle-stiffened circular hollow section tubular DT-, T-, and Y-joints subjected to brace axial compression or tension at elevated temperatures. Using well-validated finite element models, an extensive study of 640 ring-stiffened tubular joints consisting of 36 DT-joints, 36 T-joints, and 8 Y-joints at different temperatures ranging from 20°C to 800°C was conducted. The strength of the stiffened joints was obtained from finite element analysis. The joint strength reduction was compared with the reduction factors of steel yield stress and elastic modulus at elevated temperatures. The parametric study shows that the effects of joint geometric parameters and stiffening position on the strength reduction of the stiffened DT-, T-, and Y-joints at elevated temperatures fall in a narrow range. An equation for predicting the strength reduction of the stiffened DT-, T-, and Y-joints at elevated temperatures was proposed by introducing a temperature factor. The statistical analysis shows that the proposed equation could provide reasonably accurate joint strength predictions.

Keywords

circular hollow section elevated temperatures ring-stiffener tubular joint ultimate strength

Introduction

Tubular joint failure often occurs on chord surface in the vicinity of brace–chord intersection because the chord radial stiffness is relatively lower than brace axial stiffness. Internal ring stiffeners are commonly used to enhance the chord stiffness. Most research on internally ring-stiffened tubular joints has been focused on the stiffened joint strengths at ambient temperature. Thandavamoorthy et al. (1999) carried out a test study on the strength of T- and Y-joints (refer to Figure 1 for the joints typology) reinforced with triple stiffeners and compared the strength with that of corresponding unstiffened joints. It is found that the strength of the internally ring-stiffened joints was almost twice that of the unstiffened joints with the same dimensions. Lee and Llewelyn-Parry (2004, 2005) conducted a finite element (FE) study on the strength of internally saddle-stiffened T- and DT-joints subjected to brace axial compression. A methodology for predicting the joint strength was proposed. Wang et al. (2014) carried out a FE study on the strength of internally crown-stiffened tubular T- and Y-joints and proposed equations for predicting the strength of the stiffened T- and Y-joints subjected to brace axial compression or tension.

Figure 1.

Internally ring-stiffened tubular joint configuration: (a) DT-joints, (b) T-joints, and (c) Y-joints.

Fire presents one of the most severe design conditions for tubular joints in steel tubular structures, for example, offshore jacket platform. Internally ring-stiffening is one of the main reinforcing methods for tubular joints. Therefore, it is necessary to conduct related studies on internally ring-stiffened tubular joints in fire. Up to now, most work on fire resistance performance of tubular joints is focused on unstiffened tubular joints. Nguyen et al. (2010a, 2010b) conducted experimental tests and FE analysis on circular hollow section (CHS) tubular T-joints. It is found that the joint strength significantly decreases with increasing temperature, and brace–chord diameter ratio (β) and ratio of chord diameter to twice chord thickness (γ) have a significant effect on the joint strength. Feng and Young (2012) carried out a FE study on the strength of square hollow section (SHS) and rectangular hollow section (RHS) cold-formed stainless steel tubular T- and X-joints at elevated temperatures. Design equations for the joints at elevated temperatures were proposed by introducing a temperature factor. Xu et al. (2012) used artificial neural networks to predict the strength of CHS tubular T-joints at elevated temperatures, and the feasibility of this method was proved. Tests and numerical analysis were both conducted by Tan et al. (2013) to investigate the structural behavior of CHS tubular T-joints in fire conditions. It was observed that joint strength decreased as temperature increased due to both the reduction in steel strength and the changes in localized plastification area around the brace–chord intersection. Ozyurt et al. (2014) focused on the strength of SHS and CHS T-, Y-, X-, N-, and nonoverlapped K-joints at elevated temperatures using FE method. The results show that for nonoverlapped K- and N-joints and T-, Y-, and X-joints subjected to brace axial tension, using design equations in design guides EN 1993-1-8:2005 (2005) and CIDECT (2010) but replacing the yield stress of steel at ambient temperature with that at elevated temperatures is suitable. However, this approach overestimates the strength of CHS T-, Y-, and X-joints subjected to brace axial compression. The aforementioned studies were focused on investigating the ultimate strengths of tubular joints at steady elevated temperatures. Some studies on fire resistance of tubular joints under transient state process instead of steady-state process were also carried out in recent years. Nguyen et al. (2012) conducted experimental tests and FE analysis on the fire resistance of CHS tubular T-joints subjected to brace axial compression and in-plane bending. The results show that the brace–chord diameter ratio has a significant effect on the joint fire resistance. Yu et al. (2011) tested the fire resistance of impacted CHS tubular T-joints and found that the impact could improve the fire resistance of the joint. Test and FE studies were both carried out by Chen et al. (2013) to investigate failure modes and fire resistance of CHS tubular T-joints. It is found that the failure mode is the plastic failure of the chord face around the brace–chord intersection, and the joints failed suddenly at critical temperature. He et al. (2013b) conducted test and FE studies on the critical temperature of CHS gap tubular K-joints and found that loading ratio and initial chord stress have a significant effect on the joint critical temperature. Yang et al. (2014) carried out a test study on fire resistance of SHS tubular T-joints. The results show that the failure mode is local buckling on the chord wall. Some studies have been conducted to investigate the fire resistance performance of internally ring-stiffened tubular joints. Liu et al. (2009) numerically investigated the strength of internally ring-stiffened tubular T-joints at elevated temperatures and compared the stiffened joint strength with corresponding unstiffened joint strength. It is found that the internal ring could effectively enhance the joint strength, and the enhancement increases with increasing stiffener number, width, and thickness. Chen et al. (2015) studied the fire resistance of CHS tubular T-joints with and without internal rings by experimental test and FE analysis. It is found that the internal ring-stiffener could effectively improve the joint fire resistance performance by decreasing chord temperature and prolonging fire resistance time. Although some important conclusions for internally ring-stiffened tubular joints were obtained, there is a lack of research on the determination of the stiffened tubular joint strength at elevated temperatures.

In this article, FE analysis on the ultimate strength of internally crown- and saddle-stiffened DT-, T-, and Y-joints (see Figure 1) subjected to brace axial compression or tension at different elevated temperatures was carried out. Based on the comparison between the joint strength reduction and the reduction factors of steel yield stress and elastic modulus at elevated temperatures, an equation was proposed for predicting the strength reduction of the stiffened DT-, T-, and Y-joints at elevated temperatures by introducing a temperature factor.

FE modeling validation

The general FE program ABAQUS was used to carry out the FE analysis. For validation, the test results of Tan et al. (2013) on unstiffened T-joints (see Figure 2(a)) at 20°C, 550°C, and 700°C and the FE results of Lee and Llewelyn-Parry (2005) on internally saddle-stiffened DT-joints (see Figure 2(b)) at ambient temperature were used. Tan et al. (2013) carried out the test under steady state in which the joint temperature was raised to the required level and then the load was applied. The geometric parameters of the DT- and T-joints are shown in Table 1.

Figure 2.

Joints used for validation: (a) PT-joint (Tan et al., 2013) and (b) DT-joint (Lee and Llewelyn-Parry, 2005).

Table 1.

Joints used for FE model validation.

Joint name	d (mm)	t (mm)	d ₁ (mm)	t ₁ (mm)	b _w (mm)	t _w (mm)	T (°C)
DT-W (0.4, 0.2)^a (Lee and Llewelyn-Parry, 2005)	1500	50	750	35	300	20	20
DT-W (0.8, 0.2)^a (Lee and Llewelyn-Parry, 2005)	1500	50	750	35	300	40	20
PT3-20 (Tan et al., 2013)	244.5	6.3	168.3	6.3	–	–	20
PT3-550 (Tan et al., 2013)	244.5	6.3	168.3	6.3	–	–	550
PT3-700 (Tan et al., 2013)	244.5	6.3	168.3	6.3	–	–	700
PT1-550 (Tan et al., 2013)	244.5	6.3	195.6	6.3	–	–	550

Part of the name of joint specimens in Lee and Llewelyn-Parry (2005).

Material properties

The steel grade of the tubular T-joint tested by Tan et al. (2013) was S355 with a yield strength f_y = 380.3 N/mm² and an ultimate strength f_u = 519.1 N/mm² from the coupon test at ambient temperature. The steel elastic modulus was 201.2 GPa. The elevated temperature stress–strain curves were based on EN 1993-1-2:2005 (2005). The true stress–strain curve was input in the FE analysis after converting the engineering stress–strain curve using the following equations (Ozyurt et al., 2014)

ε_{T} = \ln (1 + ε)

(1)

σ_{T} = σ (1 + ε)

(2)

where ε_T and ε are true and engineering strains, respectively, and σ_T and σ are true and engineering stresses, respectively. For the tubular DT-joints investigated by Lee and Llewelyn-Parry (2005), the adopted material curve of steel is an elastic–perfectly plastic curve with f_y = 254 N/mm² and E = 195.4 GPa. In ABAQUS simulation, von Mises yield surface criterion and isotropic strain hardening rules were used.

FE type

A four-node quadrilateral shell element S4R5 with 5 degrees of freedom per node—3 translational and 2 out-of-plane rotations—from the ABAQUS library was used to model the internally ring-stiffened DT-joint at ambient temperature and unstiffened tubular T-joints at different temperatures. A five-point integration through the shell thickness was adopted. The weld modeling was not included due to its insignificant effect on joint strengths (Lee and Llewelyn-Parry, 2005).

Mesh convergence

A mesh convergence study was carried out to determine suitable mesh size for the FE modeling. The tubular joint PT3-700 tested by Tan et al. (2013) was selected for this case. The mesh sensitivity study results are shown in Table 2. It was found that mesh sizes of 10 and 20 mm for the joint zone and tubular members outside the joint zone, respectively, were suitable. Similar mesh convergence study was conducted for the DT-joints studied by Lee and Llewelyn-Parry (2005). It was found that mesh sizes of 20 and 40 mm for the joint zone and tubular members outside the joint zone, respectively, were suitable. The mesh layout is shown in Figure 3.

Table 2.

Mesh sensitivity study results.

Mesh size (mm)		FE (kN)	FE/test	Relative CPU time
Joint zone	Outside joint zone
30	60	59	1.07	0.13
20	40	58	1.05	0.39
10	20	54	0.98	1.00
5	10	54	0.98	9.46

FE: finite element; CPU: central processing unit.

Figure 3.

Mesh layout: (a) PT-joint (Tan et al., 2013) and (b) DT-joint (Lee and Llewelyn-Parry, 2005).

Validation against literature results

The results of this validation study are shown in Table 3. It can be seen that the strengths of all six joints have been predicted to well within 10% of the literature results. This level of accuracy confirms the validity of the FE modeling adopted. It should be noted that the strength of tubular joints was determined by the peak load or deformation limit (3%d) in load–displacement curves. If the deformation at the peak load is smaller than 3%d, the peak load was considered to be the joint strength. If the deformation at the peak load is larger than 3%d, the load at the deformation of 3%d was considered to be the joint strength.

Table 3.

Results of the validation study.

Joint name	Joint strength
	FE (kN)	Published data (kN)	FE/published data
DT-W (0.4, 0.2)^a (Lee and Llewelyn-Parry, 2005)	9288	8616	1.08
DT-W (0.8, 0.2)^a (Lee and Llewelyn-Parry, 2005)	11250	10543	1.07
PT3-20 (Tan et al., 2013)	311	338	0.92
PT3-550 (Tan et al., 2013)	169	175	0.97
PT3-700 (Tan et al., 2013)	54	55	0.98
PT1-550 (Tan et al., 2013)	213	217	0.98

FE: finite element.

Part of the name of joint specimens in Lee and Llewelyn-Parry (2005).

Parametric study

The geometric parameters in the parametric study are shown in Table 4. The range of these parameters is common in joints that occur in engineering practice. In this article, 12 cases, as shown in Table 5, were investigated. It should be noted that two stiffeners were positioned at the two crown positions for all crown-stiffened joints and one stiffener was placed at the saddle position for all saddle-stiffened joints.

Table 4.

Geometric parameters for parametric study.

Joint number	t (mm)	d ₁ (mm)	b _w (mm)	t _w (mm)	θ (°)	γ	β	η	τ
1	40	500	160	32	90	10	0.625	0.2	0.8
2	20	500	160	16	90	20	0.625	0.2	0.8
3	13.3	500	160	10.6	90	30	0.625	0.2	0.8
4	20	300	160	16	90	20	0.375	0.2	0.8
5	20	700	160	16	90	20	0.875	0.2	0.8
6	20	500	80	16	90	20	0.625	0.1	0.8
7	20	500	240	16	90	20	0.625	0.3	0.8
8	20	500	160	20	90	20	0.625	0.2	1.0
9	20	500	160	24	90	20	0.625	0.2	1.2
10	20	500	160	16	30	20	0.625	0.2	0.8
11	20	500	160	16	60	20	0.625	0.2	0.8

All joints with l = 4800 mm, l₁ = 2400 mm, d = 800 mm, and t = t₁.

Table 5.

Summary of analyzed cases.

Case studies	Joint type	Brace loading	Stiffening position
Case 1	DT-joint	Compression	Crown
Case 2	DT-joint	Compression	Saddle
Case 3	DT-joint	Tension	Crown
Case 4	DT-joint	Tension	Saddle
Case 5	T-joint	Compression	Crown
Case 6	T-joint	Compression	Saddle
Case 7	T-joint	Tension	Crown
Case 8	T-joint	Tension	Saddle
Case 9	Y-joint	Compression	Crown
Case 10	Y-joint	Compression	Saddle
Case 11	Y-joint	Tension	Crown
Case 12	Y-joint	Tension	Saddle

Numerical simulation

Figure 4 shows the loading modes and boundary conditions of the stiffened DT-, T-, and Y-joints. The brace compression and tension loadings were applied by displacement at brace ends. In the numerical simulation, steady-state analysis was used in which the mechanical properties of steel were changed to those at elevated temperatures. The elevated temperature stress–strain curves of the steel Q345B which is commonly used in Chinese construction industry were based on EN 1993-1-2:2005 (2005) as shown in Figure 5. The steel yield stress and elastic modulus were 345 N/mm² and 206 GPa, respectively, and Poisson’s ratio was 0.3. In the FE modeling, the true stress–strain curve was input after conversion from the engineering stress–strain curve. The temperature distribution was assumed to be uniform throughout the joint. A mesh convergence study similar to that in section “Mesh convergence” was conducted. It was found that mesh sizes of 30 and 60 mm for the joint zone and tubular members outside the joint zone, respectively, were suitable.

Figure 4.

Boundary conditions and loading modes: (a) DT-joints, (b) T-joints, and (c) Y-joints.

Figure 5.

Steel stress–strain curves at elevated temperatures.

Simulation results

The ratio (R_T) of joint strength at elevated temperature T (N_T) to that at ambient temperature of around 20°C (N₂₀) was used to evaluate the effect of elevated temperatures on the joint strength. The ratios (R_T) at different elevated temperatures were compared with the reduction factors of steel yield strength (k_y,_T) and elastic modulus (k_E,T), and the proposed joint strength reduction factor (k₁) at elevated temperatures. The comparison for DT-, T-, and Y-joints in cases 1–12 is shown in Figures 6 to 9. It should be noted that the name of each joint consists of three letters and one number for identification. The first letter (X, T, or Y) represents the joint type (DT-joint, T-joint, or Y-joint). The second letter (C or T) indicates the brace loading direction (compression or tension). The third letter (S or C) represents the stiffening position (saddle or crown). Finally, the number refers to the joint number as shown in Table 4. For example, XCS1 denotes saddle-stiffened DT-joints subjected to brace axial compression. The geometric parameters of XCS1 are listed in Table 4.

Figure 6.

Comparison for crown-stiffened DT-joints with (a) brace in compression (case 1) or (b) brace in tension (case 3).

Figure 7.

Comparison for saddle-stiffened DT-joints with brace in compression (case 2) or (b) brace in tension (case 4).

Figure 8.

Comparison for crown-stiffened T- and Y-joints with (a) brace in compression (cases 5 and 9) or (b) brace in tension (cases 7 and 11).

Figure 9.

Comparison for saddle-stiffened T- and Y-joints with (a) brace in compression (cases 6 and 10) or (b) brace in tension (cases 8 and 12).

Stiffened joint strength prediction

Figures 6 to 9 show that the strength of the stiffened DT-, T- and Y-joints reduces with decreasing steel yield stress and elastic modulus at elevated temperatures. The parametric study covers a wide range of joint parameters (30° ≤ θ ≤ 90°, 10 ≤ γ ≤ 30, 0.1 ≤ η ≤ 0.3, 0.8 ≤ τ ≤ 1.2, 0.375 ≤ β ≤ 0.875). The effects of these joint parameters on the strength reduction of the stiffened DT-, T-, and Y-joints at elevated temperatures fall in a narrow range as shown in Figures 6 to 9. A similar observation that the unstiffened joint strength reduction at elevated temperatures is not influenced by joint geometry was reported by Ozyurt et al. (2014). Figures 6 to 9 also indicate that the stiffening position has relatively small influence on the strength reduction of the stiffened DT-, T-, and Y-joints. The effect of loading direction on the strength reduction of the stiffened DT- and T-joints is negligible. For the stiffened Y-joints subjected to brace tension, the joint strength reduction follows the yield stress reduction at elevated temperatures. The values of R_T of the Y-joints subjected to brace axial compression are lower than those of Y-joints with brace member in tension (see Figures 8 and 9). Therefore, it could be concluded that the strength reduction of the stiffened DT-, T-, and Y-joints is relatively independent of joint geometric parameters and stiffening position but is related to the deterioration of mechanical properties of steel at elevated temperatures and loading direction.

Based on the above observations, a unified strength equation for the stiffened DT-, T-, and Y-joints at elevated temperatures is proposed as follows

N_{T} = k_{T} N_{20}

(3)

where N_T is the joint strength at temperature T, N₂₀ is the joint strength at ambient temperature which could be obtained from design equations in existing literatures (e.g. Lee and Llewelyn-Parry, 2004, 2005; Wang et al., 2014), and k_T is the temperature factor at temperature T, and the proposed equation for temperature factor is as follows

k_{T} = {\begin{matrix} k_{1} = \frac{k_{y, T} + k_{E, T}}{2} for cases 1 - 10 \\ k_{y, T} for cases 11 - 12 \end{matrix}

(4)

The reduction factors of steel yield stress (k_y,T) and elastic modulus (k_E,T) specified in EN 1993-1-2:2005 (2005) and the values of k₁ are given in Table 6. The curves of k_y,T, k_E,T, and k₁ are shown in Figures 6 to 9. It is shown in Figures 6 to 9 that the joint strength reduction in the parametric study could be accurately predicted by the proposed temperature factor as calculated by equation (4).

Table 6.

Values of k_y,T, k_E,T, and k_1.

T (°C)	k _y,T	k _E,T	k ₁
20	1.00	1.00	1.00
200	1.00	0.90	0.95
300	1.00	0.80	0.90
400	1.00	0.70	0.85
500	0.78	0.60	0.69
600	0.47	0.31	0.39
700	0.23	0.13	0.18
800	0.11	0.09	0.10

Intermediate values of k_y,T, k_E,T, and k₁ may be determined by linear interpolation.

Assessment of the proposed equation

To assess the accuracy of the proposed equation (see equations (3) and (4)), an error analysis was conducted. The comparison between joint strength obtained from FE analysis (N_fi) and that calculated from the proposed equation (N_ei) in cases 1–12 is shown in Figure 10. It could be seen that the value of N_ei has predicted to well within 10% of the value of N_fi. The result of the statistical analysis of strength ratio, r_si (=N_fi/N_ei, i = 1–n, where n is the number of joints analyzed in each case), for cases 1–12 is shown in Table 7. The relative error $(e_{i}^{*})$ , the average relative error (e), and relative standard deviation (s*) are defined as follows (He et al., 2013a)

e_{i}^{*} = \frac{N_{fi} - N_{ei}}{N_{fi}}, (i = 1, 2, 3, \dots, n)

(5)

e = \frac{\sum_{i = 1}^{n} | e_{i}^{*} |}{n}

(6)

s^{*} = \sqrt{\frac{\sum_{i = 1}^{n} {| e_{i}^{*} |}^{2}}{n - 1}}

(7)

Figure 10.

Comparison between N_fi and N_ei for cases 1–12: (a) DT-joints (cases 1–4) and (b) T- and Y-joints (cases 5–12).

Table 7.

Statistical analysis of r_si for cases 1–12.

Case studies	Case 1	Case 2	Case 3	Case 4	Case 5	Case 6	Case 7	Case 8	Case 9	Case 10	Case 11	Case 12
n	72	72	72	72	72	72	72	72	16	16	16	16
Mean	1.00	0.96	1.02	0.96	1.01	1.01	1.00	0.98	0.98	0.99	0.97	0.97
COV	0.04	0.04	0.03	0.05	0.04	0.04	0.04	0.05	0.03	0.02	0.02	0.02
e	2.79%	4.34%	3.02%	4.67%	3.44%	2.94%	3.51%	4.25%	2.86%	2.31%	3.32%	3.83%
s*	3.82%	6.16%	3.73%	6.26%	4.39%	3.90%	4.51%	5.75%	3.77%	2.96%	3.62%	4.05%

COV: coefficient of variation.

The values of r_si at T = 20°C were excluded when calculating the values of mean, COV, e, and s*.

Table 7 shows that the values of coefficient of variation (COV), e, and s^* in cases 1–12 are lower than 0.05, 4.67%, and 6.26%, respectively. Such values indicate that the proposed equation could produce reasonably accurate stiffened joint strength predictions at elevated temperatures.

Conclusion

The results of a parametric study on the ultimate strength of internally crown- and saddle-stiffened DT-, T-, and Y-joints subjected to brace compression or tension at elevated temperatures are presented. The parametric study covers a wide range of parameters including the stiffened joint geometric parameters, stiffening position, and brace loading direction. The mechanical properties of steel at elevated temperatures were based on EN 1993-1-2:2005 (2005). Based on the comparison between the joint strength reduction and the reduction factors of steel yield stress and elastic modulus at elevated temperatures, the conclusions are summarized as follows:

The effects of joint geometric parameters and stiffening position on the strength reduction of the stiffened DT-, T-, and Y-joints at elevated temperatures fall in a narrow range. The stiffened joint strength reduction is relatively dependent of loading direction and the reduction of steel yield stress and elastic modulus at elevated temperatures.

An equation (see equation (4)) for predicting the strength reduction of the stiffened DT-, T-, and Y-joints at elevated temperatures was proposed. The statistical analysis shows that the proposed equation (see equations (3) and (4)) could produce reasonably accurate stiffened joint strength predictions.

Footnotes

Appendix 1 Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The research described in this paper was financially supported by the Open Project of State Key Laboratory of Subtropical Building Science, South China University of Technology (nos 2014KB29 and 2015ZB30).

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