Modelling and probabilistic study of the residual stress of cold-formed hollow steel sections

Engineering Structures 150 (2017) 986–995

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Modelling and probabilistic study of the residual stress of cold-formed hollow steel sections Wenyu Liu, Kim J.R. Rasmussen, Hao Zhang ⇑ The University of Sydney, Australia

a r t i c l e

i n f o

Article history: Received 16 March 2017 Revised 10 July 2017 Accepted 2 August 2017

Keywords: Cold-formed steel Residual stress 3D steel frames Advanced analysis Inelastic analysis Nonlinear frame analysis Probabilistic study

a b s t r a c t Cold-formed Hollow Steel Sections (HSS) are widely used in the construction industry. The distribution of residual stress in non-stress-relieved cold-formed HSS is complex due to the highly non-uniform variation around the cross-section and through the plate thickness. Modeling residual stress in frame analysis is therefore a difficult task. This paper presents a practical method for approximating the effects of different components of residual stress on the behavior of cold-formed HSS by modifying the steel stress-strain curve. The method proposes a convenient means of including residual stress in beam element-based nonlinear frame analysis. A probabilistic study is then carried out to study the effect of the uncertainty in residual stress on frame strength. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Cold-formed Hollow Steel Sections (HSS) are broadly used in the construction industry due to their superior mechanical properties and aesthetic appeal. A common method to produce HSS is by (1) uncoiling and levelling (flatten a sheet coil), (2) roll-forming (bend the sheet coil progressively along the width direction), (3) welding (seam-weld the flange tips of the bent strip to a closed circular hollow section (CHS)), and (4) sizing (finish the exact shape to form Rectangular Hollow Section (RHS) or Square Hollow Section (SHS)) [1]. Since there is a large permanent local deformation of the walls of HSS due to the longitudinal and transverse bending effects, residual stresses often exist in different parts of the section. The strength and behaviour of cold-formed steel structural members may be greatly influenced by the presence of residual stress [2,3]. The residual stresses in a cold-formed steel sections have a substantial bending part and a comparatively small membrane part [2], different from the thermally induced residual stress in hot-rolled and fabricated sections [4]. The distribution of residual stresses in cold-formed structural steel members have been investigated experimentally and analytically. Davison and Birkemoe [5] presented the statistical data for the yield stress and

⇑ Corresponding author. E-mail addresses: [email protected] (W. Liu), kim.rasmussen@sydney. edu.au (K.J.R. Rasmussen), [email protected] (H. Zhang). http://dx.doi.org/10.1016/j.engstruct.2017.08.004 0141-0296/Ó 2017 Elsevier Ltd. All rights reserved.

residual stress of HSS. The data was based on experimental results obtained for stub and full-sized HSS to study the behaviour of cold-formed heat-treated and non-heat-treated HSS columns. Longitudinal bending residual stresses were measured by Rasmussen and Hancock [6,7] when investigating stainless steel sections subjected to compression and bending. The residual stress distributions in cold-bent thin steel plates were investigated experimentally and numerically by various researchers [8–10]. Key and Hancock [11] performed experiments on cold-formed Square Hollow Sections (SHS) and proposed a model for the longitudinal and transversal residual stress patterns. Li et al. [1,12] investigated the distribution and magnitude of the residual stress due to the cold-forming process for HSS members. Jandera et al. [13] explored the presence and influence of residual stress in cold-formed stainless steel box sections using experimental and numerical techniques and found that the effect of residual stress on the column buckling of SHS varies with the slenderness of the section. The effect of residual stresses on cold-formed HSS members has also been studied numerically using shell-element based finite element analysis [13,14]. Previous studies have used the Ramberg-Osgood material models to approximate the effects of residual stress on the behavior of cold-formed high strength steel sections [4,11,15–18]. The parameters of the Ramberg-Osgood model (e.g., the parameter controlling the transition from elastic to plastic state) needs to be determined from measured material properties typically obtained

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stress around cross-section in the longitudinal and transversal directions shown in Figs. 1 and 2, respectively. Somodi & Kövesdi [15] and Ma et al. [21] examined the residual stress distribution of cold-formed high strength steel sections. Although the steel grades were different from those examined in [11], the residual stress patterns observed in these studies are comparable, and some common observations are:

from coupon tests [17]. Moreover, this approach tends to overestimate the resistance of the member in the low slenderness range [17]. For hot-rolled I-section members, one practical way to include the effect of thermal residual stress on the behavior of highly compressed members is to use a tangent modulus approximation, i.e., to reduce the Young’s modulus based on the member’s axial force (e.g., using the AISC column curve). The Young’s modulus may be further reduced based on major and minor axis bending moments (e.g., see [19]). However, such an approach cannot be used for cold-formed HSS, since the residual stress of coldformed HSS is completely different from the thermal residual stress in hot-rolled sections. This paper develops a practical method to approximate the effects of different components of residual stress on the behavior of cold-formed HSS by modifying the stress-strain curve, so that it can be conveniently used in beam element-based nonlinear analysis. The influence of different components of residual stress on frame ultimate strength is investigated numerically. A probabilistic study is then carried out to quantify the effect of the uncertainty in residual stress on frame strength.

1. The longitudinal bending residual stresses in the corner zone are significantly smaller than the values measured in the middle of the plates. 2. The longitudinal membrane residual stress is in compression at the corner and tension in the middle of the plate. 3. The average magnitude of transverse membrane residual stress is zero. 2.2. Residual stress through the section wall thickness The model for the variation of residual stress through the section wall thickness is based on the experimental measurements in [11], which consists of three components in the longitudinal and transversal directions. The panel removal residual stress was modelled as the membrane and bending components of residual stress in the longitudinal and transversal directions, whereas the released residual stresses from layering was modelled as layering residual stress in the longitudinal and transversal directions. The analytical model satisfies the requirement of zero net axial force and moment [11]. The distributions of through-wall thickness residual stress in the longitudinal and transversal directions are shown in Figs. 3 and 4, respectively.

2. Residual stress in cold-formed steel tubes 2.1. Residual stress around the cross-section The residual stress pattern for cold-formed steel tubes has been reported in the literature. The residual stress involves two parts, including: (1) variation of the residual stress through crosssection wall thickness, and (2) magnitude and distribution of the residual stress around the cross-section. Three components of through-thickness residual stress in each direction exist, including: (1) membrane, (2) bending, and (3) layering residual stress. Descriptions of the different residual stress components around cross-section are presented in Table 1. Based on the experimental investigation of four cold-formed normal strength HSS, Key and Hancock [11] proposed the distribution patterns of the residual

3. Simplified models of residual stress The full residual stress pattern shown in Figs. 1–4 can only be incorporated into Shell Finite-Element or Finite-Strip models. To incorporate the effect of residual stress in beam element-based

Table 1 Descriptions of different residual stress components around cross-section. Longitudinal membrane (rl;m ) Longitudinal bending (rl;b ) Longitudinal layering (rl;l ) Transversal membrane (rt;m ) Transversal bending (rt;b ) Transversal layering (rt,l)

Stresses varying from maximum tensile at the centre of each face to maximum compressive at the corner Uniform stress over each flat face of the section with half the face value at the corners Same distribution as longitudinal bending residual stress, as suggested in [5] Magnitude of transversal membrane residual stress is assumed to be zero as suggested by Key and Hancock [11] Uniform distribution across each section face as proposed in [20] Same distribution as transversal bending residual stress, as suggested in [5]

l,m

2t

2t 2t

l,m

l,b/2

2t

2t 2t

l,l/2 l,l

l,b

t: wall thickness 2t

2t

Longitudinal membrane

Longitudinal bending residual

Longitudinal layering residual

stress

stress

Fig. 1. Residual stress distribution around cross-section in longitudinal direction [11].

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t,m=0

t,l

t,b

Transverse membrane

Transverse bending

Transverse layering

Fig. 2. Residual stress distribution around cross-section in transversal directions [11].

Outside

Analytical Model: 1. Membrane 2. Bending

Analytical model Through thickness (t)

Through thickness (t)

Outside

Measured longitudinal panel removal residual stress

Measured transverse layering residual stress

Inside -400

-300

-200

-100

0

100

200

300

-200

Stress (MPa)

(a) Longitudinal direction

-100

0 Stress (MPa)

100

(a) Longitudinal direction Outside

Through thickness (t)

Through thickness (t)

Outside

Bending analytical model and membrane transverse panel removal residual stress

Membrane = 0

Inside 200

Analytical model

Measured transverse layering residual stress

Inside -400

-300

-200

-100

0

100

200

300

Stress (MPa)

Inside

(b) Transverse direction

-200

Fig. 3. Through-wall thickness membrane and bending residual stress (redrawn from [11]). Tension is positive.

finite element analysis, it is proposed to approximate the effects of different components of residual stress by modifying the elasticperfectly-plastic stress-strain curve. According to von Mises yield criterion, a plate in a state of plane stress (rz = 0) will yield when a combination of in-plane stresses (rx, ry, sxy) reaches the uniaxial yield stress (Fy) as follows:

F 2y ¼ r2x þ r2y  rx ry þ 3s2xy

ð1Þ

where rx and ry are the normal stresses in the longitudinal and transversal directions, respectively, and sxy is the shear stress. Let x in Fig. 5 denote the longitudinal direction of a cold-formed steel tube and assume a stress (r) is applied, which is constant through

-100

0 Stress (MPa)

100

200

(b) Transversal direction Fig. 4. Through-wall thickness layering residual stress (redrawn from [11]). Tension is positive.

the thickness. In the presence of residual stress, the stress components (rx, ry, sxy) are:

rx ¼ r þ rl ; rl ¼ rl;m þ rl;b þ rl;l ;

ð2Þ

ry ¼ rt ; rt ¼ rt;m þ rt;b þ rt;l ;

ð3Þ

sxy ¼ 0;

ð4Þ

where rl and rt denote the total residual stresses in the longitudinal and transversal directions, respectively.

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x

xy

(a ) C old- fo r me d s te el tu be

(b) Stress components

Fig. 5. Stress components in SHS.

The present study develops a simple method for modifying the stress-strain curve to account for the residual stresses in the longitudinal and transversal directions for the flat material. The procedure is summarized as follow: 1. Apply an increment of strain (Dei ) to the section. 2. Divide the section into a number of layers (e.g. thirteen layers as suggested in [11]) and monitor the stress at each node between the layers as shown in Fig. 6. 3. At each monitoring node, assume first the material is elastic and calculate the change in stress (Dri ) as :

Dri ¼ EDei

ð5Þ

where E is the Young’s modulus. 4. Determine the total stress in the longitudinal and transversal directions as:

rx

X ¼ Dri þ rl ; ry ¼ rt ; sxy ¼ 0:

ð6Þ

i

5. Substitute (rx, ry, sxy) into Eq. (7) for the effective stress (re):

r2e ¼ r2x þ r2y  rx rx þ 3s2xy :

ð7Þ

6. Check if re < Fy. 7. If the effective stress is less than the yield stress, the increment in applied stress calculated as Dri ¼ EDei is correct and the total applied stress is determined as:

r¼

X

Dri :

ð8Þ

i

8. If the effective stress exceeds the yield stress, it must be scaled back in order to satisfy re ¼ F Y . This can be done

rigorously using the Prandtl-Reuss flow rules and would imply changes to both rx and ry. For simplicity, without little loss in accuracy, it is proposed to simply scale back the applied stress (Dri ), i.e. write (Dri ) as:

Dri ¼ aEDei :

ð9Þ

9. Determine a so that re ¼ F Y . Note that since the applied strain (e) is increased monotonically, once yielding has commenced at a given monitoring node, the effective stress will remain at the yield stress and hence Dri remains zero as the strain is further increased at that node. 10. Determine the stress at each monitoring node as described in Step 2. Then calculate the average applied stress (rav g ) as follows:

rav g ¼

R P t



ri dt

iD

t

:

ð10Þ

11. Repeat Steps 1–10 to obtain a series of points of applied stress and applied strain (rav g ; eÞ where:

e¼

X

Dei :

ð11Þ

12. Based on Steps 1–11, the stress-strain curve which approximates the effect of residual stress in the longitudinal and transversal directions can be obtained, as illustrated in Fig. 7. Since (i) the residual stress in the longitudinal direction is generally larger than the residual stress in the transversal direction, and (ii) the residual stress in the longitudinal direction is dominated by the bending component, it is expected that the residual stresses can be modelled with a reasonable accuracy by only considering the longitudinal bending component of the residual stress.

350 300

σavg(MPa)

250 200 150

Without considering residual stress Considering all six components of residual stress

100 50 0 0.000

0.001

0.002

0.003

ε Fig. 6. Monitoring nodes through the section thickness.

Fig. 7. Comparison of the stress-strain curves with and without residual stresses.

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To verify this assertion, Steps 1–12 were repeated but calculating the stresses using Eq. (12) instead of Eq. (6):

rx ¼

X Dri þ re;b ; ry ¼ 0; sxy ¼ 0:

ð12Þ

i

The stress-strain curve with the longitudinal bending component of residual stress is plotted in Fig. 8 and compared with the stress-strain curve considering all components of residual stresses in both directions. As can be seen in Fig. 8, the approximate stressstrain curve accounting for only the longitudinal bending residual stress is reasonably close to the ‘‘accurate” stress-strain curve. Thus in engineering practice, it is proposed to only consider the longitudinal bending residual stress. The approximate stress-strain curve considering only the longitudinal bending residual stress as shown in Fig. 8 can be fitted with a cubic polynomial function as follows:

r ¼ X1 e3 þ X2 e2 þ X3 e1 þ X4 ; for e1 < e < e2 ;

ð13Þ

in which X1, X2, X3 and X4 are constants, e1 and e2 are used to define the nonlinear transition from the elastic range to the yield plateau of the stress-strain curve, and e1 and e2 can be solved using Eqs. (14) and (15), respectively,

e1 ¼

F y  rb ; E

2ðe21 þ e1 e2 þ e22 ÞE  3ðe1 þ e2 ÞF y ðe1  e2 Þ3

 X3 ¼

4e21 E þ e1 e2 E þ e22 E  6e1 F y ðe2  e1 Þ3



e2

;

ð21Þ

;

ð22Þ

and

 X4 ¼

2e22 E þ e1 F y  3e2 F y



e21

ðe1  e2 Þ

3

:

ð23Þ

With X1, X2, X3 and X4 determined, the polynomial function can be expressed as:

r ¼ Ee for 0 < e < e1 ; r¼

ð14Þ

ð24Þ 



e21 þ e1 e2 þ e22 E  3ðe1 þ e2 ÞF y 2 e 3 ðe1  e2 Þ ðe1  e2 Þ3 ð4e21 E þ e1 e2 E þ e22 E  6e1 F y Þe2 1 þ e ðe2  e1 Þ3  2  2 2e2 E þ e1 F y  3e2 F y e1 þ for e1 < e < e2 ; ð25Þ ðe1  e2 Þ3 2F y  ðe1 þ e2 ÞE

e3 þ

2

and

and

e2 ¼

F y þ rb ; E

ð15Þ

in which rb is the magnitude of the longitudinal bending residual surface stress. To solve the four constants in Eq. (13), four initial conditions are defined in Eqs. (16)–(19).

dr ¼ E at e ¼ e1 ; de

ð16Þ

dr ¼ 0 at e ¼ e2 ; de

ð17Þ

e¼

X2 ¼

r E

at e ¼ e1 ;

ð18Þ

and

r ¼ F y at e ¼ e2 :

ð19Þ

By substituting Eqs. (16)–(19) into Eq. (13), the four constants can be obtained as:

X1 ¼

2F y  ðe1 þ e2 ÞE ðe1  e2 Þ3

;

ð20Þ

r ¼ F y for e > e2 :

Fig. 9 compares the stress-strain curves obtained by Eqs. (7)– (11) and by the cubic polynomial function (Eqs. (24)–(26)). It can be seen that the two stress-strain curves are almost identical, indicating that the proposed approximate cubic stress-strain curve is sufficiently accurate for engineering practice. Based on the proposed stress-strain curves, the influence of residual stress on the ultimate strength of steel frames was studied by using beam element-based geometric and material nonlinear analysis (advanced analysis). Two sample frames with the configurations shown in Fig. 10 were considered. Concentrated vertical reference loads of 500 kN were applied on each beam-column joint. The Young’s modulus and yield stress are 200 GPa and 350 MPa, respectively. The 3D plastic zone beam-column element (B31) in ABAQUS is used to trace the spread of plasticity through the cross-section and along the member length. Using the incremental load deflection response, the element geometry in each load increment is updated to capture the second-order effects. Based on a convergence study, it was found that a mesh size of 200 mm is sufficient, resulting in typically 20–40 elements per beam/column member. The frames are deliberately designed such

400

350

350

300

300

Fy

250

200 150 Without residual stress Longitudinal bending residual stress only Considering all six components of residual stress

100 50 0.001

ε

0.002

Fig. 8. Comparison of stress-strain curves.

0.003

σ (ΜPa)

σavg(MPa)

250

0 0.000

ð26Þ

σb

200 150 100 50

Considering the cubic polynomial function Considering the longitudinal bending residual stress only

Ε

0 0.000

ε1

ε

0.002

ε2

Fig. 9. Comparison of the stress-strain curve considering only longitudinal bending residual stress and the approximate cubic polynomial function.

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W. Liu et al. / Engineering Structures 150 (2017) 986–995

H=3x 6 m=18 m

y 4m x

z

8m

y

8m z

6m

x

6m Frame 1

Frame 2 Fig. 10. Frame configurations.

rffiffiffiffiffi Fy E

ð27Þ

where r is radius of gyration. Note that when kc = 1.0, the squash load and the elastic buckling load of the column coincide and this would produce the greatest, or almost the greatest, sensitivity to residual stress [14]. The cross-sections used for the two sample frames can be found elsewhere [22]. Four cases for residual stress modelling are considered and summarised in Table 2. The load-displacement curves for Frame 1 and Frame 2 for the four cases are shown in Figs. 11 and 12, respectively. The ultimate load factors of the two frames for the four cases are presented in Table 3. It can be seen from Table 3 that ignoring residual stress (Case 1) leads to significant errors, i.e., 39.14% for Frame 1 and 22.71% for Frame 2. By including the longitudinal bending residual stress (Case 3), the accuracy of results is significantly improved, with a relative error of less than 4% for both frames compared to Case 2 where all residual stress components are included. This result confirms that the effect of residual stress in HSS can be accounted for with reasonable accuracy by only modelling the longitudinal bending component of residual stress. Table 3 also shows that the results from Case 3 and Case 4 are very similar, suggesting that the polynomial function of Eqs. (24)–(26) can sufficiently accurately represent the longitudinal bending residual stress. It should be mentioned that the two frames were both designed with the column slenderness parameter (kc) close to unity to maximize the effect of residual stress. In practical situations, the effect of residual stress would be less severe since it is unlikely that all columns have a slenderness ratio close to unity. To represent a more practical situation, the point loads on Frame 2 were replaced by a uniformly distributed load of 50 kN/m applied along the

Table 2 Descriptions of the four cases for residual stress modelling. Case 1 Case 2 Case 3 Case 4

No residual stress The stress-strain curve accounting for all six components of residual stress The stress-strain curve accounting for the longitudinal bending residual stress only The stress-strain curve using the approximate polynomial function of Eqs. (24)–(26)

7

Case Case Case Case

6

Applied load ratio

kL kc ¼ pr

beams, requiring the frame to be redesigned. The cross-sections for the redesigned Frame 2 can be found elsewhere [17]. A comparison of the load-displacement curves from the four cases is presented in Fig. 13. It can be seen that the influence of the residual stress on the frame’s ultimate strength is less pronounced; however, the residual stress still has a significant impact on the stiffness of the frame.

5 4

1 2 3 4

3 2 1 0 0

100

200

300

400

Displacement (mm) Fig. 11. Load-deflection curves for Frame 1.

3.0 2.5 Applied load ratio

that the slenderness parameters (kc) of all columns are close to unity, with kc defined as:

2.0

Case 1

1.5

Case 2 1.0

Case 3 Case 4

0.5 0.0 0

50

100

150

Displacement (mm) Fig. 12. Load-displacement curves for Frame 2.

200

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Table 3 Comparison of the ultimate load factors of Frames 1 and 2.

Case Case Case Case

1 2 3 4

Frame 1

Error (%)

Frame 2

Error (%)

Mean

COV

Distribution

6.77 4.86 5.05 5.10

39.14  3.91 4.94

2.54 2.07 2.15 2.15

22.71  3.86 3.86

0.7Fy

0.05

Lognormal

1.0

Applied load ratio

0.8 Case 1 0.6

Case 2 Case 3

0.4

Case 4

0.2

0.0 0

50

100

150

200

250

Displacement (mm) Fig. 13. Load-displacement curves for the redesigned Frame 2 subject to uniformly distributed beam loads.

Based on the results shown in Figs. 11–13 and Table 3, it can be concluded that (i) the effect of residual stress on the ultimate strength of HSS frames is particular significant when the columns have a slenderness (kc) close to unity, (ii) the effect of residual stress in HSS can be accounted for with sufficient accuracy by only modelling the longitudinal bending residual stress, and (iii) the proposed stress-strain curve based on a quadratic polynomial function can be used to model the longitudinal bending residual stress. 4. Probabilistic study of the residual stress in HSS frames In the new paradigm of advanced analysis-based steel structural design [23,24], all sources of uncertainties in material strength, stiffness, and imperfections need to be accounted for [25–27]. Therefore, it is necessary to investigate the uncertainty in the residual stress and its influence on the probabilistic characteristics of structural strength and stiffness. Davison and Birkemoe [5] reported the experimental measurements of the surface residual stress of 35 cold-formed HSS samples. It was found that the longitudinal bending residual stress has a mean of 0.7Fy with a coefficient of variation (COV) of 0.05. Li et al. [1] conducted experiments on ten normal strength coldformed steel sections. It was found that the magnitude of bending residual stress varies between 0.65Fy to 0.7Fy. Based on the experimental measurement of 13 samples, Somodi & Kövesdi [15] suggested that the mean value of the bending residual stress can be estimated by Eq. (28)

rb ¼ ð0:8F y  67 MPaÞ:

with a COV of 0.05 in the present study. A lognormal distribution is assumed for the longitudinal bending residual stress. Table 4 summarises the statistics for longitudinal bending residual stress. A probabilistic study is carried out to investigate the effect of the uncertainty in residual stress on the distribution of frame ultimate strength. The HSS used in this study was assumed to be C450 grade, with a nominal yield stress of 450 MPa and a nominal Young’s modulus of 200 GPa. The nominal longitudinal bending residual stress (rb) is 315 MPa. The stress-strain curve from Eqs. (24)–(26) using the stated nominal material properties is shown in Fig. 14. Eight moment-resisting space frames with the configurations shown in Fig. 15 were studied. The cross-section sizes for each frame can be found in [22]. A concentrated vertical load of 500 kN is applied at each beam-column joint throughout the frames. The frames were selected and designed such that the column slenderness parameter (kc) (Eq. (27)) was close to unity to maximize the effect of residual stress. Beam element-based advanced analyses were conducted for each frame with and without considering the longitudinal bending residual stress. The nominal ultimate load factor for the frames is denoted by k0n if the residual stress is not included in the analysis, and k1n otherwise. The effects of the uncertainties in stress-strain relationship on the ultimate strengths of the frames were investigated using a Monte Carlo simulation method. For each frame, 200 random stress-strain curves were sampled using the Latin Hypercube technique, which essentially is an efficient Monte Carlo simulation technique. For each stress-strain curve, an advanced analysis was performed to find the corresponding ultimate load factor. Thus, a sample of 200 frame strength was obtained, based on which the probabilistic characteristics (e.g., mean, standard deviation, and distribution type) of the frame strength could be estimated. In the probabilistic study, two cases were examined: (1) only the longitudinal bending residual stress (i.e., rb in Eqs. (13)–(15) is treated as a random variable while the Young’s modulus and yield stress are assumed to be deterministic using their nominal values, and (2) the Young’s modulus, yield stress and the longitudinal bending residual stress are all considered to be random variables. The

500

Fy 400

Stress (MPa)

Case

Table 4 Statistical data for longitudinal bending residual surface stress (rb).

300

σb

200

100

Without considering residual stress Considering the cubic polynomial function

Ε

ð28Þ

This equation suggests that rb =F y ratio is between 0.52 and 0.65, assuming the yield stress of normal strength cold-formed steel varies from 235 to 460 MPa. By synthesizing all these results, the mean value of the bending residual stress rb is taken as 0.7Fy,

0 0.000

ε10.001

0.002

0.003

Strain

Fig. 14. Nominal stress-strain curve.

ε20.004

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W. Liu et al. / Engineering Structures 150 (2017) 986–995

residual stress. It can be seen that k0n is about 20–47% higher than k1n. This again confirms that ignoring the residual stress may significantly overestimate a frame ultimate strength. The mean (l(kr)) and COV of the ultimate load factors for the eight frames are

L =8 m

H= 3 × 6 m=18 m

H= 4m

statistics for the Young’s modulus and yield stress are given in Table 5. The first two rows of Table 6 compare the nominal strengths of the eight frames with (k1n) and without (k0n) including the effect of

L =8 m L =2× 6 m=12m

H= 8 m

Frame 2

H= 3 x 6 = 18 m

Frame 1

L =6 m

L =6 m L =6 m L =6 m

L =6 m

Frame 4

H=6 x 4m=24m

H= 4 x 6m= 24 m

Frame 3

L =6 m

L =6 m L =6 m L =6 m

L =6 m Frame 5

Frame 6

Fig. 15. Eight 3D frames with cold-formed HSS.

L =6 m

W. Liu et al. / Engineering Structures 150 (2017) 986–995

H= 4×6m=16 m

H= 3 × 6 m=18 m

994

L =4 × 6 m=24 m

L =6 m

L =4 × 6 m=24 m

L =6 m

L =6 m

L =6 m Frame 7

Frame 8 Fig. 15 (continued)

Table 5 Statistical data of material properties for HSS [28–30].

120

Mean

COV

Distribution type

E Fy

1.0 En 1.1Fyn

0.06 0.1

Normal Lognormal

100

Density

Random variable

Note: Fyn = nominal yield stress; En = nominal Young’s modulus.

80

σb is random σb , Fy and E are random

60 40

presented in Table 6. The Table shows that the mean-to-nominal ratios, l(kr)/k1n, for the eight frames are all around 1.0. The COVs of kr for the eight frames are about 0.01–0.02. From a statistical point of view, a COV of such order represents very small uncertainty. This result demonstrates that while it is important to consider the effect of residual stress, the residual stress can statistically be modelled as deterministic. Table 7 shows the results when the Young’s modulus, yield stress and residual stress are all modelled as random variables. The mean-to-nominal ratios l(kr)/k1n for the eight frames are around 1.05, with COVs of about 0.06–0.07. The uncertainty in kr mainly arises from the randomness in the yield stress and Young’s modulus. Fig. 16 shows the sample distributions of kr for a typical frame (Frame 7). It can be seen that the scatter of kr is very small when only the residual stress rb is modelled as random. When the yield stress, Young’s modulus, and residual stress are all modelled as random variables, the frame strength has a COV of 0.06, and can be modelled as a log-normal distribution.

20 0

1.4

1.5

1.6 1.7 1.8 1.9 2.0 Frame ultimate load factor

2.1

2.2

2.3

Fig. 16. Histograms of the ultimate load factor for Frame 7.

5. Conclusion In this paper, a simple practical method is introduced to approximate the effect of residual stress in cold-formed HSS by modifying the stress-strain relationship. This provides a convenient way of incorporating the effect of residual stress in beam element-based finite element analysis. It was found that in general, the longitudinal bending residual stress has a more dominant effect on the behavior of the frame (ultimate strength and stiffness)

Table 6 Strength of the eight frames with random longitudinal bending residual stress (rb). Frame

1

2

3

4

5

6

7

8

k0n k1n k0n/k1n

8.043 5.630 1.43

3.140 2.463 1.275

2.912 2.146 1.36

12.21 8.278 1.47

1.551 1.115 1.39

2.526 2.076 1.22

2.38 1.729 1.38

3.188 2.553 1.25

l(kr) l(kr)/k1n

5.640 1.002 0.016

2.466 1.001 0.017

2.147 1.001 0.010

8.30 1.00 0.010

1.116 1.00 0.013

2.083 1.003 0.022

1.75 1.01 0.020

2.553 1.00 0.014

COV(kr)

Table 7 Strengths of the eight frames with random Young’s modulus, yield stress and residual stress. Frame

1

2

3

4

5

6

7

8

k1n l(kr) l(kr)/k1n COV(kr)

5.630 5.836 1.037 0.060

2.463 2.564 1.04 0.061

2.146 2.228 1.038 0.059

8.278 8.62 1.04 0.06

1.115 1.158 1.04 0.064

2.076 2.175 1.05 0.069

1.73 1.83 1.06 0.06

2.553 2.665 1.04 0.065

W. Liu et al. / Engineering Structures 150 (2017) 986–995

than other components of the residual stress, and that with sufficient accuracy, the effect of residual stress in HSS can be accounted for by only modelling the longitudinal bending residual stress. A stress-strain relationship based on a cubic polynomial function is proposed for use in engineering practice to account for the effect of longitudinal bending residual stress in cold-formed HSS. The accuracy of the proposed polynomial function is verified. It was found that ignoring residual stress may lead to significant errors in predicting a frame’s ultimate strength, particularly when the slenderness ratios of columns are close to unity. For the two frames examined in this study, relative errors of 23% and 39% in the frame ultimate strengths were observed when ignoring the effect of residual stress. The two frames were deliberately designed such that the slenderness ratios of all columns were close to unity to maximize the effect of residual stress. In a more practical design with less slender columns, the role of residual stress on the frame strength would be less important; however, it still has a significant impact on the overall stiffness of the frame. A probabilistic study was performed for eight regular and irregular 3D HSS frames to study the effect of the uncertainties in residual stress on the probabilistic characteristics of frame strengths. It was found that the randomness in the magnitude of residual stress leads to very small uncertainty in the frame strength. Therefore, while it is important to include the effect of residual stress in structural analysis, the residual stress can be treated as deterministic. Acknowledgement The research described in this paper is supported by the Australian Research Council under Discovery Project Grant DP110104263. This support is gratefully acknowledged. However, any opinions and findings expressed herein are solely those of the authors, and may not necessarily reflect the positions of the sponsoring organization. References [1] Li SH, Zeng G, Ma YF, Guo YJ, Lai XM. Residual stresses in roll-formed square hollow section. Thin-Wall Struct 2009;47(5):505–13. [2] Jandera M, Gardner L, Machacek J. Residual stresses in cold-rolled stainless steel hollow sections. J Constr Steel Res 2008;64(11):1255–63. [3] Chan TM, Gardner L. Compressive resistance of hot-rolled elliptical hollow sections. Eng Struct 2008;30(2):522–32. [4] Rasmussen KJR, Hancock GJ. Deformations and residual stresses induced in channel section columns by presetting and welding. J Constr Steel Res 1988;11 (3):175–204. [5] Davison TA, Birkemoe PC. Column behavior of cold-formed hollow structural steel shapes. Can J Civ Eng 1982;10:125–41.

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