Strength of internally ring-stiffened tubular DT-joints subjected to brace axial loading

Journal of Constructional Steel Research 125 (2016) 88–94

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Journal of Constructional Steel Research

Strength of internally ring-stiffened tubular DT-joints subjected to brace axial loading Xiaoyi Lan, Fan Wang ⁎, Chen Ning, Xiaofeng Xu, Xiaorong Pan, Zhifeng Luo a b

State Key Laboratory of Subtropical Building Science, South China University of Technology, Guangzhou 510640, China Architectural Design and Research Institute, South China University of Technology, Guangzhou 510640, China

a r t i c l e

i n f o

Article history: Received 8 November 2015 Received in revised form 17 May 2016 Accepted 6 June 2016 Available online 19 June 2016 Keywords: Circular hollow section Internal ring-stiffener Tubular DT-joint Plastic hinge Joint strength

a b s t r a c t This paper presents the results of numerical and theoretical studies to obtain static strength equations for internally ring-stiffened circular hollow section (CHS) tubular DT-joints. An extensive study of 1264 unstiffened and ring-stiffened DT-joints subjected to brace axial compression or tension was conducted. The numerical analysis shows that failure mechanism of crown- and saddle-stiffened DT-joints under brace axial loading is formation of plastic hinges in the stiffener and chord wall yielding near the brace-chord intersection. Based on the identified failure mechanism, theoretical models and corresponding equations for predicting the stiffener strength in crown- and saddle-stiffened DT-joints subjected to brace axial compression or tension were proposed. The accuracy of the proposed stiffener strength equations was evaluated by an error analysis. A chord stress function was proposed to consider chord axial stress effect on the stiffened DT-joint strength. In conjunction with existing unstiffened DT-joint strength formulae and considering chord axial stress effect, a strength equation for crown- and saddle-stiffened DT-joints subjected to brace axial loading was proposed. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Tubular joints are critical load-carrying components in steel tubular structures. There is relatively extensive research on the strength of unstiffened tubular joints (e.g. Kim et al. [9], Shen et al. [14,15]). Strength equations for unstiffened tubular joints are available in major design codes for steel structures, e.g. CIDECT [3], CECS 280:2010 [4] and EN 1993-1-8 [5]. Internal ring stiffeners are commonly used to strengthen tubular joints in onshore and offshore structures, e.g. offshore platforms, lattice girders and long-span roof systems. However, there is relatively less research on internally ring-stiffened tubular joints. Vegte et al. [20] conducted a numerical study on the strength of X-joints stiffened with T-shaped ring stiffeners under brace axial compression. Lee et al. [10–12] proposed equations for predicting the strength of saddlestiffened tubular T-joints and DT-joints (i.e. X-joints with in-plane brace angle θ = 90°) subjected to brace axial compression without considering chord axial stress effect. Thandavamoorthy et al. [16–19] conducted experimental and numerical studies on the strength of internally ring-stiffened tubular T- and Y-joints. Gandhi et al. [6,7] experimentally investigated the fatigue performance of internally ring-stiffened tubular T- and Y-joints. Mei et al. [13] conducted experimental and numerical studies on the strength of internally ringstiffened X-joints subjected to brace axial compression. Ahmadi et al. ⁎ Corresponding author. E-mail address: [email protected] (F. Wang).

http://dx.doi.org/10.1016/j.jcsr.2016.06.012 0143-974X/© 2016 Elsevier Ltd. All rights reserved.

[1,2] numerically investigated the SCFs of internally ring-stiffened tubular KT-joints. Wang et al. [21] proposed equations for predicting the strength of crown-stiffened tubular T- and Y-joints subjected to brace axial compression or tension considering chord axial stress effect. Although some studies have been conducted to investigate mechanical performance of internally ring-stiffened tubular joints, there is a lack of research on the determination of strength of crown- and saddlestiffened tubular DT-joints subjected to brace axial compression or tension considering chord axial stress effect. In this paper, finite element and theoretical studies on static strength of internally ring-stiffened DT-joints (see Fig. 1) subjected to brace axial compression or tension were carried out. Based on the identified collapse mechanism, theoretical models and corresponding equations for stiffener strength prediction in crown- and saddle-stiffened DT-joints subjected to brace axial loading were presented. Considering chord axial stress effect, an equation for predicting strength of the stiffened joints was proposed. 2. Finite element modeling Using the non-linear finite element software ABAQUS/Standard v6.10-1 [22], the finite element (FE) analysis was carried out. A shell element S4R from the ABAQUS library was used to model unstiffened and internally ring-stiffened DT-joints. The mesh size of the joints analyzed in Sections 3, 4 and 5 was carefully determined by a convergence study. It was found that the mesh size of 20 × 20 mm (length by width) for all joints could produce accurate joint strength predictions with reasonably

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Nomenclature d t l bw θ τ η Nfi COV d1 t1 l1 tw r γ β Nei rsi

chord outer diameter chord wall thickness chord length depth of stiffener in-plane brace angle stiffener thickness to chord wall thickness ratio (=tw/t) stiffener depth to chord outer diameter ratio (=bw/d) stiffener strength obtained from finite element analysis coefficient of variation brace outer diameter brace wall thickness brace length thickness of stiffener joint strength enhancement rate chord outer diameter to twice chord wall thickness ratio (=d/2 t) brace to chord outer diameter ratio (=d1/d) stiffener strength calculated from proposed equations strength ratio (=Nei/Nfi)

computational cost. Both geometric and material nonlinearities were considered. The nonlinear geometric parameter (*NLGEOM) in ABAQUS was used to consider the effect of large displacement. The yield stress (fy), elastic modulus (E) and the Poisson's ratio (v) of steel Q345B which is commonly used in Chinese construction industry are 345 N/mm2, 206GPa, and 0.3 respectively. The material curve used was an elastic-perfectly plastic curve. In ABAQUS simulation, Von-Misses yield criterion was adopted. The weld was not included in the modeling due to its insignificant effect on the stiffened joint strength [10–12]. The joint strength was determined by the peak load or 3% indentation (i.e. deformation limit up to 3% of chord outer diameter, d) in load-displacement curves. If the peak load at a deformation smaller than 3%d, then the peak load is considered to be the joint strength; if the peak load at a deformation larger than 3%d, then the load at the deformation of 3%d is considered to be the joint strength. Fig. 2 shows the boundary conditions and loading modes. One end of the chord was given a fixed boundary condition and the other end was constrained to displace along the chord axis. The brace axial translation and three rotations were unconstrained with the other two translations being restricted. The brace axial compression and tension loadings were applied by displacement at brace ends. The loads were applied in increments by using the “Static” method in the ABAQUS library.

Fig. 2. Boundary conditions and loading modes.

To validate the numerical modeling of the stiffened DT-joints adopted in this paper, the published internally ring-stiffened DT-joint strength data [12,13] were used. The result of the validation study is shown in Table 1. It can be seen that the strengths of all six joints have been predicted to well within 10% of the published data. It therefore confirms the accuracy of the finite element model adopted.

3. Effect of stiffener position A numerical study on 264 ring-stiffened DT-joints and corresponding unstiffened joints was conducted to investigate the effect of stiffener position on the enhancement of joint strengths. The joint dimensions are chord outer diameter d = 800 mm, chord wall thickness t = 20 mm, chord length l = 4800 mm; brace outer diameter d1 = 300 mm, 350 mm, 400 mm, 450 mm, 500 mm, 550 mm, 600 mm, 650 mm, 700 mm, 750 mm, 790 mm, brace wall thickness t1 = 16 mm, brace length l1 = 2400 mm; stiffener width bw = 120 mm. The chord length is equal to 6 times of chord outer diameter (l = 6d) to ensure that the stresses at the brace-chord intersection are not influenced by the ends of the chord, and the brace length is chosen to prevent occurrence of local buckling of brace members [10–12]. The ratio (β) of brace outer diameter (d1) to chord outer diameter (d) ranges from 0.375 to 0.9875. Stiffeners were positioned at a distance of x(=αlsc, where α = 0,0.1,…,1.0, lsc is the distance between the saddle and the crown) from the saddle position (see Fig. 1 for the joint parameter definition). For saddle-stiffened joints (α = 0), only one stiffener with stiffener thickness tw = 32 mm was used. For the crown position (α = 1.0) and other stiffener positions (α = 0.1,0.2,…,0.9), two stiffeners each with tw = 16 mm were used to ensure the same steel consumption. The joint strength enhancement rate is defined as r = (Ns − Nu) / Nu, where Ns is stiffened joint strength, Nu is unstiffened joint strength. Curves of r-α are shown in Fig. 3. The relationships between joint strength enhancement rate (r) and stiffener position (α) are summarized as follows: Table 1 Results of the validation study. Specimen a

Fig. 1. Internally ring-stiffened DT-joint configuration.

DT-W(0.4,0.2) [12] DT-W(0.6,0.2)a [12] DT-W(0.8,0.2)a [12] Specimen 1 [13] Specimen 2 [13] Specimen 3 [13]

FE/kN

Published Data/kN

FE/Published Data

9288 10460 11250 3580 5778 5904

8616 9868 10543 3900 5300 6200

1.08 1.06 1.07 0.92 1.09 0.95

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Fig. 3. Relationships between r and α.

(1) Fig. 3(a) shows that the stiffener could provide higher r values when 0 ≤ α b 0.5. The value of r is relatively lower and decreases with increasing α when 0.5 ≤ α ≤ 1. (2) Fig. 3(b) shows that for 0.375 ≤ β ≤ 0.75, the relationships between r and α in the case of brace in tension are generally the same as the case of brace in compression. For 0.75 b β ≤ 0.9875, the value of r is relatively lower, indicating that comparatively the stiffener could not enhance the joint strength effectively. Thus, this joint set (0.75 b β ≤ 0.9875) was excluded in the case of brace in tension in the following analysis. 4. Stiffener strength The parameter ranges of analyzed joints, which are common in practice and recommended by CECS 280:2010 [4] and EN 1993-1-8 [5], are shown in Table 2. The stiffened joint strength was defined as the sum of unstiffened joint strength and stiffener strength [12]. This section describes the theoretical and numerical investigations on the stiffener strength. Four cases as shown in Table 3 were studied. 4.1. Failure mechanism Fig. 4(a)–(d) show typical yielding patterns within stiffeners in Cases 1–4 (see Table 3). The distinct highly strained zones (in red and green colors) which become plastic indicate the existence of plastic hinges in the stiffeners. These plastic hinges therefore suggest that bending action is the main load-carrying mechanism of the stiffeners. Fig. 4(e) shows that an area (in a red color) in vicinity of brace-chord intersection yields which extends along the length of the chord. Crown- and saddle-stiffened DT-joints subjected to brace axial compression or tension fail when the chord wall yields and plastic hinges form in those highly strained areas in the stiffener. It is therefore reasonable to assume that the stiffener and chord wall interact in providing joint strength enhancement. The cross section of the plastic hinge is postulated to be that of a T-section (see Fig. 5).

Table 2 Ranges of joint parameters analyzed. Joint parameter

Parameter range

d(mm) t(mm) l(mm) d1(mm) t1(mm) l1(mm) bw(mm) tw(mm)

800 20 4800 300, 350, 400, 450, 500, 550, 600, 650, 700 16 2400 80, 120, 160, 200, 240 16, 18, 20, 22, 24

4.2. Stiffener strength prediction for Case 1 4.2.1. Theoretical model The collapse of the stiffened joints in Case 1 is assumed to be by a four-hinge failure mechanism which has both vertical and horizontal planes of symmetry. Fig. 6(b) shows a forth model, above the horizontal plane of symmetry. The moment equilibrium equation in Fig. 6(b) is given by 2M P ¼ P=2  d=2

ð1Þ

where Mp is full plastic moment resistance of the T-section (see Fig. 5), P is the strength of a stiffener. The total stiffener strength (ΔNX) is given by ΔNX ¼ 2P ¼

16M P : d

ð2Þ

4.2.2. Deduction for Mp Detailed deduction for the full plastic moment of resistance (Mp) was carried out by Lee et al. [12]. Herein a brief summary is presented. In Fig. 5, the depth of the centroidal axis X-X (ye) is given by

ye ¼


 2 0:5 bc t 2 þ t w bw þ tt w bw

ð3Þ

A

where the area of the T-section (A) is given by A ¼ bc t þ bw t w :

ð4Þ

The position of neutral axis (yp) is given by
 A 0byp;c ≤t or yp;w 2bc  A þ 2t ðt w −bc Þ
¼ tbyp;w ≤t þ bw : 2t w

yp;c ¼

ð5Þ

Table 3 Summary of analyzed cases. Case studies

Brace loading

Stiffener position

Case 1 Case 2 Case 3 Case 4

compression compression tension tension

crown saddle crown saddle

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Fig. 6. A four-hinge failure mechanism.

to stiffener strength obtained from finite element analysis for the corresponding joint. To readily calculate the value of bc, an equation for predicting the value of bc was obtained by investigating the effects of non-dimensional geometric parameters, β(=d1/d), η(=bw/d) and τ(=tw/t) on non-dimensional bc,th/d. Near quadratic relationships between bc,th/d and β, η as well as τ, respectively were found. The relationship between bc,th/d and γ(=d/2 t, 10 ≤ d/2 t ≤ 30) is linear [12]. Based on above observations, multiple regression analyses were carried out using the values of bc,th of 225 ring-stiffened joints. An equation for predicting the value of bc is given by     bc ¼ −2:005  τ 2 −2:613τ þ 1:165  η2 −0:407η þ 0:023 d
  β2 −0:958β þ 0:273  ð2γ−266:947Þ Fig. 4. Highly strained zones in the stiffened joints at the ultimate load.

The validity ranges of Eq. (8) are 0.375 ≤ β ≤ 0.875, 0.8 ≤ τ ≤ 1.2, 0.1 ≤ η ≤ 0.3, 10 ≤ γ ≤ 30, θ = 90°.

Respectively, the plastic section modulus (Zp) is given by. Z p;c ¼

  A ye − A or 4bc

Z p;w ¼ ðt w −bc Þt 2 þ Aye −t w y2p;w :

ð6Þ

or M P ¼ Z p;w f y :

ð7Þ

4.2.3. Calculation of effective flange width The effective flange width (bc) has to be determined to obtain the stiffener strength using Eqs. (2)–(7). The value of bc (bc,th) was calculated by an iterative method using Excel VBA program. The value of bc was iterated until the stiffener strength calculated from Eqs. (2)–(7) is equal

Fig. 5. T-section of the plastic hinges.

4.2.4. Stiffener strength prediction Given the geometric parameters of the stiffened joints, the effective flange width (bc) can be determined using Eq. (8). This allows the determination of the plastic moment of resistance (Mp) using Eq. (7). By substituting Mp into Eq. (2), the stiffener strength could be obtained. 4.3. Stiffener strength predictions for Cases 2–4

Plastic moment of resistance (Mp) is given by MP ¼ Z p;c f y

ð8Þ

Fig. 4(b)–(d) show that the stiffened DT-joint in Cases 2–4 (see Table 3) fails when six, four and six plastic hinges form in the stiffener, respectively. Thus, the collapse of the stiffened joints in Cases 2–4 is assumed to be by six-, four- and six-hinge failure mechanism, respectively. The same procedures were carried out as in Section 4.2 to determine the stiffener strength in Cases 2–4. Table 4 shows a brief summary. Forth models are shown in the schematic diagram column. The unstiffened and corresponding stiffened joints fail through the same mechanism [12]. Thus, the value of c1 is the same as the coefficient in the strength equation for unstiffened DT-joints subjected to brace axial loading in CECS 280:2010 [4] and EN-1993-1-8 [5] (i.e. c1 = 0.81). The procedure of stiffener strength calculation in Cases 2–4 is similar with that described in Section 4.2.4. Given the geometric parameters of the stiffened DT-joints, the value of bc could be determined using the equation of bc as shown in Table 4. Thus, the plastic moment of resistance (Mp) could be determined using Eqs. (3)–(7). The total stiffener strength could be obtained by substituting the value of Mp into the equation of ΔNX (see Table 4). The validity ranges for Case 2 are 0.375 ≤ β ≤ 0.875, 0.8 ≤ τ ≤ 1.2, 0.1 ≤ η ≤ 0.3, 10 ≤ γ ≤ 30, θ = 90°, for Case 3 are 0.375 ≤ β ≤ 0.75, 0.8 ≤ τ ≤ 1.2, 0.1 ≤ η ≤ 0.3, 10 ≤ γ ≤ 30, θ = 90°, for Case 4 are 0.375 ≤ β ≤ 0.75, 0.8 ≤ τ ≤ 1.2, 0.1 ≤ η ≤ 0.2, 10 ≤ γ ≤ 30, θ = 90°.

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Table 4 Stiffener strength predictions for Cases 2–4. bc

ΔNX

Case 2

bc d

¼ 0:241  ðτ þ 2:673Þ ðη2 −0:727η þ 0:144Þ ðβ 2 −1:397β þ 0:533Þ ð2γ−9:331Þ

8MP ΔN X ¼ P ¼ d−c 1 d1

Case 3

bc ¼ −2:192  ðτ 2 −2:767τ þ 1:107Þ d ðη2 −0:408η þ 0:022Þ ðβ 2 −0:502β þ 0:096Þ ð2γ−202:659Þ

P ΔN X ¼ 2P ¼ 16M d

Case 4

bc d

8MP ΔN X ¼ P ¼ d−c 1 d1

Case studies

Schematic diagram

¼ −0:203  ðτ−3:860Þ ðη2 −1:162η þ 0:276Þ ðβ 2 −0:320β þ 0:068Þ ð2γ−18:594Þ

4.4. Assessment of proposed equations The statistical analysis of strength ratio rsi (=Nei/Nfi, where Nfi and Nei denote numerical and theoretical stiffener strength of joint i, respectively) for the stiffened joints calculated in Cases 1–4 is shown in Table 5. The total numbers of the stiffened joints (n) analyzed in Cases 1–4 are 225, 225, 175, and 105, respectively. The relative error (ei ), the average relative error (e), relative standard deviation (s⁎) and average relative standard deviation (s) are defined as follows [8]: ei ¼

Nfi −Nei ði ¼ 1; 2; 3; …; nÞ Nfi

ð9Þ

s ¼

i¼1

ð12Þ

n−1

The mean values of rsi are 0.99, 0.99, 1.01 and 1.00 with corresponding coefficients of variation (COV) of 0.06, 0.09, 0.09 and 0.09 for Cases 1–4, respectively (see Table 5). The values of e, s⁎ and s in Cases 1–4 are lower than 7.32%, 9.28% and 6.76%, respectively. Such values indicate that the theoretical models and corresponding equations could provide stiffener strengths with acceptable accuracy. 5. Stiffened joint strength 5.1. Effect of chord axial stress

n   X e  i

e¼

s¼

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n   u e −e 2 u i t

i¼1

ð10Þ

n vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi uX n   u e 2 u i t i¼1

The effect of chord axial stress on unstiffened X-joint strength is significant. Thus, design codes (e.g. CECS 280:2010 [4], EN 1993-1-8 [5]) propose a chord stress function for unstiffened X-joints to consider such effect. The effect of chord axial stress on joint strength is defined by ψn = NX1/NX2, where NX1 and NX2 denote the strength of joints

ð11Þ

n−1

Table 6 Parameter ranges of stiffened joints analyzed. Table 5 Statistical analysis of rsi. Case studies

Case 1

Case 2

Case 3

Case 4

Mean COV e s⁎

0.99 0.06 3.55% 5.77% 5.04%

0.99 0.09 6.79% 8.67% 5.48%

1.01 0.09 6.30% 9.25% 6.76%

1.00 0.09 7.32% 9.28% 5.99%

s

Joint parameter

Parameter range

d(mm) t(mm) d1(mm) t1(mm) bw(mm) tw(mm) σ/fy

800 20 300, 400, 500, 600, 700 16 120 16 ±0.8, ±0.6, ±0.4, ±0.2, 0

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Fig. 7. Curves of ψn −σ/fy of joint set with β = 0.625.

with and without chord axial stress, respectively. In CECS 280:2010 [4] and EN 1993-1-8 [5], the chord axial compression stress function is as follow: ! !2 σ σ ψn ¼ 1−0:3 −0:3 fy fy

ð13Þ

where σ and fy are chord axial compression stress and steel yield stress, respectively. For chord axial tension, ψn = 1. A parametric study on 270 stiffened and corresponding unstiffened DT-joints was carried out to investigate the effect of chord axial stress on the joint strength. Table 6 shows parameter ranges of crown-stiffened, saddlestiffened and corresponding unstiffened joints analyzed. Negative and positive values of σ/fy refer to chord axial compression and tension, respectively. Herein, the definition of ψn (=NX1/NX2) was also adopted to evaluate the effect of chord axial stress on the stiffened joint strength. Fig. 7 shows curves of ψn − σ/fy of the joint set with β = 0.625. It could be seen that the ψn value decreases with increasing chord axial compression stress. The ψn value slightly increases and then decreases with increasing chord axial tension stress. Curves of ψn −σ/fy of other joint sets generally exhibited the same trends as the joint set with β = 0.625 and are therefore not shown herein. The parametric study shows that all curves of ψn − σ/fy are on the top of the curve of Eq. (13). As a conservative purpose for engineering design, Eq. (13), where σ denotes the absolute value of chord axial stress, is adopted to consider the effect of chord axial stress on the stiffened joint strength. 5.2. Stiffened joint strength

Acknowledgments The authors gratefully acknowledge the financial support provided by Guangzhou Science Technology and Innovation Commission (No. 201506010038) and the State Key Laboratory of Subtropical Building Science, South China University of Technology (No. 2014KB29, 2015ZB30).

References

The static strength of unstiffened DT-joints under brace axial loading (Nu) could be calculated from parametric equations in existing literatures (e.g. design codes [3–5]). In conjunction with the published unstiffened DT-joint strength equations, an equation for predicting the strength of the stiffened DT-joints (Ns) in Cases 1–4 is proposed as follow: Ns ¼ ψn ðNu þ ΔNX Þ

brace axial compression or tension. The conclusions are summarized as follows: (1) In the joint parameter ranges of Cases 1–4, the stiffener could provide higher joint strength enhancement rate (r) when 0 ≤ α b 0.5, and r is relatively lower and decreases with increasing α when 0.5 ≤ α ≤ 1. (2) The failure mechanism of crown- and saddle-stiffened joints subjected to brace axial compression or tension could be characterized as formation of plastic hinges in highly strained areas of the stiffener and chord wall yielding in the vicinity of brace-chord intersection. The proposed theoretical models and corresponding stiffener strength equations could produce reasonably accurate estimation of the stiffener strength. (3) Chord axial stress has a significant effect on the stiffened DT-joint strength. A chord stress function was proposed to consider the effect of chord axial stress. In conjunction with existing unstiffened DT-joint strength equations, a strength equation for the stiffened DT-joints was proposed.

ð14Þ

6. Conclusions This paper presents the results of numerical and theoretical studies on static strength of internally ring-stiffened DT-joints subjected to

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