Residual stresses in press-braked stainless steel sections, I: Coiling and uncoiling of sheets

Journal of Constructional Steel Research 65 (2009) 1803–1815

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Residual stresses in press-braked stainless steel sections, I: Coiling and uncoiling of sheets W.M. Quach a , J.G. Teng b,∗ , K.F. Chung b a

Department of Civil and Environmental Engineering, University of Macau, Macau, China

b

Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hong Kong, China

article

info

Article history: Received 7 July 2008 Accepted 12 April 2009 Keywords: Residual stresses Stainless steel Material anisotropy Plastic bending Coiling Uncoiling Finite element simulation

abstract The manufacturing process of cold-formed thin-walled steel members induces cold work which can be characterized by the co-existent residual stresses and equivalent plastic strains and has a significant effect on their structural behaviour and strength. The present paper and the companion paper are concerned with the prediction of residual stresses and co-existent equivalent plastic strains in stainless steel sections formed by the press-braking method. This manufacturing process consists of the following two distinct stages: (i) coiling and uncoiling of the sheets, and (ii) press-braking operations. This paper presents an analytical solution for the residual stresses and co-existent equivalent plastic strains that arise from the first stage. In the analytical solution, the coiling–uncoiling stage is modelled as an inelastic plane strain pure bending problem; the stainless steel sheets are assumed to obey Hill’s anisotropic yield criterion with isotropic hardening to account for the effects of material anisotropy and nonlinear stress–strain behaviour. The accuracy of the solution is demonstrated by comparing its predictions with those obtained from a finite element analysis. The present analytical solution and the corresponding analytical solution for press-braking operations presented in the companion paper form an integrated analytical model for predicting residual stresses and equivalent plastic strains in press-braked stainless steel sections. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction The manufacturing process of a cold-formed thin-walled steel section induces significant cold work in the section and has a significant effect on its structural behaviour and strength. This cold work can be characterized by the co-existent residual stresses and equivalent plastic strains in the section, with the latter representing the strain hardening of the steel induced by cold work. For sections produced using the press-braking method, the manufacturing process consists of the following two distinct stages, (i) coiling and uncoiling of sheets (i.e. the coiling–uncoiling stage), and (ii) press-braking operations (i.e. the cold bending stage). The effects of cold work have traditionally been represented by idealized residual stress distributions based on limited laboratory measurements and by specifying different mechanical properties for the flat portions and the corner regions of sections. As a result, laboratory measurements of residual stresses and appropriate tensile tests of coupons cut from sections have been widely used in the past to quantify the effects of cold work in cold-formed sections.

∗

Corresponding author. Tel.: +852 2766 6012; fax: +852 2334 6389. E-mail address: [email protected] (J.G. Teng).

0143-974X/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2009.04.007

Laboratory measurements of residual stresses are, however, time-consuming, difficult and of limited accuracy; the experimental quantification of equivalent plastic strains is even more problematic. For example, due to the thinness of cold-formed sections, variations of residual stresses across the plate thickness cannot be determined with reasonable accuracy. Indeed, in most experimental studies on cold-formed thin-walled sections [1–5], only surface residual stresses were measured while the variations across the plate thickness (with a thickness generally less than 4 mm) were assumed to be linear. Moreover, the measured residual stresses do not provide the information required to understand the relationships between residual stresses and various steps of the manufacturing process. Given the difficulties with experimental measurements of residual stresses and the amount of plasticization induced by the manufacturing process, theoretical quantification of the effects of cold work has been considered to be an attractive alternative and has been attempted by some researchers. Ingvarsson [6] and Kato and Aoki [7] modelled cold-bending operations as the plane strain pure plastic bending of a wide plate by means of an incremental numerical process. Rondal [8] presented a similar numerical analysis of the pure plastic bending of a wide plate, and then proposed an approximate approach for deriving residual stresses in channel sections based on results from his pure bending analysis. Recently, Quach et al. [9,10] presented a finite element-

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based method for predicting residual stresses in press-braked carbon steel sections, in which the two stages of the manufacturing process are modelled by an analytical solution and a finite element simulation respectively. The validity and accuracy of this finite element-based approach [10] have been demonstrated using results from laboratory measurements [1,11,12]. A significant conclusion from these studies is that complex residual stress variations exist across the plate thickness. Such through-thickness variations are difficult to quantify experimentally for sections with a thickness less than 4 mm, but have been successfully measured in cold-bent plates with a thickness of 25 and 38 mm by Weng and White [11,12] and in steel tubes with a thickness of 6 mm by Key and Hancock [13]. Subsequent to the present study which was initially documented in Ref. [14], significant effort has been made by some researchers [15,16] to improve the accuracy of residual stress measurement in thin cold-formed stainless steel sections. In their work, residual stresses were assumed to be uniform over each half thickness and to change sign at the mid-depth [15,16] in order to account for the effect of large plastic bending deformation on residual stresses more accurately. The X-ray diffraction method was employed to measure the half-thickness residual stress value [16] with some limited success (e.g. only two successful measurements at two discrete locations were reported with eight other measurements providing unreliable results). The important role of the forming parameters of the manufacturing process in the assessment of the cold work effect examined in Ref. [14] was also observed in a recent study [17], where empirical relations were proposed to link the strength enhancement in cold-formed stainless steel sections to section geometric and sheet material properties. Despite this significant progress, accurate experimental measurement of residual stresses in thin cold-formed section remains a highly challenging topic that needs much further research. To predict residual stresses in press-braked stainless steel sections, the cold work effects of both stages of the manufacturing process need to be captured. This paper is concerned with the prediction of residual stresses and equivalent plastic strains in stainless steel sheets arising from the coiling–uncoiling stage while a corresponding study for the cold bending stage involving pressbraking operations is presented in the companion paper [18]. In the present solution, coiling and uncoiling of sheets is modelled as plane strain pure bending and the effect of material anisotropy is duly accounted for. The validity and accuracy of the analytical solution are demonstrated by comparing its predictions with those from finite element simulations. It should be noted that the closedform analytical solution of Quach et al. [9] for residual stresses due to the coiling–uncoiling process of carbon steel sheets is based on the assumptions of material isotropy and elastic–perfectly plastic stress–strain behaviour. Therefore, it cannot be used to predict residual stresses in stainless steel sheets which exhibit material anisotropy and nonlinear stress–strain behaviour. 2. Fabrication process and terminology Cold-formed members are usually manufactured by either roll forming or press braking. Only press braking is considered in the present study. Modelling of residual stresses in cold-formed sections requires knowledge of the fabrication process, which is briefly summarized here for press-braked sections. Before press-braking operations are applied, a flat steel sheet has already experienced the coiling, uncoiling and flattening process (simply referred to as the coiling–uncoiling process hereafter). That is, a cold-rolled steel sheet is first coiled into a roll for storage, and is subsequently uncoiled from the roll and forced to become a flat sheet before press-braking operations. It should be noted that the natural uncoiling of coiled sheets leads to sheets

(a) Coiling.

(b) Uncoiling including flattening.

(c) Press braking.

Fig. 1. Manufacturing process of press-braked sections.

with a small residual curvature, but in practice this curvature is removed either before press braking by the imposition of necessary forces/restraints or during press braking as a result of the outof-plane stiffness associated with a fold. In this study, flattening, corresponding to the imposition of necessary forces/restraints, is assumed to take place before press braking, and is implemented by the application of a curvature equal in magnitude but opposite in sign to the coiling curvature. During the press-braking process, flat steel sheets (or strips) cut from a coil are fed into a press brake and a complete fold is produced along the full length of a section (Fig. 1). The complete forming process of a section generally requires the press-braking operation to be repeated several times. In the present study, it is assumed that any residual stresses due to the cold-rolling process have been removed during annealing. That is, the flat steel sheets are assumed to be free from residual stresses before they are coiled for storage and transportation. Residual stresses due to coiling and uncoiling are however considered in the study. As a result, the final residual stresses in a cold-formed section arise from two distinct stages: the coiling–uncoiling stage and the press-braking stage. Before proceeding further, the terminology adopted in this paper in referring to stresses in various directions should be noted first. The direction of bending is referred to as the longitudinal direction (z direction) and the width direction of the sheet is referred to as the transverse direction (x direction), while the direction normal to the sheet is referred to as the throughthickness direction (y direction). The present terminology has the advantage that the longitudinal direction of a stainless steel sheet remains the longitudinal direction of a cold-formed member produced from the sheet. It should also be noted that, in the manufacturing process of a press-braked lipped channel section (see Fig. 1), the outer surface of a coiled sheet becomes the inner surface of the lipped channel section produced from the sheet. 3. Analytical solution for the coiling–uncoiling process 3.1. Assumptions and overview of the analytical solution Stainless steel alloys are characterized by a nonlinear stress– strain relationship and material anisotropy, and have different properties in tension and compression as well as different properties in the longitudinal and the transverse directions. Although the effect of material anisotropy is small and can be ignored for austenitic alloys, it is more pronounced for other stainless steel alloys, such as duplex and ferritic alloys. A flat stainless steel sheet can be assumed to possess a stress–strain curve of the virgin material before it is coiled for storage, since the effect of cold work prior to coiling is assumed to have been removed during annealing. In the present analytical solution, the tensile stress–strain behaviour of a stainless steel is assumed to be the same as the compressive stress–strain behaviour, and a stress–strain curve from either tension or compression may be used in the analytical solution. The state of anisotropy is described by Hill’s anisotropic yield criterion [19], while nonlinear strain hardening is modelled using the uniaxial tensile or compressive

W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

stress–strain relationship of the virgin material in the longitudinal direction, which can generally be defined by a function:

σyL = F εyL




or εyL = f σyL




(1)

where σyL and εyL are the instantaneous yield stress and the corresponding strain of the uniaxial stress–strain relationship of the material in the longitudinal direction. For brevity, σyL and εyL are often referred to simply as the instantaneous yield stress and the corresponding strain and Eq. (1) as the uniaxial stress–strain curve or the stress–strain curve in the remainder of the paper. The coiling of a stainless steel sheet into a curvature κc and its subsequent uncoiling and flattening can be modelled as plane strain pure bending in the y–z plane (see Fig. 1). This section presents the analytical solution for this inelastic plane strain bending problem. In this analytical solution, the nonlinear strainhardening behaviour of the anisotropic stainless steel sheet is described by the relationship between the equivalent stress σ¯ and the equivalent plastic strain ε¯ p which can be defined using the uniaxial stress–strain relationship of Eq. (1). Section 3.2 explains how the equivalent stress-equivalent plastic strain relationship is established. In the present solution, residual stresses and equivalent plastic strains are expressed using four parameters: the instantaneous yield stresses σyLc and σyLr for coiling and uncoiling respectively, and their corresponding stress ratios ωc and ψr . Sections 3.3.1 and 3.4.1 present the general equations for coiling and uncoiling respectively, while Sections 3.3.2 and 3.4.2 deal with the determination of the respective instantaneous yield stresses and stress ratios. Section 3.5 deals with the implementation of a full-range stress–strain relationship developed by Quach et al. [20,21] for Eq. (1) in the present analytical solution. To define the material anisotropy of stainless steels, the initial yield stresses (0.2% proof stresses) in the three principal directions of anisotropy are needed but the initial yield stress in the through-thickness direction cannot be easily determined. Section 3.6 is concerned with the approximation of this initial yield stress in the throughthickness direction.

The nonlinear strain-hardening behaviour of anisotropic stainless steel can be represented by the relationship between the equivalent stress σ¯ and the equivalent plastic strain ε¯ p . This relationship can be established from the uniaxial stress–strain relationship represented by Eq. (1). The definitions of σ¯ and ε¯ p as well as their relationship are discussed in this subsection. According to the hypothesis of work equivalence, when the principal axes of stresses are coincident with the axes of anisotropy, the equivalent stress σ¯ and the equivalent plastic strain increment dε¯ p are given by the following equations [19]:

dε¯ p =

2 3

  F × 

F =

G=

H =


2

+ G (σz − σx )2 + H σx − σy F +G+H

1

"

2 σyN

2 1

"

1

"

Gdεpy − Hdεpz FG + GH + HF

 +H

+G

Hdεpz − F dεpx FG + GH + HF

¯ εpy F dεpx − Gd

2

1 2 σyT

2

+

+

+

1

1

1

#

1

(4b)

2 σyN

−

2 σyN

(4a)

2 σyT

−

σyL2

#

1

−

σyL2

1 2 F¯ = F σyL =

"

1

"

2

¯ = Gσ G

2 yL

¯ = Hσ H

#

1

(4c)

σyL2

=

2 yL

=

2 1

σyL σyN

2

σyL σyT

2

"

2

 +

σyL σyT

 +

2

σyL σyL

2

σyL σyL

2

 +

 −  −

σyL σyN

2

σyL σyT

2 #

σyL σyN

2 #

 −

(2)

2 1/2   

(3)

FG + GH + HF

where σx , σy and σz are the principal stresses in the x, y and z directions respectively; dεpx , dεpy and dεpz are the principal plastic strain increments along the x, y and z directions respectively. In

(5a)

σyL σyL

(5b)

2 # (5c)

¯ and H¯ are dimensionless anisotropy parameters. in which F¯ , G By assuming that changes in material anisotropy due to cold work are negligible, the yield stresses of the material in the three different directions then increase in a proportional manner as the material deforms. That is, σyL : σyT : σyN = σ0L : σ0T : σ0N

(6a)

and (6b)

where σ0L , σ0T and σ0N are the initial yield stresses (taken as 0.2% proof stresses) in the longitudinal (z), transverse (x) and throughthickness (y) directions respectively. By substituting Eq. (6a) into Eq. (5), the dimensionless ani¯ and H¯ can then be defined by the ratios sotropy parameters F¯ , G of the initial yield stresses in the principal directions of anisotropy, and are given by F¯ =


2 !

(F + G + H ) 

1 2 σyT

2

1

"

2 1

σ0L σ0N

2

σ0L σ0T

2

σ0L σ0T

2

"

2

¯ = 1 H

"

2

2

1

where σyL , σyT and σyN are the instantaneous yield stresses in the longitudinal (z ), transverse (x) and through-thickness (y) directions respectively. By taking the yield stress σyL in the longitudinal direction as the 2 reference yield stress and multiplying both sides of Eq. (4) by σyL , Eq. (4) becomes

¯= G F σy − σz

2

r

Eqs. (2) and (3), F , G and H are the anisotropy parameters defining the current state of anisotropy, and are defined by

¯ : H¯ F : G : H = F¯ : G

3.2. Equivalent stresses and equivalent plastic strains

v u u3 σ¯ = t

1805

 +  +  +

σ0L σ0L

2

σ0L σ0L

2

 −

σ0L σ0N

 −

2

σ0L σ0T

2 #

σ0L σ0N

2 #

σ0L σ0L

2 #

 −

(7a)

(7b)

.

(7c)

The relationship between the equivalent stress and the equivalent plastic strain does not vary with stress states, so this relationship can be established using the uniaxial stress–strain curve of the longitudinal direction Eq. (1). For the state of uniaxial stresses in the longitudinal (z ) direction as experienced by the tensile test of a coupon cut along the longitudinal direction, σx = 0, σy = 0, and σz = σyL , Eq. (2) can be combined with Eq. (7) to become

σ¯ =

s  3

¯ F¯ + G

2

¯ + H¯ F¯ + G

 σyL .

(8)

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W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

The plastic strain increment in the longitudinal (z ) direction dεpz is now the same as the plastic strain increment dεpL of the uniaxial stress–strain curve. According to the flow rule, the plastic strain increments are given by

 
) ( H σx − σy
+ G (σx − σz
)  dεpx dεpy = dλ F σy − σz + H σy − σx
  dεpz G (σz − σx ) + F σz − σy

(9)

in which dλ is a positive scalar representing the magnitude of the plastic strain increment vector. For the state of uniaxial stresses in the longitudinal (z ) direction, the ratios between the plastic strain increments are then obtained from Eq. (9) as

¯ : −F¯ : G¯ + F¯ . dεpx : dεpy : dεpL = −G

(10)

with

σ¯ =

s  3

F σz2 + G (σz − σx )2 + H σx2

2

F +G+H

(15b)

where {σ } is the stress vector representing the state of stresses; k is the hardening parameter; σz and σx are the stresses in the longitudinal (z ) and the transverse (x) directions respectively during the coiling process at an arbitrary location y from the neutral axis of the section. By substituting Eq. (5) into Eq. (15), Hill’s anisotropic yield criterion can be written as

s f ({σ } , k) = σ¯ −

Using Eqs. (3), (7) and (10), the equivalent plastic strain increment dε¯ p is related to dεpL through

with

s  ¯ + H¯  2 F¯ + G dε¯ p = dεpL . ¯ 3 F¯ + G

v u u3 σ¯ = t

(11)



3

¯ + H¯ 2 F¯ + G


σyL = 0

F σz2 + G (σz − σx )2 + H σx2

¯ + H¯ F¯ + G

2

(16a)

! .

(16b)

Integration of Eq. (11) leads to

s  ¯ + H¯  2 F¯ + G εpL ε¯ p = ¯ 3 F¯ + G

(12a)

σyL

(12b)

E0z

where E0z is the initial elastic modulus in the longitudinal direction. By combining the differentiation of Eq. (8) with Eq. (11), the slope of the equivalent stress–equivalent plastic strain relationship H 0 can be determined as 0

H =

dσ¯ dε¯ p

dσyL

¯ 3 F¯ + G




=

¯ + H¯ dεpL 2 F¯ + G


.

(13)

By differentiating Eq. (12b) and then combining it with Eq. (13), the slope H 0 is found to be

¯ 3 F¯ + G




H0 =

f σyL (see Eq. (1)). Eq. (16) can be expressed in a simple form as




2 F σz2 + G (σz − σx )2 + H σx2 = σyL .

with

εpL = εyL −

In Eq. (16), the nonlinear strain hardening of the material is
represented by the stress–strain relationship σyL = F εyL or εyL =

¯ + H¯ 2 F¯ + G




dεyL dσyL

−

1

−1 (14)

E0z

in which dεyL /dσyL can be obtained by differentiating Eq. (1) and can be expressed in terms of σyL . Eq. (14) is applicable to both the coiling and the uncoiling deformations. 3.3. Coiling

s f ({σ } , k) = σ¯ −

By defining the following stress ratio:

ω = σx /σz

3 2 (F + G + H )

=0

(15a)

(18)

and combining it with Eq. (17), the coiling stresses of any material point (such as point P in Fig. 2) at the end of coiling can then be obtained as

σ z ,c = ± q σx,c = ± q

σyLc

(19a)

¯ − 2G¯ ωc + G¯ + H¯ ωc2 F¯ + G





ωc σyLc

(19b)

¯ − 2G¯ ωc + G¯ + H¯ ωc2 F¯ + G





in which the subscript c is used to refer to the end of coiling, σx,c and σz ,c ≥ 0 (i.e. tensile stresses) when y ≥ 0, and σyLc is the instantaneous yield stress at the end of coiling for this material point. Similarly to Eq. (12), the equivalent plastic strain due to the applied coiling curvature κc can be obtained as

ε¯ p,c

3.3.1. General equations Due to the ‘‘roundhouse’’ type of stress–strain behaviour of stainless steel alloys, purely elastic straining does not occur during the coiling process, so no elastic core of the sheet thickness exists. During the coiling of a stainless steel sheet, an arbitrary point in the sheet undergoes inelastic straining, and the amount of straining depends on the given coiling curvature κc and its location y away from the neutral axis of the section. Under a plane strain condition in the transverse direction (x direction) and a plane stress condition in the through-thickness direction (y direction), stresses at material points need to satisfy Hill’s anisotropic yield criterion [19] given by

(17)

s  ¯ + H¯  2 F¯ + G = εpL,c ¯ 3 F¯ + G

(20a)

with

εpL,c = εyLc −

σyLc E0z

(20b)

in which εyLc is the strain corresponding to the instantaneous yield stress σyLc at the end of coiling, and εpL,c is the corresponding plastic strain, both defined by the uniaxial stress–strain curve of Eq. (1). Therefore, if σyLc is known, εyLc can be determined from Eq. (1) and the equivalent plastic strain ε¯ p,c due to coiling can then be calculated from Eq. (20). To determine σx,c , σz ,c and ε¯ p,c (see Eqs. (19) and (20)) at any location y for a given coiling curvature κc , the values of σyLc and ωc need to be calculated numerically for each value of y. The determination of σyLc and ωc is discussed in the next subsection.

W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

4H 0 σ¯ 2

(1 − νxz νzx ) 9E0z "   4σ¯ 2 E0x ∂f 2

S3 =

∂f ∂σx ∂f ∂σz

1807

∂f ∂f + + 2νxz + 9 E0z ∂σx ∂σx ∂σz  ¯ 3 H σx + G¯ (σx − σz ) = ¯ + H¯ 2σ¯ F¯ + G ¯  ¯ 3 F σz + G (σz − σx ) = . ¯ + H¯ 2σ¯ F¯ + G



∂f ∂σz

2 # (22d)

(22e)

(22f)

Due to the plane strain condition, the transverse strain increment dεx is zero during coiling: dεx = dεex + dεpx = 0

(23)

where dεex and dεpx are the elastic strain increment and the plastic strain increment respectively corresponding to the transverse coiling strain increment dεx . By defining the following ratio of stress increments: Fig. 2. Stress path of a surface point of a stainless steel strip during the coiling– uncoiling process.

3.3.2. Determination of σyLc and ωc σyLc and ωc are related to each other. To establish their relationship, the stress ratio ω and its increment dω can be related to the instantaneous yield stress σyL and its increment dσyL . Due to the nonlinear material properties, it is difficult to obtain closedform analytical expressions for σyLc and ωc . Instead, their values can be determined numerically, using their inter-relationship and the known boundary values. During coiling, stress increments are related to strain increments through



dσz dσx

     dεz = [De ] − Dp d εx



E0z / (1 − νxz νzx ) [De ] = νxz E0z / (1 − νxz νzx )

νxz E0z / (1 − νxz νzx ) E0x / (1 − νxz νzx )

n

∂f

[De ] ∂{σ }

 

Dp =

H0 +

n

on

∂f ∂{σ }

oT

∂f ∂{σ }

oT n

 (21b)

[De ] ∂f

[De ] ∂{σ }

(21c)

o

dσz dσx

 =

E0z

×

1 − S12 /S3 νxz − S1 S2 /S3

νxz − S1 S2 /S3 E0x /E0z − S22 /S3

dεz d εx



S2 =




σyL



¯ − G¯ ω + Ω F¯ + G

2σ¯ 3 2σ¯ 3



νxz

∂f ∂f + ∂σx ∂σz

E0x ∂ f





 dσyL

¯ + H¯ ω − G¯ G


dεpz dεpx



  ¯ (σz − σx ) F¯ σ + G = dλ¯ ¯ z ¯ σx G (σx − σz ) + H

(27)

¯ is a positive scalar. in which dλ Substituting Eq. (18) into Eq. (27) yields dεpz dεpx

=

¯ − G¯ ω F¯ + G
. ¯G + H¯ ω − G¯

(28)

The longitudinal coiling strain increment dεz consists of an elastic strain increment dεez and a plastic strain increment dεpz : (29)

(22a)

"



∂f + νxz E0z ∂σx ∂σz

(26)

in which Ω is given by Box I. Eq. (26) can then be used to solve numerically for the values of σyLc and the corresponding stress ratio ωc at each location y. Furthermore, due to the plane stress condition in the throughthickness direction (y direction) in coiling, the flow rule given by Eq. (9) reduces to

dεz = dεez −






Substitution of Eqs. (23) and (28) into Eq. (29) leads to



with S1 =

¯ − 2G¯ ω + G¯ + H¯ ω2 (Ω − ω) F¯ + G

dεz = dεez + dεpz .

(1 − νxz νzx )



dω =

(21d)

in which dσz and dσx are the longitudinal and the transverse stress increments, and dεz and dεx are the longitudinal and the transverse strain increments; νxz is Poisson’s ratio for a strain in the z direction due to a uniaxial stress in the x direction; E0x is the initial elastic modulus in the transverse direction; and f is the yield function given by Eq. (16). By substituting Eq. (16) after differentiation into Eq. (21), the incremental stress–strain relationship can be rewritten as



By combining Eqs. (24) and (25), the following equation is obtained:



νzx = νxz E0z /E0x

(24)

and making use of Eqs. (18), (22) and (23), the ratio of stress increments is obtained as the equation given in Box I in which H 0 is given by Eq. (14). As H 0 is given in terms of σyL , the ratio Ω given in Box I becomes a function of ω and σyL . Differentiating and combining Eqs. (17) and (18) leads to
 
¯ − G¯ ω dωσ yL + F¯ + G¯ − 2G¯ ω + G¯ + H¯ ω2 ωdσyL F¯ + G d σx

  . (25) =  ¯ − G¯ + H¯ ω dωσyL + F¯ + G¯ − 2G¯ ω + G¯ + H¯ ω2 dσyL d σz G

 (21a)

with



Ω = dσx /dσz

(22b)



#

¯ + H¯ ωc − G¯ G


dεex .

(30)

The incremental elastic strains are given by

 (22c)

¯ − G¯ ωc F¯ + G

dεez dεex



1/E0z = −νzx /E0z



−νxz /E0x 1/E0x

dσz . dσx





(31)

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W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

Ω=

¯ + H¯ 2H 0 νxz (E0z /E0x ) F¯ + G

¯ − 2G¯ ω + G¯ + H¯ ω2 − 3E0z F¯ + G¯ − G¯ ω F¯ + G




¯ + H¯ 2H 0 (E0z /E0x ) F¯ + G






¯ − 2G¯ ω + G¯ + H¯ ω2 + 3E0z F¯ + G











¯ + H¯ ω − G¯ G









2

¯ + H¯ ω − G¯ G


Box I.

nE 



o 0z ¯ − G¯ ω 2 + G¯ − G¯ + H¯ ω 2 − 2νxz E0z F¯ + G¯ − G¯ ω G¯ − G¯ + H¯ ω σyL  F¯ + G  E E 0x 0x   dω  

  ¯ ¯ ¯ ¯ ¯ ¯ ω + G¯ + H¯ ω2 3/2 E G − G + H ω F + G − 2 G 0z  o n d εz = ± 

E  ¯ − G¯ + H¯ ω + E0z (ω − νxz ) F¯ + G¯ − G¯ ω G 1 − ων xz E0z   E 0x 0x   dσyL + 

  ¯ − G¯ + H¯ ω F¯ + G¯ − 2G¯ ω + G¯ + H¯ ω2 1/2 E0z G

            

Box II.

elastic, until the reverse bending curvature exceeds a threshold curvature value. The elastic uncoiling stresses are given by

Eq. (31) is then substituted into Eq. (30) to arrive at

( d εz =

"

#)

¯ − G¯ ω F¯ + G dσ z
¯G + H¯ ω − G¯ E0z " # ¯ − G¯ ω F¯ + G dσx
− νxz + . ¯ + H¯ ω − G¯ E0x G 1 + νxz

E0z

E0z

σ z ,u =

E0x

(32)

By substituting Eqs. (17) and (18) after differentiation into Eq. (32), the equation in Box II can be obtained. In Box II, the +ve sign of ‘‘±’’ applies when y ≥ 0. Integrating the left-hand side of Box II from zero to the longitudinal coiling strain εz ,c (where εz ,c = κc y) and the righthand side from zero to the instantaneous yield stress σyLc at the end of coiling for dσyL and from Poisson’s ratio νxz to the stress ratio ωc corresponding to εz ,c for dω results in the equation in Box III. In Box III, as the value of εz ,c is known and equal to κc y at any arbitrary location y, the integration can be performed numerically by the Euler forward method. The lower limits (i.e. νxz for dω and zero for dσyL ) for the integration of Box III are treated as initial conditions. Starting with these initial conditions, the value of dω can be calculated from Eq. (26) for a small assigned value of dσyL , and the values of ω and σyL can then be updated for each step. After the numerical integration (see Box III) is done, the values of σyLc and the corresponding stress ratio ωc , which are the upper limits of the integration, can be determined.

νzx E0x

σx,u =

(1 − νxz νzx )

E0z

κu y =

(1 − νxz νzx )

2 E /E 1 − νxz 0z 0x


κu y

(35a)

νxz E0z
κu y . 2 E /E 1 − νxz 0z 0x

κu y =

(35b)

For any material point, an uncoiling curvature limit κuy , beyond which uncoiling stresses are no longer elastic, can be defined to indicate the onset of reverse yielding. If reverse yielding is attained at the limit κuy as a result of uncoiling, then the total stresses should satisfy the yield surface developed at the end of coiling. That is, F σz2,r + G σz ,r − σx,r


2

2 + H σx2,r = σyLc .

(36)

Hence, such an uncoiling curvature limit κuy can be determined by substituting Eqs. (19), (34) and (35) into Eq. (36) as κuy = −

2 2σyLc 1 − νxz E0z /E0x

¯ − G¯ νxz + F¯ + G




2 ¯ − 2G¯ νxz + G¯ + H¯ νxz E0z |y| F¯ + G








¯ + G¯ νxz − G¯ ωc H




q



¯ − 2G¯ ωc + G¯ + H¯ ωc2 F¯ + G


.

(37) The total longitudinal strain of a point at the onset of reverse yielding (shown as point UE in Fig. 2) during uncoiling is then


εz ,uy = κc + κuy y.

(38)

The corresponding uncoiling stresses are 3.4. Uncoiling including flattening 3.4.1. General equations In the present study, flattening, corresponding to the imposition of necessary forces/restraints, is assumed to take place before cold forming through press-braking operations, and is implemented by the application of a curvature equal in magnitude but opposite in direction to the coiling curvature κc . That is, the uncoiling curvature κu satisfies the following condition:

κu = −κc .

(33)

After such uncoiling (including flattening), the total stresses (i.e. σz ,r and σx,r ) of any point can be found by adding the uncoiling stresses (i.e. σz ,u and σx,u ) to the coiling stresses (point UP in Fig. 2):

σz ,r = σz ,c + σz ,u σx,r = σx,c + σx,u

E0z

σz ,uy =

(34a) (34b)

where the subscript r refers to the total value of a quantity at the end of the uncoiling stage while the subscript u refers to the amount due to the uncoiling process. The unloading stresses are


κuy y

(39a)

νxz E0z
κuy y 2 E /E 1 − νxz 0z 0x

(39b)

2 E /E 1 − νxz 0z 0x

σx,uy =

and the corresponding stress ratio is

ψuy =

σx,c + σx,uy . σz ,c + σz ,uy

(40)

Making use of Eqs. (19), (37) and (39), Eq. (40) can be re-written as

 ψuy =

2 ¯ − H¯ + G¯ νxz F¯ + G ωc − 2 F¯ + G¯ − G¯ νxz νxz









2 ¯ − H¯ + G¯ νxz ωc − F¯ + G¯ − H¯ + G¯ νxz 2 G













.

(41)

Therefore, when κc ≤ κuy ,



σ z ,u = − σx,u = −

E0z 2 E /E 1 − νxz 0z 0x

νxz E0z 2 E /E 1 − νxz 0z 0x




κc y

(42a)


κc y.

(42b)

W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

εz ,c

1809

n  



o Z ωc E0z F¯ + G¯ − G¯ ω 2 + G¯ − G¯ + H¯ ω 2 − 2νxz E0z F¯ + G¯ − G¯ ω G¯ − G¯ + H¯ ω σyL   E0x E0x   dω  

  ν ¯ ¯ ¯ ¯ ¯ ¯ ω + G¯ + H¯ ω2 3/2 E G − G + H ω F + G − 2 G xz 0z  n =±
o
 E Z σyLc  ¯ − G¯ + H¯ ω + E0z (ω − νxz ) F¯ + G¯ − G¯ ω G 1 − ων xz E0z   E 0x 0x   dσyL + 

  ¯ − G¯ + H¯ ω F¯ + G¯ − 2G¯ ω + G¯ + H¯ ω2 1/2 0 E0z G

      

.

     

Box III.

Ψ =

¯ + H¯ 2H 0 νxz (E0z /E0x ) F¯ + G

¯ − 2G¯ ψ + G¯ + H¯ ψ 2 − 3E0z F¯ + G¯ − G¯ ψ F¯ + G







¯ + H¯ 2H 0 (E0z /E0x ) F¯ + G



¯ − 2G¯ ψ + G¯ + H¯ ψ 2 + 3E0z F¯ + G














¯ + H¯ ψ − G¯ G




2

¯ + H¯ ψ − G¯ G


.

Box IV.

When κc > κuy , reverse yielding occurs. The total stresses after

with

uncoiling (including flattening) are constrained by Hill’s anisotropic yield criterion:

εpL,r = εyLr −

F σz2,r + G σz ,r − σx,r


2

2 + H σx2,r = σyLr

(43)

in which σyLr is the instantaneous yield stress after uncoiling (including flattening). From Eq. (43), the total stresses after uncoiling (such as point UP in Fig. 2) can be obtained as

σz ,r = ∓ q σx,r = ∓ q

σyLr

(44a)

¯ − 2G¯ ψr + G¯ + H¯ ψr2 F¯ + G





ψr σyLr

(44b)

¯ − 2G¯ ψr + G¯ + H¯ ψr2 F¯ + G





with (44c)

in which σx,r and σz ,r ≤ 0 (i.e. compressive stresses) when y ≥ 0. Hence, from Eqs. (19), (34) and (44), the uncoiling stresses at the end of uncoiling for κc > κuy are calculated as

 σz ,u = ∓  q

σyLc ¯ − 2G¯ ωc + G¯ + H¯ ωc2 F¯ + G







+ q

σyLr 

¯F + G¯ − 2G¯ ψr + G¯ + H¯ ψr2

 σx,u = ∓  q

+ q

(45a)

ωc σyLc ¯ − 2G¯ ωc + G¯ + H¯ ωc2 F¯ + G







ψr σyLr ¯ − 2G¯ ψr + G¯ + H¯ ψr2 F¯ + G







(45b)

ε¯ p,r

s  ¯ + H¯  2 F¯ + G = εpL,r ¯ 3 F¯ + G

(46a)

.

(46b)

3.4.2. Determination of σyLr and ψr Following the same procedure as explained in Section 3.3.2 (see Eqs. (21)–(32), Box I and Box II), the equation for the increment of the stress ratio during uncoiling can be obtained as dψ =

¯ − 2G¯ ωu + G¯ + H¯ ψ 2 (Ψ − ψ) F¯ + G

σyL






¯ − G¯ ψ + Ψ F¯ + G






 dσyL

¯ + H¯ ψ − G¯ G


(47)

in which H 0 is given by Eq. (14), and Ψ is the ratio of stress increments dσx /dσz , for the uncoiling process, given by the equation shown in Box IV. The equation for the increment of the longitudinal strain during uncoiling can also be found as the equation given in Box V, where the +ve sign of ‘‘±’’ applies when y ≥ 0. Box V is the same as Box II, except for a change in sign since uncoiling causes material yielding in the opposite direction and a different symbol for the stress ratio. Integrating the left-hand side of Box V from the longitudinal strain εz ,uy at the onset of reverse yielding to the final longitudinal strain εz ,r , and the right-hand side from the instantaneous yield stress σyLc due to coiling to the instantaneous yield stress σyLr after uncoiling for dσyL and from the stress ratio ψuy at the onset of reverse yielding to the stress ratio ψr corresponding to εz ,r for dψ results in the equation in Box VI. Uncoiling including flattening enforces the final longitudinal strain εz ,r to become zero at the end of the process. That is,

εz ,r = 0.

where σx,u and σz ,u ≤ 0 (i.e. compressive stresses) when y ≥ 0. Similarly to Eq. (12), the total equivalent plastic strain after uncoiling can be determined as

E0z

The relationship between σyLr and εyLr is given by Eq. (1). Therefore, once σyLr is obtained, εyLr can be determined from Eq. (1) and the total equivalent plastic strain ε¯ p,r after uncoiling can then be calculated from Eq. (46). In order to determine the uncoiling stresses, the total residual stresses and the total equivalent plastic strain (see Eqs. (44)–(46)) at any location y, and the values of σyLc , σyLr , ωc and ψr need to be calculated numerically for each value of y. The values of σyLc and ωc can be determined numerically from Eq. (26) and Box III. The determination of σyLr and ψr is discussed in the next subsection.





ψr = σx,r /σz ,r = σx,c + σx,u / σz ,c + σz ,u

σyLr

(48)

After σyLc and ωc are determined from the numerical integration of Box III for the coiling stage, the value of ψuy can be calculated using Eq. (41). In Box VI, as the values of εz ,uy and εz ,r are given by Eqs. (38) and (48) respectively, the integration can be performed numerically again by the Euler forward method. The lower limits (i.e. ψuy for dψ and σyLc for dσyL ) for the integration of Box VI are treated as initial conditions. Starting with these initial conditions, the value of dψ can be calculated from Eq. (47) for a small assigned

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W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

 nE
2
2 

o E0z ¯ 0z ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯  − 2 ν + G − G + H ψ F + G − G ψ F + G − G ψ G − G + H ψ σyL xz  E0x E0x   dψ  

3/2 − ¯ − G¯ + H¯ ψ F¯ + G¯ − 2G¯ ψ + G¯ + H¯ ψ 2 E G 0z  n d εz = ±
o
 E  ¯ − G¯ + H¯ ψ + E0z (ψ − νxz ) F¯ + G¯ − G¯ ψ G 1 − ψ ν xz E0z   E 0x 0x   dσyL − 

  ¯ − G¯ + H¯ ψ F¯ + G¯ − 2G¯ ψ + G¯ + H¯ ψ 2 1/2 E0z G

            

Box V.

εz ,r − εz ,uy

n  



o Z ψr E0z F¯ + G¯ − G¯ ψ 2 + G¯ − G¯ + H¯ ψ 2 − 2νxz E0z F¯ + G¯ − G¯ ψ G¯ − G¯ + H¯ ψ σyL   E E 0x 0x   dψ  

  ψ ¯ ¯ ¯ ¯ ¯ ¯ ψ + G¯ + H¯ ψ 2 3/2 E G + H ψ − G F + G − 2 G uy 0z  o n =± 

 E Z σyLr  ¯ − G¯ + H¯ ψ + E0z (ψ − νxz ) F¯ + G¯ − G¯ ψ G 1 − ψ ν xz E0z   E0x 0x   dσyL + 

  ¯ + H¯ ψ − G¯ F¯ + G¯ − 2G¯ ψ + G¯ + H¯ ψ 2 1/2 σyLc E0z G

      

.

     

Box VI.

value of dσyL , and the values of ψ and σyL can then be updated for each step. After the numerical integration in Box VI is completed, the values of σyLr and the corresponding stress ratio ψr , which are the upper limits of the integration, can be determined.

becomes

3.5. Description of stress–strain behaviour For the application of the above analytical solution to a specific stainless steel alloy, the general representation of the uniaxial stress–strain relationship of stainless steel alloys Eq. (1) needs to be replaced by a specific uniaxial stress–strain relationship. Several stress–strain relationships to describe the nonlinear strainhardening behaviour of stainless steel alloys exist, as discussed in Refs. [20,21], but each of them has its own limitations. A new improved stress–strain relationship, the so-called three-stage fullrange stress–strain model (Eq. (49)), has recently been proposed by Quach et al. [20,21] for stainless steel alloys and is described by the following equation:

  σ σ n  , σ ≤ σ0.2  + 0.002   E0   σ0.2     σ − σ0.2 1 1   + 0.008 + (σ1.0 − σ0.2 ) −  E0.2 E0 E0.2 ε= n00.2,1.0   σ − σ  0 . 2  + ε0.2 , σ0.2 < σ ≤ σ2.0  ×     σ − a σ1.0 − σ0.2   , σ > σ2.0 b∓σ

(49)

where the upper sign corresponds to tension, and the lower sign corresponds to compression. In Eq. (49), E0 is the initial elastic modulus; σ0.2 is the 0.2% proof stress; n is the strain-hardening exponent; and the parameters E0.2 , σ1.0 , σ2.0 , n00.2,1.0 , a and b are expressed in terms of the basic Ramberg–Osgood parameters (E0 , σ0.2 and n) which are generally available in the current design codes (e.g. [22,23]). The development of the three-stage full-range stress–strain relationship and the expressions for these parameters can be found in Refs. [20,21]. This new stress–strain relationship is adopted herein to describe the stress–strain behaviour in the longitudinal direction in this analytical solution. The initial elastic modulus E0 in Eq. (49) is now replaced by E0z with the subscript z referring to the longitudinal (z ) direction, and the stress σ of the stress–strain curve (Eq. (49)) in the longitudinal direction is now the same as the instantaneous yield stress σyL . By substituting Eq. (49) after differentiation into Eq. (14), the slope of the equivalent stress–equivalent plastic strain relation H 0

H0 =

!
 n ¯ 3 F¯ + G σ  0.2 
 , σyL ≤ σ0.2  n −1   2 F¯ + G¯ + H¯ ( 0.002nσyL  
  ¯ 3 F¯ + G 1 1   
− + n00.2,1.0   ¯ ¯ ¯ E E F + G + H 2  0 . 2 0z         × 0.008 + (σ − σ ) 1 − 1 1.0

0.2

E

E

0z 0.2
n00.2,1.0 −1 )−1    σyL − σ0.2    , σ0.2 < σyL ≤ σ2.0 ×  n00.2,1.0   (σ 1.0 − σ0.2 )   #−1
"

  ¯  3 F¯ + G b ∓ σyL ± σyL − a 1  
− , 
2  ¯ ¯ ¯  E0z b ∓ σyL  2 F + G + H σyL > σ2.0

(50)

where the upper sign corresponds to tension, and the lower sign corresponds to compression. The parameters E0.2 , σ1.0 , σ2.0 , n00.2,1.0 , a and b in Eq. (50) are the material parameters in the longitudinal direction and are described in terms of the basic Ramberg–Osgood parameters (E0z , σ0.2 and n) [20,21]. The residual stresses and the equivalent plastic strains due to the coiling–uncoiling process can be obtained by replacing Eqs. (1) and (14) with Eqs. (49) and (50) and then following the procedure explained in Subsections 3.2–3.4. 3.6. Initial yield stress in the through-thickness direction To define the material anisotropy of stainless steels, the initial yield stresses (0.2% proof stresses) in the three principal directions of anisotropy are needed. The initial yield stresses in the longitudinal and the transverse directions can be taken as the 0.2% proof stresses determined from coupon tests for both directions. However, the initial yield stress in the through-thickness direction cannot be easily determined and is generally not available. Hence, it is necessary to relate the through-thickness yield stress to the yield stresses in the other two directions on the plane, so that its value can be determined. If a stainless steel sheet is subjected to stresses in its plane (x–z plane), the only non-zero stress components are σx , σz and τxz , and the yield criterion [19] at initial yielding is given by 2 F0 σz2 + G0 (σz − σx )2 + H0 σx2 + 2M0 τxz =1

(51a)

with F0 =

1 2



1 2 σ0N

+

1 2 σ0L

−

1 2 σ0T

 (51b)

W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

G0 =

1



H0 =

1

1

+

2 σ0T

2



2

1

+

2 σ0T

1 2 σ0L

1 2 σ0N

− −

1

(51c)

2 σ0N

1

 (51d)

2 σ0L

2 M0 = 1/2τ0XZ

(51e)

in which F0 , G0 and H0 are the anisotropy parameters defined by the initial yield stresses; and τ0XZ is the initial shear yield stress (0.2% proof stress) for the x–z plane. For a uniaxial stress σ applied in a direction making a counterclockwise angle α with the longitudinal direction (z direction), the stress components corresponding to the uniaxial stress σ are

σz = σ cos2 α

(52a)

σx = σ sin2 α τxz = σ sin α cos α.

(52b) (52c)

When the uniaxial stress σ is applied in the diagonal direction (α = 45◦ ) and reaches the diagonal initial yield stress σ0D , that is σ = σ0D , the through-thickness initial yield stress σ0N can then be determined by substituting Eq. (52) into Eq. (51), as

σ0N =

1 2



1 2 σ0D

−

1 2 4τ0XZ

−1/2

.

(53)

The initial shear yield stress τ0XZ can be approximated by the following relationship [24]:

√ τ0XZ ∼ = σ0D / 3.

(54)

Substitution of Eq. (54) into Eq. (53) leads to the following approximation:

σ0N ∼ = σ0D ∼ =

1811



√

3τ0XZ .

(55)

Therefore, the initial yield stress σ0N in the through-thickness direction can be determined either from the diagonal initial yield stress σ0D or from the initial shear yield stress τ0XZ . 4. Finite element simulation of the coiling–uncoiling process

Fig. 3. Mesh and boundary conditions.

compression tests of Rasmussen et al. [24] were thus employed in the present study. Poisson’s ratio νxz (or ν31 ) was assumed to be 0.3. The material modelling of the duplex stainless steel strip is explained in the next two subsections. 4.2. Nonlinear strain hardening In the finite element analysis, the uniaxial stress–strain curve for longitudinal compression (LC) was adopted to describe the hardening behaviour. Considering that strains from large coiling curvatures can exceed the total strain at the 0.2% proof stress substantially, the three-stage full-range stress–strain model (Eq. (49)) was employed to define the nominal stress–strain relationship over the whole range of compressive strains in the longitudinal direction. The modelling of the nonlinear hardening behaviour of anisotropic metals requires the definition of a relationship between the equivalent true stress σ¯ and the equivalent true plastic strain ε¯ p as input data [26]. In the present study, the equivalent true stress σ¯ and the equivalent true plastic strain ε¯ p were converted from the true stress σt and the true plastic strain εtp for longitudinal compression (LC) by the following equation [19]:

σ¯ =

4.1. General In order to verify the analytical solution for the coiling–uncoiling process presented in Section 3, the coiling–uncoiling process of stainless steel sheets was also simulated using the finite element package ABAQUS [25]. A duplex stainless steel strip of 60 mm in length, having a thickness of 2 mm, was modelled with one end fixed and the other end free (Fig. 3). Both geometrical and material nonlinearities were considered. Coiling was simulated as pure bending of the cantilever steel strip to a coil radius r (= D/2) of 100 mm, and uncoiling including flattening was simulated as reverse bending of the strip to the initial zero curvature. Further details of the finite element model are available elsewhere [14]. To define the material anisotropy of the duplex stainless steel strip in both the analytical solution and the finite element model, the through-thickness initial yield stress σ0N is needed and is approximated by the initial yield stress σ0D in the diagonal direction (Eq. (55)) in the present work. Both the tensile and the compressive yield stresses in the diagonal direction of stainless steel sheets are generally not available in the existing literature and design codes but can be found in Ref. [24], which reports tension and compression tests of coupons cut in the longitudinal, the transverse and the diagonal directions to obtain the mechanical properties of a duplex stainless steel plate (grade UNS31803 duplex alloy) (Table 1). The mechanical properties from the

ε¯ p =

s  3

F0 + G0

2

F0 + G0 + H0

 σt

s  2

F0 + G0 + H0

3

F0 + G0



(56a)

εtp

(56b)

in which F0 , G0 and H0 are the anisotropy parameters defined by Eqs. (51b)–(51d). In order to determine the equivalent true stress σ¯ and the equivalent true plastic strain ε¯ p from Eq. (56), the true stress σt and the true plastic strain εtp were first converted from the nominal stress–strain data for longitudinal compression (LC), which were defined by the three-stage full-range stress–strain model (Eq. (49)), using the following equations [27]:

σt = σn (1 ± εn ) εt = ± ln (1 ± εn ) σt σn (1 ± εn ) εtp = εt − = ± ln (1 ± εn ) − E0

E0

(57a) (57b) (57c)

where σn and εn are the nominal stress and the nominal strain respectively, εt is the true strain, the upper sign corresponds to tension, the lower sign corresponds to compression, and σn , εn , σt , εt and εtp are absolute values for both tension and compression. The nominal stress–strain curve for longitudinal compression, the true stress–strain curve for longitudinal compression, and the equivalent stress–equivalent plastic strain curve are shown in

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W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

Table 1 Mechanical properties of grade UNS31803 duplex stainless steel alloy tested by Rasmussen et al. [24]. Specimen

E0 (GPa)

σ0.01 (MPa)

σ0.2 (MPa)

e(=σ0.2 /E0 )

n

LT TT DT LC TC DC

200.00 215.25 195.00 181.65 210.00 205.00

310 430 376 275 380 460

575 635 565 527 617 610

0.00288 0.00295 0.00290 0.00290 0.00294 0.00298

4.8 7.7 7.4 4.6 6.2 10.6

LT = Longitudinal tension coupon; TT = Transverse tension coupon; DT = Diagonal tension coupon. LC = Longitudinal compression coupon; TC = Transverse compression coupon; DC = Diagonal compression coupon.

(a) Full stress–strain curves.

(b) Initial stress–strain curves. Fig. 4. Stress–strain curves for longitudinal compression.

Fig. 4. It is worth noting that, for isotropic materials, F = G = H and Eq. (56) leads to σ¯ = σt and ε¯ p = εtp . Thus, for isotropic materials, the true stress–true plastic strain relationship can be directly used as input data.

values based on information from various sources. To summarize, the values of the engineering constants are as follows:

4.3. Material anisotropy

ν12 = 0.30; ν13 = ν31 E1 /E3 = 0.26; G12 = 75.00 GPa; G13 = 75.00 GPa;

To describe the material anisotropy in the finite element model, a local coordinate system, which initially coincides with the global coordinate system, was used to define the material directions of each plane strain element. The material directions of each element rotated as the stainless steel strip was deformed during the finite element simulation. In this local coordinate system, the 1-direction is the longitudinal direction, the 2-direction is the through-thickness direction and the 3-direction is the transverse direction. In the finite element analysis, the material anisotropy was described by the orthotropic elasticity model and the anisotropic metal plasticity model. The orthotropic elasticity model is defined by the following engineering constants: elastic moduli E1 , E2 , E3 ; Poisson’s ratios ν12 , ν13 , ν23 ; and shear moduli G12 , G13 and G23 . For these engineering constants, the subscripts 1, 2 and 3 are used to refer to the local 1-, 2- and 3-directions respectively for each plane strain element. The moduli E1 and E3 were taken to be the initial elastic moduli E0z and E0x for longitudinal compression (LC) and transverse compression (TC) respectively. Since the modulus E2 is the initial elastic modulus E0y in the through-thickness direction, which is not available and inconsequential, E2 was assumed to have the same value as the initial elastic modulus E3 (i.e. E0x ). The Poisson’s ratios ν12 and ν23 were assumed to be 0.3. The Poisson’s ratio ν31 was also assumed to be 0.3, leading to a value of 0.26 for the Poisson’s ratio ν13 according to Eq. (21d). The shear moduli G12 , G13 and G23 were taken as the initial shear elastic modulus of grade UNS31803 duplex alloy given in Appendix B of the AS/NZS 4673 Standard [22]. It should be noted that the values of ν12 , ν23 , G12 , G13 and G23 are inconsequential, even though they were assigned reasonable

E1 = E0z = 181.65 GPa;

E2 = E0y = 210.00 GPa;

E3 = E0x = 210.00 GPa;

(58)

ν23 = 0.30; G23 = 75.00 GPa.

The ABAQUS anisotropic metal plasticity model is characterized by Hill’s yield criterion for anisotropic materials and the flow rule with isotropic hardening. While nonlinear strain hardening is modelled by specifying a ‘‘reference’’ stress–strain curve, the state of plastic anisotropy is defined in ABAQUS by means of six yield stress ratios, Rij . The six yield stress ratios are defined as

σ0,22 σ0,33 σ0,11 ; R22 = ; R33 = ; σ0 σ0 σ0 τ0,12 τ0,13 τ0,23 R12 = ; R13 = ; R23 = (59) τ0 τ0 τ0 where σ0,ii is the measured initial yield stress in the i-direction, τ0,ij is the measured initial shear yield stress √ for the i–j plane, σ0 is the reference yield stress and τ0 = σ0 / 3. The initial yield stresses σ0,ii were taken as the compressive R11 =

0.2% proof stresses in the three principal directions (see Table 1), in which the through-thickness 0.2% proof stress σ0N (i.e. σ0,22 ) is approximated by the diagonal 0.2% proof stress σ0D (see Eq. (55)). That is,

σ0,11 = σ0L = 527 MPa;

σ0,22 = σ0N = 610 MPa;

σ0,33 = σ0T = 617 MPa.

(60)

The values of shear yield stresses τ0,ij are inconsequential for this plane strain bending problem. Nevertheless,√the shear yield stress τ0,13 , which is τ0XZ , is approximated by σ0D / 3 (see Eq. (54)). Both τ0,12 and τ0,23 were taken as τ0 such that R12 = 1 and R23 = 1. That is,

τ0,12 = τ0 ;

√ τ0,13 = τ0XZ = 610/ 3 MPa;

τ0,23 = τ0 . (61)

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(a) Longitudinal coiling stress after coiling.

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(b) Transverse coiling stress after coiling.

(c) Final longitudinal residual stress after uncoiling and flattening.

(d) Final transverse residual stress after uncoiling and flattening.

Fig. 5. Comparison of residual stresses between the analytical solution and finite element analysis.

(a) After coiling.

(b) After uncoiling and flattening.

Fig. 6. Comparison of equivalent plastic strains between the analytical solution and finite element analysis.

In the present study, the stress–strain curve for longitudinal compression (LC) was employed to describe the material hardening behaviour. Thus, the equivalent stress–equivalent plastic strain relationship converted from the stress–strain curve for longitudinal compression (LC) was used to define the ‘‘reference’’ stress–strain curve. As the reference yield stress σ0 is the yield stress of the ‘‘reference’’ stress–strain curve, the reference yield stress σ0 was taken as the equivalent stress σ¯ converted from the compressive 0.2% proof stress σ0L in the longitudinal direction, and is hence given by

σ0 =

s  3

F0 + G0

2

F0 + G0 + H0

 σ0L

(62)

where F0 , G0 and H0 are the anisotropy parameters defined by Eqs. (51b)–(51d). Its value was thus calculated to be σ0 = 580 MPa.

4.4. Comparison between analytical predictions and finite element results Since the residual stresses predicted by the finite element model are uniform along the whole length of the stainless steel strip, only the stress distributions at the fixed end are compared with the predictions of the analytical solution. Results from the analytical solution and the finite element simulation are compared in Figs. 5 and 6. Fig. 5 shows comparisons for both the longitudinal and the transverse residual stresses while Fig. 6 shows comparisons for the equivalent plastic strain. The close agreement between the results from the two approaches demonstrates the validity and accuracy of both approaches. The residual stresses are seen to vary in a nonlinear manner across the thickness (Fig. 5). A similar observation has been made by Quach et al. [9] on residual stresses induced by the coiling–uncoiling process in carbon steel sheets.

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W.M. Quach et al. / Journal of Constructional Steel Research 65 (2009) 1803–1815

(a) End of coiling.

(b) Onset of reverse surface yielding.

(c) End of flattening. Fig. 7. Residual stresses in the stainless steel sheet during the coiling–uncoiling process.

Using the predictions from the analytical solution presented and verified above, the stainless steel sheet described in this section is examined to illustrate the development of stresses in different stages of the coiling–uncoiling process. Fig. 2 shows the stress path of the extreme tension fibre of the stainless steel sheet during the entire coiling–uncoiling process while Figs. 7 and 8 present the distributions of stresses and equivalent plastic strain, respectively, corresponding to different deformation states. In Fig. 2, the stress path of the extreme tension fibre during coiling is represented by O–P, with the point P denoting the end of the coiling stage. During this stage, except for the point at the middle surface of the sheet, the material fibres over the whole thickness are subjected to inelastic straining. As a result, the distributions of coiling stresses and equivalent plastic strain ε¯ p,c over the thickness are nonlinear (Figs. 7(a) and 8). The coiling stresses and the corresponding equivalent plastic strain have their maximum magnitudes at the sheet surfaces, and decrease to zero at the middle surface. At the end of the coiling stage, the maximum longitudinal and transverse coiling stresses at the sheet surfaces are 1.451σ0L and 0.865σ0L respectively. Elastic uncoiling which follows inelastic coiling is represented by the stress path P–UE in Fig. 2. During elastic uncoiling, no additional plastic strains are induced. The stress distributions at the onset of reverse yielding are shown in Fig. 7(b). The next stage, involving reverse yielding, is represented by the stress path UE–UP in Fig. 2. During this stage, new plastic strains are induced. At the end of this stage, the residual stresses are as shown in Fig. 7(c), while the distribution of the equivalent plastic strain ε¯ p,r is shown in Fig. 8. The maximum longitudinal and transverse residual stresses are 1.402σ0L and 0.433σ0L respectively, and they occur at the sheet surfaces. At the end of uncoiling, the two zones of material fibres subjected to reverse yielding are located near the two surfaces of the sheet respectively, and each has a

Fig. 8. Equivalent plastic strains in the stainless steel sheet.

size of approximately 0.2t. This reverse yielding results in two corresponding zones of high tensile and compressive longitudinal residual stresses near the two sheet surfaces respectively, with magnitudes exceeding σ0L . The equivalent plastic strains also further increase in the two reverse yielding zones, with the maximum magnitudes at both surfaces being increased by approximately 40% to around 0.009. Therefore, it is seen that the magnitudes of residual stresses and equivalent plastic strains in flattened sheets are related to the amount of material subjected to reverse yielding during uncoiling, which is in turn dependent on the coiling curvature and the 0.2% proof stress. 5. Conclusions Cold work in a cold-formed steel section is in general due to the following two stages of the manufacturing process: (a) the coiling–uncoiling stage; and (b) the subsequent cold-forming

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stage (press braking or cold rolling). The amount of cold work can be characterized by the co-existent residual stresses and equivalent plastic strains in the section. In this paper, an analytical solution for residual stresses and equivalent plastic strains due to the coiling–uncoiling stage has been presented in which coiling, uncoiling and flattening are all taken into account in a plane strain pure inelastic bending model. The analytical solution has been verified using numerical results from a finite element simulation. Results from both methods have shown that through-thickness variations of residual stresses are nonlinear. The analytical solution was employed to generate numerical results to illustrate the development process of residual stresses and equivalent plastic strains in stainless steel sheets. These numerical results has indicated that the magnitudes of residual stresses and equivalent plastic strains in flattened sheets are dependent on the amount of material subjected to reverse yielding during uncoiling, which is in turn dependent on the coiling curvature and the 0.2% proof stress. The analytical solution presented in this paper can be combined with another analytical solution for the press-braking stage to form an analytical model for predicting residual stresses and equivalent plastic strains in press-braked stainless steel sections. Such exploitation of the present analytical solution will be reported in the companion paper [18]. Acknowledgements The authors would like to thank The Hong Kong Polytechnic University (Project No. G-V864), the Research Grants Council of the Hong Kong S.A.R. (Project No. PolyU5056/02E), the University of Macau (Ref. No. RG051/06-07S/QWM/FST) and the Fundo para o Desenvolvimento das Ciências e da Tecnologia (FDCT) of the Macao S.A.R. (Ref. No. 011/2007/A1) for their financial support. References [1] Weng CC, Peköz T. Residual stresses in cold-formed steel members. Journal of Structural Engineering, ASCE 1990;116(6):1611–25. [2] Batista EM, Rodrigues FC. Residual stresses measurements on cold-formed profiles. Experimental Techniques 1992;(September/October):25–9. [3] Rasmussen KJR, Hancock GJ. Design of cold-formed stainless steel tubular members, I: Columns. Journal of Structural Engineering, ASCE 1993;119(8): 2349–67. [4] Abdel-Rahman N, Sivakumaran KS. Material properties models for analysis of cold-formed steel members. Journal of Structural Engineering, ASCE 1997; 123(9):1135–43.

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