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Residual stresses in steel sheets due to coiling and uncoiling: a closed-form analytical solution
Engineering Structures 26 (2004) 1249–1259 www.elsevier.com/locate/engstruct
Residual stresses in steel sheets due to coiling and uncoiling: a closed-form analytical solution W.M. Quach a, J.G. Teng b,, K.F. Chung b a
b
Department of Civil and Environmental Engineering, University of Macau, Macau, China Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China Received 15 October 2003; received in revised form 14 April 2004; accepted 15 April 2004
Abstract Residual stresses in cold-formed members may play a significant role in determining their behaviour and strength, and have traditionally been obtained by laboratory measurements. This paper presents the results of research which forms part of a larger study on the theoretical predictions of residual stresses in cold-formed sections. The paper is concerned with residual stresses that arise from the coiling and uncoiling process experienced by steel sheets before they are cold formed into sections. In this paper, a closed-form solution is presented for these residual stresses, in which the coiling and uncoiling process is modelled as an elastic– plastic plane strain pure bending problem with the steel assumed to obey the von Mises yield criterion and the Prandtl–Reuss flow rule. To facilitate its application in subsequent finite element simulation of the cold-forming process, the prediction of equivalent plastic strains is also addressed in this solution. The accuracy of the solution is demonstrated by comparing its predictions with those from a finite element simulation. Numerical results from the analytical solution are also presented to illustrate the development process of residual stresses and the effects of coiling curvature and yield stress on the final residual stresses. # 2004 Elsevier Ltd. All rights reserved. Keywords: Residual stresses; Cold-formed sections; Plastic bending; Coiling; Uncoiling; Flattening; Finite element simulation
1. Introduction Residual stresses in cold-formed members may play a significant role in determining their behaviour and strength, and have traditionally been obtained by laboratory measurements. Various destructive and nondestruction methods for residual stress measurements are available and each has its own limitations [1]. Laboratory measurements of residual stresses in cold-formed thin-walled sections are time-consuming, difficult and have limited accuracy. For example, due to the thinness of cold-formed sections, variations of residual stresses across the plate thickness generally cannot be obtained. Indeed, in most existing experimental studies on cold-formed thin-walled sections [2–4], only surface residual stresses were measured, with the variations across the plate thickness being assumed to be linear. Residual stresses measured on Corresponding author. Tel.: +852-2766-6012; fax: +852-23346389. E-mail address: [email protected] (J.G. Teng).
0141-0296/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.04.005
thicker plates [5–7] have shown that they vary across the plate thickness in a complex manner. Moreover, clear relationships between residual stresses and various steps of the fabrication process cannot be established by an examination of the measurement results. Accurate theoretical predictions of residual stresses in cold-formed sections require the modelling of the cold-forming process and are not yet available. Nevertheless, a limited amount of effort has been made in modelling residual stresses due to cold bending [8–10]. Ingvarsson [8] and Kato and Aoki [9] modelled the pure plastic bending of a wide plate as a plane strain problem by means of an incremental numerical process, with the steel assumed to obey the von Mises yield criterion and the Prandtl–Reuss flow rule. Rondal [10] presented a similar numerical analysis of the pure plastic bending of wide plates, and then proposed an approximate approach of deriving residual stresses in channel sections based on the results from his pure bending analysis. Numerical results from all three studies also showed complex residual stress variations through the plate thickness.
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To accurately predict residual stresses in cold-formed members, their manufacturing process needs to be closely modelled. Cold-formed members are usually manufactured by either roll forming or press-braking. In both roll forming and press-braking, before the coldforming process is applied to the flat steel sheet, the flat steel sheet has already experienced the coiling, uncoiling and flattening process. That is, cold rolled steel sheets are first coiled into rolls for storage. They are subsequently uncoiled from the roll and forced to become a flat sheet before cold-forming forces are applied. The residual stresses in a cold-formed section are therefore derived from two sources: the coiling, uncoiling and flattening process (referred to simply as the coiling–uncoiling process) and the cold-forming process. Such a two-stage fabrication process is illustrated in Fig. 1 for the press-braking method. The predictions of residual stresses in cold-formed sections can thus be separated into two tasks: the prediction of residual stresses from the coiling and uncoiling process involving pure bending of the steel sheet and the prediction of the residual stresses from the cold-bending process. This paper is concerned only with the first task by presenting a closed-form analytical solution for the pure bending of wide plates. The pure bending of wide plates into the plastic range has been studied by many researchers [8–16]. Among these existing studies, several were based on the deformation theory [13–15] so that the effect of deformation history on residual stresses was ignored. This effect was shown to be important by Zhang and Hu [16], so for accurate predictions, the flow theory should be used. The studies based on the flow theory [8–12] have all been numerical, and some of them [8,10] ignored part of the elastic deformation which is important for the coiling and uncoiling process of steel sheets where the curvature involved is not so large. This paper therefore presents an accurate closed-form analytical solution for the coiling–uncoiling process modelled as a plane strain pure bending problem with all factors appropriately included. To facilitate its application in subsequent finite element simulation of the cold-forming process, the prediction of equivalent plastic strains is also addressed in addition to residual stresses in both the longitudinal and transverse directions.
Fig. 1.
2. Analytical solution 2.1. Assumptions and terminology In the present study, it is assumed that any residual stresses due to cold rolling have been removed in the annealing furnace. That is, the flat steel sheet is assumed to be free from residual stresses before it is coiled for storage. Furthermore, as the effect of cold work is deemed to have been removed in the annealing furnace, a steel sheet before coiling can also be assumed to possess a stress–strain curve of the sharpyielding type (i.e. a yield plateau exists before strain hardening). The strains induced by the coiling and uncoiling process are relatively small, so it is reasonable to assume that strain hardening is not involved. That is, the steel can be assumed to possess an elastic– perfectly plastic stress–strain curve, and obey the von Mises yield criterion and the Prandtl–Reuss flow rule. The coiling of a steel sheet into a curvature jc and its subsequently uncoiling and flattening can thus be modelled as plane strain pure elastic–plastic bending in the y–z plane (Fig. 1). This section presents a closed-form analytical solution for this elastic–plastic bending problem. 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, the width direction of the plate is referred to as the transverse direction, while the direction normal to the curved plate surface is referred to as the radial direction. In terms of the bent plate, the longitudinal direction is the circumferential direction. The present terminology has the advantage that the longitudinal direction remains the longitudinal direction of a cold-formed member produced from a steel sheet after the coiling and uncoiling process being discussed here. 2.2. Coiling During the coiling of the steel sheet, an arbitrary point in the steel sheet undergoes elastic or elastic–plastic deformation, depending on the given coiling curvature jc and its location y away from the neutral axis of
Manufacturing process of press-braked sections. (a) coiling; (b) uncoiling including flattening; (c) press-braking.
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
the section. For elastic material points across the thickness, under a plane strain condition in the width direction (x direction) and a plane stress condition in the through-thickness direction (y direction), the in-plane strains are given by ðrz;c mrx;c Þ E ðrx;c mrz;c Þ ¼0 ¼ E
ez;c ¼
ð1aÞ
ex;c
ð1bÞ
where E is the elastic modulus, rx,c and rz,c are the stresses in the x and z directions, respectively, due to coiling while ex,c and ez,c are the corresponding strains. The von Mises yield criterion is given by r2z;c
þ
r2x;c
rx;c rz;c ¼
r2y
ð2Þ
where ry is the yield stress. The elastic coiling stresses at any arbitrary location y from the neutral axis of the section are then related to the longitudinal strain due to bending ez,c by rz;c
E ez;c ¼ ð1 m2 Þ
ð3aÞ
mE ez;c ð1 m2 Þ
ð3bÞ
rx;c ¼ with
ez;c ¼ jc y
ð3cÞ
The above relationships are valid only for points under elastic straining. By substituting Eq. (3) into Eq. (2), the longitudinal strain at which a material point starts to yield is found to be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez;cy ¼ ry ð1 m2 Þ E 1 m þ m2 ð4Þ
Fig. 2.
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in which ez;cy > 0 if the strain is tensile (i.e. y > 0). Under a coiling curvature jc, the central core of material of the section remains elastic and the size of this central core is twice the value given by the following expression: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ycy ¼ ry ð1 m2 Þ Ejc 1 m þ m2 ð5Þ Therefore, for j yj ycy , the stresses are given by Eq. (3). Points located at j yj > ycy are in plastic flow and the stresses obey the von Mises yield criterion. By defining the following stress ratio xc ¼ rx;c =rz;c
ð6Þ
and combining it with Eq. (2), the coiling stresses of any point undergoing plastic straining (such as point P in Fig. 2) can then be obtained as ry rz;c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xc þ x2c xc ry rx;c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xc þ x2c
ð7aÞ ð7bÞ
in which rx,c and rz;c 0 when y 0. In order to determine the coiling stresses at any location y for a given coiling curvature jc, it is necessary to relate the stress ratio xc to the value of y as detailed below. The Prandtl–Reuss flow rule is given by 3 dezp;c sz;c ¼ dep;c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ dexp;c 2 r2z;c þ r2x;c rx;c rz;c sx;c in which the subscript c denotes that the variables are associated with the coiling stage, dezp,c and dexp,c are the longitudinal and transverse plastic strain increments, dep;c is the equivalent plastic strain increment,
Stress path of the extreme fiber of a steel strip during coiling–uncoiling process.
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and sz,c and sx,c are the deviatoric stresses and are given by sz;c ¼ ð2rz;c rx;c Þ=3
ð9aÞ
sx;c ¼ ð2rx;c rz;c Þ=3
ð9bÞ
Eq. (8) implies that dezp;c sz;c 2rz;c rx;c ¼ ¼ dexp;c sx;c 2rx;c rz;c
ð10Þ
By substituting Eq. (6) into Eq. (10), the following equation is obtained: dezp;c 2 xc ¼ dexp;c 2xc 1
ð11Þ
The longitudinal coiling strain increment dez,c consists of an elastic strain increment deze,c and a plastic strain increment dezp,c dez;c ¼ deze;c þ dezp;c
ð12Þ
The transverse coiling strain increment is zero due to the plane strain condition, therefore, dex;c ¼ dexe;c þ dexp;c ¼ 0
ð13Þ
Substitution of Eqs. (11) and (13) into Eq. (12) leads to
2 xc ð14Þ dez;c ¼ deze;c dexe;c 2xc 1 The incremental elastic strains are given by 1 1 drz;c deze;c m ¼ dexe;c drx;c E m 1
ð15Þ
On the other hand, differentiation of Eq. (7a) gives ry ð1 2xc Þ
drz;c ¼
2ð1 xc þ x2c Þ3=2
dxc
ð16Þ
while differentiating Eq. (6) yields drx;c ¼ xc drz;c þ rz;c dxc to drx;c ¼
2ð1 xc þ x2c Þ3=2
dxc
ð18Þ
Eqs. (15), (16) and (18) can now be substituted into Eq. (14) to arrive at dez;c ¼
m
By substituting Eqs. (3c) and (4) into Eq. (20), the following expression is obtained: " ry ð1 m2 Þ ry xc ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j yj ¼ 2 Ejc 1 xc þ x2c Ejc 1 m þ m 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xc pffiffiffi 3 4ð1 xc þ x2c Þ A5 coth1 @ þ ð21Þ 2 3 m
which relates the stress ratio xc to the value of y. Therefore, for a given coiling curvature jc, the stress ratio xc can be determined from Eq. (21) at each location of j yj ycy , and then the corresponding inelastic coiling stresses can be determined from Eq. (7). There exists a limit value jcy for the coiling curvature at which the extreme surfaces of the steel sheet start to yield (shown as point E in Fig. 2), and this limit is found by substituting Eqs. (3a) and (3b) into Eq. (2) and noting that ez;c ¼ jcy t=2: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jcy ¼ 2ry ð1 m2 Þ Et 1 m þ m2 ð22Þ The value of jcy depends only on the material properties of the steel sheet. Obviously, if the coiling curvature jc < jcy , no plastic bending is involved during coiling and hence no residual stresses exist following uncoiling. When jc jcy , yielding of material due to coiling will result in the development of residual stresses at the end of the coiling–uncoiling process.
ð17Þ
Substitution of Eqs. (7a) and (16) into Eq. (17) leads ry ð2 xc Þ
and the right-hand side from Poisson’s ratio m to the stress ratio xc corresponding to ez,c results in " ry xc ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez;c ¼ ez;cy E 1 xc þ x2c 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xc pffiffiffi 4ð1 xc þ x2c Þ A5 3 coth1 @ þ ð20Þ 3 2
ry 2Eð1 xc þ x2c Þ3=2 "
# ð2 xc Þ2 ð1 2xc Þ 2mð2 xc Þ dxc ð2xc 1Þ ð19Þ Integrating the left-hand side of Eq. (19) from the onset yield strain ez,cy to an arbitrary coiling strain ez,c
2.3. Uncoiling including flattening It should be noted that, when jc jcy , the natural uncoiling of a coiled sheet leads to a sheet with a small residual curvature, but in practice this curvature is removed either before cold forming by the imposition of necessary restraints or during cold forming as a result of the out-of-plane stiffness associated with a fold. In the present study, flattening, corresponding to the imposition of necessary restraints, is assumed to take place before cold forming, and implemented by the application of a curvature equal in magnitude but opposite in direction to the coiling curvature. That is, the uncoiling curvature ju satisfies the following condition: ju ¼ jc
ð23Þ
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
After such uncoiling, the final stresses of any point can be found by adding the uncoiling stresses to the coiling stresses (point UP in Fig. 2): rz;r ¼ rz;c þ rz;u
ð24aÞ
rx;r ¼ rx;c þ rx;u
ð24bÞ
The unloading stresses are elastic, until the reverse bending curvature exceeds a threshold curvature value. The elastic uncoiling stresses are given by rz;u
E ju y ¼ ð1 m2 Þ
ð25aÞ
mE ju y ð1 m2 Þ
ð25bÞ
rx;u ¼
An uncoiling curvature limit juy, beyond which uncoiling stresses are no longer elastic, can be defined to indicate the onset of reverse yielding (point UE in Fig. 2). If reverse yielding occurs as a result of uncoiling, then the final stresses should also obey the von Mises yield criterion. That is, r2z;r þ r2x;r rx;r rz;r ¼ r2y
ð26Þ
Such an uncoiling curvature limit juy can be determined by substituting Eqs. (6), (7a), (24) and (25a) into Eq. (26) as juy ¼
ry ð1 m2 Þ½2 m þ ð2m 1Þxc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E j yjð1 m þ m2 Þ 1 xc þ x2c
ð27Þ
The total longitudinal strain of any point at the onset of reverse yielding during uncoiling is then ez;uy ¼ ðjc þ juy Þy
ð28Þ
The corresponding uncoiling stresses are rz;uy ¼
E juy y ð1 m2 Þ
ð29aÞ
rx;uy ¼
mE juy y ð1 m2 Þ
ð29bÞ
and the corresponding stress ratio is xuy ¼
rx;c þ rx;uy rz;c þ rz;uy
ð30Þ
Making use of Eqs. (7), (27) and (29), Eq. (30) can be re-written as ½ð1 m2 Þxc mð2 mÞ ½ð1 2mÞxc ð1 m2 Þ Therefore, when jc juy , xuy ¼
ð31Þ
rz;u ¼
E jc y ð1 m2 Þ
ð32aÞ
rx;u ¼
mE jc y ð1 m2 Þ
ð32bÞ
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When jc > juy , reverse yielding occurs. The final stresses after uncoiling, being constrained by the von Mises yield criterion, are given by ry rz;r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xu þ x2u
ð33aÞ
xu ry rx;r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xu þ x2u
ð33bÞ
with xu ¼ rx;r =rz;r ¼ ðrx;c þ rx;u Þ=ðrz;c þ rz;u Þ
ð33cÞ
in which rx,r and rz;r 0 when y 0. Hence, from Eqs. (7), (24) and (33), the uncoiling stresses are calculated as ! ry ry rz;u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð34aÞ 1 xc þ x2c 1 xu þ x2u rx;u
xc ry xu ry ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xc þ x2c 1 xu þ x2u
! ð34bÞ
where rx,u and rz;u 0 when y 0. To determine the uncoiling stresses or total stresses after uncoiling at an arbitrary location y, it is again necessary to first relate the stress ratio xu to the value of y. Following the same procedure as explained in the preceding subsection (Eqs. (8)–(19)), the following equation for the longitudinal strain increment during uncoiling can be obtained: " ry dez;u ¼ ð1 2xu Þ 2mð2 xu Þ 2Eð1 xu þ x2u Þ3=2 # ð2 xu Þ2 dxu ð35Þ ð2xu 1Þ Eq. (35) is the same as Eq. (19), except for a change in sign since uncoiling causes material yielding in the opposite direction and a different subscript to indicate that this equation is for uncoiling. Integrating the left-hand side of Eq. (35) from the longitudinal strain ez,uy at the onset of reverse yielding to the final longitudinal strain ez,r and the right-hand side from the stress ratio xuy at the onset of reverse yielding to the stress ratio xu corresponding to ez,r results in " ry xu ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez;r ez;uy ¼ E 1 xu þ x2u 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xu pffiffiffi 4ð1 xu þ x2u Þ A5 3 coth1 @ þ ð36Þ 3 2 xuy
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Since uncoiling including flattening enforces ez;r ¼ 0 at the end of the process, substitution of Eq. (28) into Eq. (36) yields " ry xu ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j yj ¼ Eðjc þ juy Þ 1 xu þ x2u 0 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xu pffiffiffi 4ð1 xu þ x2u Þ A5 3 coth1 @ þ ð37Þ 3 2 xuy
where the values of juy and xuy can be determined from Eqs. (27) and (31), respectively. With Eq. (37), the stress ratio xu can be determined at any location y for jc > juy , and then the corresponding uncoiling stresses and total residual stresses can be determined from Eqs. (34) and (24), respectively. 2.4. Equivalent plastic strain Apart from the residual stresses, the equivalent plastic strain is another quantity of interest as it reflects the total deformation state of a point. It can be derived from the equivalent plastic strain increment dep . During the coiling stage, the equivalent plastic strain increment dep;c is given by dep;c ¼
2ry ðsz;c þ msx;c Þ dez;c 3ðs2x;c þ s2z;c þ 2msx;c sz;c Þ
ð38Þ
where sz,c and sx,c are the deviatoric stresses and are given by Eq. (9). By substituting Eqs. (9) and (19) into Eq. (38), one obtains dep;c ¼
ry ½ð2 xc Þ þ mð2xc 1Þ dxc Eð1 xc þ x2c Þð1 2xc Þ
ð39Þ
Integrating the left-hand side from zero to the corresponding equivalent plastic strain ep;c experienced during coiling and the right-hand side of Eq. (39) from Poisson’s ratio m to the stress ratio xc results in " ! ry ð1 xc þ x2c Þ1=2 ep;c ¼ ln E 2xc 1 !#xc pffiffiffi pffiffiffi 3 3 ð2x 1Þ c tan1 þð1 2mÞ ð40Þ 3 3 m
Similarly, the equivalent plastic strain increment dep;u during the uncoiling stage can be obtained as dep;u ¼
ry ½ð2 xu Þ þ mð2xu 1Þ dxu Eð1 xu þ x2u Þð1 2xu Þ
ð41Þ
Integrating the left-hand side of Eq. (41) from the equivalent plastic strain ep;c at the end of coiling to the final equivalent plastic strain ep;r after flattening, and the right-hand side from the stress ratio xuy at the onset of reverse yielding to the stress ratio xu
corresponding to ez,r or ep;r , results in " ! ry ð1 xu þ x2u Þ1=2 ep;r ¼ ep;c þ ln E 2xu 1 !#xu pffiffiffi pffiffiffi 3 3ð2xu 1Þ 1 tan þð1 2mÞ 3 3
ð42Þ
xuy
With the stress ratios xc and xu determined at each location y from Eqs. (21) and (37), the equivalent plastic strain during coiling and uncoiling can be calculated from Eqs. (41) and (42), respectively. 3. Finite element simulation In order to verify the analytical model presented in the preceding section, the coiling and uncoiling process of steel sheets was also simulated using the finite element package ABAQUS [17]. A flat steel strip of 60 mm in length was modelled with one end fixed and the other end free (see Fig. 3). The steel was assumed to be elastic–perfectly plastic with the following properties: yield stress ry ¼ 250 MPa, Young’s modulus E ¼ 200 GPa and Poisson’s ratio m ¼ 0:3. The sheet had a thickness of 2 mm. Twenty four layers of CPE4R elements, which are 2-D plane strain four-node elements with reduced integration and hourglass control, were employed to capture the through-thickness variations of stresses (Fig. 3). Two steps were required to simulate the whole coiling–uncoiling process: coiling was simulated as pure bending of the cantilever steel strip to a coiling radius r (¼D=2) of 250 mm, and uncoiling including flattening was simulated as reverse bending of the strip to the initial zero curvature. The steps involved are summarized in Fig. 4. At the fixed end, all nodes were constrained in the longitudinal direction. In addition, the node at the mid-depth was constrained also in the through-thickness direction. By means of kinematic coupling, the longitudinal displacements of all nodes at the free end were constrained to the rigid body motion of the reference node located at the mid-depth on the free end, to ensure that the plane section remained plane. Both coiling and uncoiling were achieved by specifying the desired amounts of translation and rotation of the reference node at the free end, corresponding to the desired coiling curvature and final zero curvature, respectively. Both geometrical and material non-linearities were considered. The ABAQUS metal plasticity model is characterized by the von Mises yield criterion and the Prandtl–Reuss flow rule with isotropic hardening. The finite element model predicted residual stresses which are uniform along the whole length, so only the stress
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
Fig. 3.
Mesh and boundary conditions.
distributions at the fixed end are compared in Figs. 5 and 6 with the predictions of the analytical solution. The longitudinal and transverse residual stresses as well as the equivalent plastic strains predicted by both the analytical solution and the finite element simulation are seen to be in very close agreement, which demonstrates the validity and accuracy of both approaches. The results in Figs. 5 and 6 also show that the residual stresses are not linearly distributed across the thickness. Therefore, the assumption of linear residual stress distribution adopted in existing experimentally-based residual stress studies is not appropriate. 4. Stress path of the coiling–uncoiling process Using the predictions from the analytical solution presented and verified above, the steel sheet described in the preceding section is examined here to illustrate the development of stresses at different stages of the coiling-flattening process. Fig. 2 shows the stress path of the extreme tensile fibre of the 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 path O–E represents elastic coiling, with point E being reached (attainment
Fig. 4.
1255
of yielding) when the applied curvature jcy reaches 0.00128 mm1. During this stage, the through-thickness distributions of both the longitudinal and transverse stresses are linear (Fig. 7a). With further loading, the stress path of the extreme fibre is represented by EP in Fig. 2, with point P denoting the end of the coiling stage. In this stage, the through-thickness stress distributions become non-linear and the equivalent plastic strain ep;c starts to accumulate (Figs. 7b and 8). At the end of the coiling stage, two plastic zones are developed near the top and bottom surfaces of the sheet with an elastic core in the middle (Fig. 7b). The maximum longitudinal and transverse coiling stresses are 1.154ry and 0.557ry, respectively. Elastic uncoiling 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. 7c. The next stage, involving reverse yielding, is represented by stress path UE–UP in Fig. 2. During this stage, plastic strains are induced. At the end of this stage, the residual stresses are as shown in Fig. 7d, while the distribution of equivalent plastic strain is shown in Fig. 8. The maximum longitudinal and transverse residual stresses are 1.145ry and 0.444ry, respectively. It is seen that, due to the zero curvature after flattening,
Pure bending of a flat strip.
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Fig. 5. Comparison of residual stresses between the analytical solution and finite element analysis, (a) longitudinal coiling stress; (b) transverse coiling stress; (c) final longitudinal residual stress; (d) final transverse residual stress.
no residual stresses are found in the elastic core. Two zones of high tensile and compressive longitudinal residual stresses, with magnitudes greater than ry, are located near the inner and outer surfaces of the flattened sheet, respectively, and each has a size larger than 0.25t. On the surfaces of the steel sheet, the stress ratio xc after coiling is 0.482 while the stress ratio xu after flattening is 0.387. Because elastic strains are considered in the present analysis, the stress ratios xc and xu will never reach the value of 0.5 for rigid-plastic bending. However, from Eqs. (20) and (36), it can be observed that, as the longitudinal coiling strain ez,c and the final strain after flattening ez,r approach infinity, the stress ratios xc and xu will approach 0.5. This means that the point UL on the yield envelope (Fig. 2) can only be approached but can never be reached. 5. Effect of coiling curvature
Fig. 6. Comparison of equivalent plastic strains between the analytical solution and finite element analysis, (a) after coiling; (b) after flattening.
Obviously, the residual stresses resulting from the coiling–uncoiling process depend on the coiling curvature (or coiling radius), so in this section, the effect of coiling curvature on residual stresses is explored for coiling radii ranging from 200 to 700 mm for steel sheets of 2 mm in thickness. The steel was assumed to have the same properties as given earlier in the paper except that three different yield stresses are considered in this section (ry ¼ 250, 350 and 450 MPa). The range
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
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Fig. 7. Residual stresses over thickness during the coiling–uncoiling process, (a) onset of surface yielding; (b) end of coiling; (c) onset of reverse surface yielding; (d) after flattening.
of radii was chosen to cover the lower values of a practical range as a coiling radius greater than 781 mm will not lead to any residual stresses. Fig. 9 shows that the magnitudes of residual stresses and the sizes of affected zones are quite sensitive to the coiling curvature. It is of interest to note that the coiling curvature limit jcy, which signifies the onset of yielding on sheet surfaces, depends on the material properties and the sheet thickness (refer to Eq. (22)). The uncoiling curvature limit juy, which signifies the onset of reverse yielding, depends not only on the
Fig. 8.
Equivalent plastic strain distributions.
material properties, but also on the bending history and the through-thickness location concerned (refer to Eq. (27)). Although the value of juy depends on the bending history, which is related to the applied coiling curvature jc, a change of the r=t ratio from 100 to 350 leads only to a change of juy value on the sheet surfaces from 0.00250 to 0.00256 mm1. This change in juy, which is greater than changes elsewhere across the thickness, is less than 3%. This indicates that the uncoiling curvature limit juy at which the steel starts to experience reverse yielding is not sensitive to the applied coiling curvature jc. As shown in Fig. 9 (also see Table 1), as the r=t ratio decreases, the maximum surface residual stresses initially increase rapidly until the r=t ratio is small enough for reverse yielding to occur during uncoiling. With further decreases in the r=t ratio, the maximum surface residual stresses become stable as required by the von Mises yield criterion, but the residual stress zones continuously expand. The total thickness of the two residual stress zones referred to in Table 1 is given by (t 2ycy ), where ycy is half the thickness of the central elastic core. Table 1 summarizes the magnitudes of maximum surface residual stresses and the total thickness of the residual stress zones. In addition to the r=t ratio, the effect of steel yield strength is also illustrated in Table 1,
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W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259 Table 1 Effect of r=t ratio and yield strength on the magnitudes of maximum surface residual stresses and the size of residual stress zones r=t
ry (MPa)
Total thickness of residual stress zones over sheet thickness ðt 2ycy Þ=t
Maximum rz;r =ry
Maximum rx;r =ry
100
250 350 450 250 350 450 250 350 450 250 350 450 250 350 450 250 350 450
0.74 0.64 0.54 0.62 0.46 0.31 0.49 0.28 0.08 0.36 0.10 N.A.a 0.23 N.A. N.A. 0.10 N.A. N.A.
1.153 1.136 1.095 1.127 0.943 0.485 1.047 0.428 0.091 0.612 0.124 N.A. 0.326 N.A. N.A. 0.124 N.A. N.A.
0.522 0.386 0.230 0.347 0.141 0.056 0.163 0.048 0.007 0.077 0.010 N.A. 0.033 N.A. N.A. 0.010 N.A. N.A.
150
200
250
300
350
a
Fig. 9. Effect of r=t ratio on residual stress distributions. (a) final longitudinal residual stress; (b) final transverse residual stress.
where results for three different yield stresses (ry ¼ 250, 350 and 450 MPa) are shown. As expected, as the r=t ratio and the yield stress increase, the residual stresses reduce in both value and extent. At r=t ¼ 100, more than half of the sheet thickness experiences plastic bending and develops residual stresses for all three yield strengths. On the other hand, at r=t ¼ 350, residual stresses are only induced when the yield stress is 250 MPa, and these residual stresses are small. Obviously, at high values of both the r=t ratio and the yield strength ry, the coiling curvature jc < jcy and no plastic bending is involved. 6. Conclusions Residual stresses in cold-formed sections are in general due to both the cold-forming process (press-braking or cold rolling) and the prior coiling–uncoiling process. In both processes, residual stresses are induced as a result of plastic bending. This paper has been concerned with the accurate prediction of residual stresses resulting from the coiling–uncoiling process, as such predictions are a necessary starting point for the prediction of residual stresses from the process of cold forming.
N.A.: No value is available, since no plastic bending is involved.
In this paper, an analytical solution for the residual stresses from the coiling–uncoiling process has been presented in which coiling, uncoiling and flattening are all taken into account in a plane strain pure elastic– plastic bending model. A finite element simulation of the same problem has also been presented. The analytical predictions have been shown to agree closely with finite element results, demonstrating the validity of both approaches. Results from both methods showed that through-thickness variations of residual stresses are non-linear. The analytical solution was employed to generate numerical results to illustrate the development process of residual stresses and the effects of coiling curvature and steel yield stress on residual stresses. The magnitude and extent of residual stresses were found to be sensitive to the coiling radius and the yield stress of steel. The analytical solution provides accurate residual stresses which can be specified as initial stresses in a finite element simulation of the cold-forming process. Such exploitation of the present analytical solution will be reported in a forthcoming paper [18]. It should be noted that the analytical solution presented in this paper is based on the plane strain assumption which is invalid for a narrow zone along each longitudinal edge of a wide plate of finite width. As a result, the solution is unlikely to give satisfactory predictions if applied to the coiling of a rather narrow strip (e.g. a width-to-thickness ratio below 50). It should also be noted that steel sheets may be coiled at an elevated temperature which may affect the mechanical
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
properties of the steel and hence the resulting residual stresses. Both the effects of elevated temperature and the plane strain assumption warrant attention in future research. Finally, it is worth noting that although the present solution has been presented in the context of cold-formed steel sections, it can be applied to any other situations where a wide steel plate is bent into a cylindrical surface. Acknowledgements The authors would like to thank The Hong Kong Polytechnic University (Project No. G-V864), the University of Macau (Ref. No. SD009(3)/99-00S/ QWM/FST) and the Research Grants Council of the Hong Kong S.A.R. (Project No. PolyU5056/02E) for their financial support. References [1] Rowlands RE. Residual stress. In: Kobayashi AS, editor. Handbook on experimental mechanics. 2nd revised ed. New York: VCH Publ; 1993, p. 785–828. [2] Weng CC, Peko¨z T. Residual stresses in cold-formed steel members. Journal of Structural Engineering, ASCE 1990;116(6): 1611–1625. [3] Batista EM, Rodrigues FC. Residual stresses measurements on cold-formed profiles. Experimental Techniques 1992;September/ October:25–9. [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. [5] Weng CC, White RN. Cold-bending of thick high-strength steel plates. Journal of Structural Engineering, ASCE 1990;116(1):40–54.
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[6] Weng CC, White RN. Residual stresses in cold-bent thick steel plates. Journal of Structural Engineering, ASCE 1990;116(1):24–39. [7] Key PW, Hancock GJ. A theoretical investigation of the column behaviour of cold-formed square hollow sections. Thin-Walled Structures 1993;16:31–64. [8] Ingvarsson L. Cold-forming residual stresses—effect on buckling. In: Proceedings of the Third International Specialty Conference On Cold-formed Steel Structures. University of Missouri-Rolla, USA, November. 1975, p. 85–119. [9] Kato B, Aoki H. Residual stresses in cold-formed tubes. Journal of Strain analysis 1978;13(4):193–204. [10] Rondal J. Residual stresses in cold-rolled profiles. Construction and Building Materials 1987;1(3):150–64. [11] Guggenberger W. Effect of geometric imperfections taking into account the fabrication process and consistent residual stress fields of cylinders under local axial loads. Imperfections in Metal Silos Workshop. Lyon, France, 19th April. 1996, p. 219–31. [12] Chakrabarty J, Lee WB, Chan KC. An exact solution for the elastic/plastic bending of anisotropic sheet metal under conditions of plane strain. International Journal of Mechanical Sciences 2001;43:1871–80. [13] Johnson W, Yu TX. Springback after the biaxial elastic–plastic pure bending of a rectangular plate—I. International Journal of Mechanical Sciences 1981;23(10):619–30. [14] Tan Z, Li WB, Persson B. On analysis and measurement of residual stresses in the bending of sheet metals. International Journal of Mechanical Sciences 1994;36(5):483–91. [15] Pourboghrat F, Chu E. Springback in plane strain stretch/draw sheet forming. International Journal of Mechanical Sciences 1995;36(3):327–41. [16] Zhang ZT, Hu SJ. Stress and residual stress distributions in plane strain bending. International Journal of Mechanical Sciences 1998;40(6):533–4. [17] ABAQUS. Standard user’s manual, v6.2. USA: Hibbitt, Karlsson and Sorensen, Inc; 2001. [18] Quach WM, Teng JG, Chung KF. Finite element predictions of residual stresses in press-braked thin-walled sections (in preparation).
Residual stresses in steel sheets due to coiling and uncoiling: a closed-form analytical solution W.M. Quach a, J.G. Teng b,, K.F. Chung b a
b
Department of Civil and Environmental Engineering, University of Macau, Macau, China Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China Received 15 October 2003; received in revised form 14 April 2004; accepted 15 April 2004
Abstract Residual stresses in cold-formed members may play a significant role in determining their behaviour and strength, and have traditionally been obtained by laboratory measurements. This paper presents the results of research which forms part of a larger study on the theoretical predictions of residual stresses in cold-formed sections. The paper is concerned with residual stresses that arise from the coiling and uncoiling process experienced by steel sheets before they are cold formed into sections. In this paper, a closed-form solution is presented for these residual stresses, in which the coiling and uncoiling process is modelled as an elastic– plastic plane strain pure bending problem with the steel assumed to obey the von Mises yield criterion and the Prandtl–Reuss flow rule. To facilitate its application in subsequent finite element simulation of the cold-forming process, the prediction of equivalent plastic strains is also addressed in this solution. The accuracy of the solution is demonstrated by comparing its predictions with those from a finite element simulation. Numerical results from the analytical solution are also presented to illustrate the development process of residual stresses and the effects of coiling curvature and yield stress on the final residual stresses. # 2004 Elsevier Ltd. All rights reserved. Keywords: Residual stresses; Cold-formed sections; Plastic bending; Coiling; Uncoiling; Flattening; Finite element simulation
1. Introduction Residual stresses in cold-formed members may play a significant role in determining their behaviour and strength, and have traditionally been obtained by laboratory measurements. Various destructive and nondestruction methods for residual stress measurements are available and each has its own limitations [1]. Laboratory measurements of residual stresses in cold-formed thin-walled sections are time-consuming, difficult and have limited accuracy. For example, due to the thinness of cold-formed sections, variations of residual stresses across the plate thickness generally cannot be obtained. Indeed, in most existing experimental studies on cold-formed thin-walled sections [2–4], only surface residual stresses were measured, with the variations across the plate thickness being assumed to be linear. Residual stresses measured on Corresponding author. Tel.: +852-2766-6012; fax: +852-23346389. E-mail address: [email protected] (J.G. Teng).
0141-0296/$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2004.04.005
thicker plates [5–7] have shown that they vary across the plate thickness in a complex manner. Moreover, clear relationships between residual stresses and various steps of the fabrication process cannot be established by an examination of the measurement results. Accurate theoretical predictions of residual stresses in cold-formed sections require the modelling of the cold-forming process and are not yet available. Nevertheless, a limited amount of effort has been made in modelling residual stresses due to cold bending [8–10]. Ingvarsson [8] and Kato and Aoki [9] modelled the pure plastic bending of a wide plate as a plane strain problem by means of an incremental numerical process, with the steel assumed to obey the von Mises yield criterion and the Prandtl–Reuss flow rule. Rondal [10] presented a similar numerical analysis of the pure plastic bending of wide plates, and then proposed an approximate approach of deriving residual stresses in channel sections based on the results from his pure bending analysis. Numerical results from all three studies also showed complex residual stress variations through the plate thickness.
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To accurately predict residual stresses in cold-formed members, their manufacturing process needs to be closely modelled. Cold-formed members are usually manufactured by either roll forming or press-braking. In both roll forming and press-braking, before the coldforming process is applied to the flat steel sheet, the flat steel sheet has already experienced the coiling, uncoiling and flattening process. That is, cold rolled steel sheets are first coiled into rolls for storage. They are subsequently uncoiled from the roll and forced to become a flat sheet before cold-forming forces are applied. The residual stresses in a cold-formed section are therefore derived from two sources: the coiling, uncoiling and flattening process (referred to simply as the coiling–uncoiling process) and the cold-forming process. Such a two-stage fabrication process is illustrated in Fig. 1 for the press-braking method. The predictions of residual stresses in cold-formed sections can thus be separated into two tasks: the prediction of residual stresses from the coiling and uncoiling process involving pure bending of the steel sheet and the prediction of the residual stresses from the cold-bending process. This paper is concerned only with the first task by presenting a closed-form analytical solution for the pure bending of wide plates. The pure bending of wide plates into the plastic range has been studied by many researchers [8–16]. Among these existing studies, several were based on the deformation theory [13–15] so that the effect of deformation history on residual stresses was ignored. This effect was shown to be important by Zhang and Hu [16], so for accurate predictions, the flow theory should be used. The studies based on the flow theory [8–12] have all been numerical, and some of them [8,10] ignored part of the elastic deformation which is important for the coiling and uncoiling process of steel sheets where the curvature involved is not so large. This paper therefore presents an accurate closed-form analytical solution for the coiling–uncoiling process modelled as a plane strain pure bending problem with all factors appropriately included. To facilitate its application in subsequent finite element simulation of the cold-forming process, the prediction of equivalent plastic strains is also addressed in addition to residual stresses in both the longitudinal and transverse directions.
Fig. 1.
2. Analytical solution 2.1. Assumptions and terminology In the present study, it is assumed that any residual stresses due to cold rolling have been removed in the annealing furnace. That is, the flat steel sheet is assumed to be free from residual stresses before it is coiled for storage. Furthermore, as the effect of cold work is deemed to have been removed in the annealing furnace, a steel sheet before coiling can also be assumed to possess a stress–strain curve of the sharpyielding type (i.e. a yield plateau exists before strain hardening). The strains induced by the coiling and uncoiling process are relatively small, so it is reasonable to assume that strain hardening is not involved. That is, the steel can be assumed to possess an elastic– perfectly plastic stress–strain curve, and obey the von Mises yield criterion and the Prandtl–Reuss flow rule. The coiling of a steel sheet into a curvature jc and its subsequently uncoiling and flattening can thus be modelled as plane strain pure elastic–plastic bending in the y–z plane (Fig. 1). This section presents a closed-form analytical solution for this elastic–plastic bending problem. 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, the width direction of the plate is referred to as the transverse direction, while the direction normal to the curved plate surface is referred to as the radial direction. In terms of the bent plate, the longitudinal direction is the circumferential direction. The present terminology has the advantage that the longitudinal direction remains the longitudinal direction of a cold-formed member produced from a steel sheet after the coiling and uncoiling process being discussed here. 2.2. Coiling During the coiling of the steel sheet, an arbitrary point in the steel sheet undergoes elastic or elastic–plastic deformation, depending on the given coiling curvature jc and its location y away from the neutral axis of
Manufacturing process of press-braked sections. (a) coiling; (b) uncoiling including flattening; (c) press-braking.
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
the section. For elastic material points across the thickness, under a plane strain condition in the width direction (x direction) and a plane stress condition in the through-thickness direction (y direction), the in-plane strains are given by ðrz;c mrx;c Þ E ðrx;c mrz;c Þ ¼0 ¼ E
ez;c ¼
ð1aÞ
ex;c
ð1bÞ
where E is the elastic modulus, rx,c and rz,c are the stresses in the x and z directions, respectively, due to coiling while ex,c and ez,c are the corresponding strains. The von Mises yield criterion is given by r2z;c
þ
r2x;c
rx;c rz;c ¼
r2y
ð2Þ
where ry is the yield stress. The elastic coiling stresses at any arbitrary location y from the neutral axis of the section are then related to the longitudinal strain due to bending ez,c by rz;c
E ez;c ¼ ð1 m2 Þ
ð3aÞ
mE ez;c ð1 m2 Þ
ð3bÞ
rx;c ¼ with
ez;c ¼ jc y
ð3cÞ
The above relationships are valid only for points under elastic straining. By substituting Eq. (3) into Eq. (2), the longitudinal strain at which a material point starts to yield is found to be pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez;cy ¼ ry ð1 m2 Þ E 1 m þ m2 ð4Þ
Fig. 2.
1251
in which ez;cy > 0 if the strain is tensile (i.e. y > 0). Under a coiling curvature jc, the central core of material of the section remains elastic and the size of this central core is twice the value given by the following expression: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ycy ¼ ry ð1 m2 Þ Ejc 1 m þ m2 ð5Þ Therefore, for j yj ycy , the stresses are given by Eq. (3). Points located at j yj > ycy are in plastic flow and the stresses obey the von Mises yield criterion. By defining the following stress ratio xc ¼ rx;c =rz;c
ð6Þ
and combining it with Eq. (2), the coiling stresses of any point undergoing plastic straining (such as point P in Fig. 2) can then be obtained as ry rz;c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xc þ x2c xc ry rx;c ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xc þ x2c
ð7aÞ ð7bÞ
in which rx,c and rz;c 0 when y 0. In order to determine the coiling stresses at any location y for a given coiling curvature jc, it is necessary to relate the stress ratio xc to the value of y as detailed below. The Prandtl–Reuss flow rule is given by 3 dezp;c sz;c ¼ dep;c qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð8Þ dexp;c 2 r2z;c þ r2x;c rx;c rz;c sx;c in which the subscript c denotes that the variables are associated with the coiling stage, dezp,c and dexp,c are the longitudinal and transverse plastic strain increments, dep;c is the equivalent plastic strain increment,
Stress path of the extreme fiber of a steel strip during coiling–uncoiling process.
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and sz,c and sx,c are the deviatoric stresses and are given by sz;c ¼ ð2rz;c rx;c Þ=3
ð9aÞ
sx;c ¼ ð2rx;c rz;c Þ=3
ð9bÞ
Eq. (8) implies that dezp;c sz;c 2rz;c rx;c ¼ ¼ dexp;c sx;c 2rx;c rz;c
ð10Þ
By substituting Eq. (6) into Eq. (10), the following equation is obtained: dezp;c 2 xc ¼ dexp;c 2xc 1
ð11Þ
The longitudinal coiling strain increment dez,c consists of an elastic strain increment deze,c and a plastic strain increment dezp,c dez;c ¼ deze;c þ dezp;c
ð12Þ
The transverse coiling strain increment is zero due to the plane strain condition, therefore, dex;c ¼ dexe;c þ dexp;c ¼ 0
ð13Þ
Substitution of Eqs. (11) and (13) into Eq. (12) leads to
2 xc ð14Þ dez;c ¼ deze;c dexe;c 2xc 1 The incremental elastic strains are given by 1 1 drz;c deze;c m ¼ dexe;c drx;c E m 1
ð15Þ
On the other hand, differentiation of Eq. (7a) gives ry ð1 2xc Þ
drz;c ¼
2ð1 xc þ x2c Þ3=2
dxc
ð16Þ
while differentiating Eq. (6) yields drx;c ¼ xc drz;c þ rz;c dxc to drx;c ¼
2ð1 xc þ x2c Þ3=2
dxc
ð18Þ
Eqs. (15), (16) and (18) can now be substituted into Eq. (14) to arrive at dez;c ¼
m
By substituting Eqs. (3c) and (4) into Eq. (20), the following expression is obtained: " ry ð1 m2 Þ ry xc ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j yj ¼ 2 Ejc 1 xc þ x2c Ejc 1 m þ m 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xc pffiffiffi 3 4ð1 xc þ x2c Þ A5 coth1 @ þ ð21Þ 2 3 m
which relates the stress ratio xc to the value of y. Therefore, for a given coiling curvature jc, the stress ratio xc can be determined from Eq. (21) at each location of j yj ycy , and then the corresponding inelastic coiling stresses can be determined from Eq. (7). There exists a limit value jcy for the coiling curvature at which the extreme surfaces of the steel sheet start to yield (shown as point E in Fig. 2), and this limit is found by substituting Eqs. (3a) and (3b) into Eq. (2) and noting that ez;c ¼ jcy t=2: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jcy ¼ 2ry ð1 m2 Þ Et 1 m þ m2 ð22Þ The value of jcy depends only on the material properties of the steel sheet. Obviously, if the coiling curvature jc < jcy , no plastic bending is involved during coiling and hence no residual stresses exist following uncoiling. When jc jcy , yielding of material due to coiling will result in the development of residual stresses at the end of the coiling–uncoiling process.
ð17Þ
Substitution of Eqs. (7a) and (16) into Eq. (17) leads ry ð2 xc Þ
and the right-hand side from Poisson’s ratio m to the stress ratio xc corresponding to ez,c results in " ry xc ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez;c ¼ ez;cy E 1 xc þ x2c 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xc pffiffiffi 4ð1 xc þ x2c Þ A5 3 coth1 @ þ ð20Þ 3 2
ry 2Eð1 xc þ x2c Þ3=2 "
# ð2 xc Þ2 ð1 2xc Þ 2mð2 xc Þ dxc ð2xc 1Þ ð19Þ Integrating the left-hand side of Eq. (19) from the onset yield strain ez,cy to an arbitrary coiling strain ez,c
2.3. Uncoiling including flattening It should be noted that, when jc jcy , the natural uncoiling of a coiled sheet leads to a sheet with a small residual curvature, but in practice this curvature is removed either before cold forming by the imposition of necessary restraints or during cold forming as a result of the out-of-plane stiffness associated with a fold. In the present study, flattening, corresponding to the imposition of necessary restraints, is assumed to take place before cold forming, and implemented by the application of a curvature equal in magnitude but opposite in direction to the coiling curvature. That is, the uncoiling curvature ju satisfies the following condition: ju ¼ jc
ð23Þ
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
After such uncoiling, the final stresses of any point can be found by adding the uncoiling stresses to the coiling stresses (point UP in Fig. 2): rz;r ¼ rz;c þ rz;u
ð24aÞ
rx;r ¼ rx;c þ rx;u
ð24bÞ
The unloading stresses are elastic, until the reverse bending curvature exceeds a threshold curvature value. The elastic uncoiling stresses are given by rz;u
E ju y ¼ ð1 m2 Þ
ð25aÞ
mE ju y ð1 m2 Þ
ð25bÞ
rx;u ¼
An uncoiling curvature limit juy, beyond which uncoiling stresses are no longer elastic, can be defined to indicate the onset of reverse yielding (point UE in Fig. 2). If reverse yielding occurs as a result of uncoiling, then the final stresses should also obey the von Mises yield criterion. That is, r2z;r þ r2x;r rx;r rz;r ¼ r2y
ð26Þ
Such an uncoiling curvature limit juy can be determined by substituting Eqs. (6), (7a), (24) and (25a) into Eq. (26) as juy ¼
ry ð1 m2 Þ½2 m þ ð2m 1Þxc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E j yjð1 m þ m2 Þ 1 xc þ x2c
ð27Þ
The total longitudinal strain of any point at the onset of reverse yielding during uncoiling is then ez;uy ¼ ðjc þ juy Þy
ð28Þ
The corresponding uncoiling stresses are rz;uy ¼
E juy y ð1 m2 Þ
ð29aÞ
rx;uy ¼
mE juy y ð1 m2 Þ
ð29bÞ
and the corresponding stress ratio is xuy ¼
rx;c þ rx;uy rz;c þ rz;uy
ð30Þ
Making use of Eqs. (7), (27) and (29), Eq. (30) can be re-written as ½ð1 m2 Þxc mð2 mÞ ½ð1 2mÞxc ð1 m2 Þ Therefore, when jc juy , xuy ¼
ð31Þ
rz;u ¼
E jc y ð1 m2 Þ
ð32aÞ
rx;u ¼
mE jc y ð1 m2 Þ
ð32bÞ
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When jc > juy , reverse yielding occurs. The final stresses after uncoiling, being constrained by the von Mises yield criterion, are given by ry rz;r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xu þ x2u
ð33aÞ
xu ry rx;r ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xu þ x2u
ð33bÞ
with xu ¼ rx;r =rz;r ¼ ðrx;c þ rx;u Þ=ðrz;c þ rz;u Þ
ð33cÞ
in which rx,r and rz;r 0 when y 0. Hence, from Eqs. (7), (24) and (33), the uncoiling stresses are calculated as ! ry ry rz;u ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð34aÞ 1 xc þ x2c 1 xu þ x2u rx;u
xc ry xu ry ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 xc þ x2c 1 xu þ x2u
! ð34bÞ
where rx,u and rz;u 0 when y 0. To determine the uncoiling stresses or total stresses after uncoiling at an arbitrary location y, it is again necessary to first relate the stress ratio xu to the value of y. Following the same procedure as explained in the preceding subsection (Eqs. (8)–(19)), the following equation for the longitudinal strain increment during uncoiling can be obtained: " ry dez;u ¼ ð1 2xu Þ 2mð2 xu Þ 2Eð1 xu þ x2u Þ3=2 # ð2 xu Þ2 dxu ð35Þ ð2xu 1Þ Eq. (35) is the same as Eq. (19), except for a change in sign since uncoiling causes material yielding in the opposite direction and a different subscript to indicate that this equation is for uncoiling. Integrating the left-hand side of Eq. (35) from the longitudinal strain ez,uy at the onset of reverse yielding to the final longitudinal strain ez,r and the right-hand side from the stress ratio xuy at the onset of reverse yielding to the stress ratio xu corresponding to ez,r results in " ry xu ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ez;r ez;uy ¼ E 1 xu þ x2u 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xu pffiffiffi 4ð1 xu þ x2u Þ A5 3 coth1 @ þ ð36Þ 3 2 xuy
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Since uncoiling including flattening enforces ez;r ¼ 0 at the end of the process, substitution of Eq. (28) into Eq. (36) yields " ry xu ð1 2mÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi j yj ¼ Eðjc þ juy Þ 1 xu þ x2u 0 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13xu pffiffiffi 4ð1 xu þ x2u Þ A5 3 coth1 @ þ ð37Þ 3 2 xuy
where the values of juy and xuy can be determined from Eqs. (27) and (31), respectively. With Eq. (37), the stress ratio xu can be determined at any location y for jc > juy , and then the corresponding uncoiling stresses and total residual stresses can be determined from Eqs. (34) and (24), respectively. 2.4. Equivalent plastic strain Apart from the residual stresses, the equivalent plastic strain is another quantity of interest as it reflects the total deformation state of a point. It can be derived from the equivalent plastic strain increment dep . During the coiling stage, the equivalent plastic strain increment dep;c is given by dep;c ¼
2ry ðsz;c þ msx;c Þ dez;c 3ðs2x;c þ s2z;c þ 2msx;c sz;c Þ
ð38Þ
where sz,c and sx,c are the deviatoric stresses and are given by Eq. (9). By substituting Eqs. (9) and (19) into Eq. (38), one obtains dep;c ¼
ry ½ð2 xc Þ þ mð2xc 1Þ dxc Eð1 xc þ x2c Þð1 2xc Þ
ð39Þ
Integrating the left-hand side from zero to the corresponding equivalent plastic strain ep;c experienced during coiling and the right-hand side of Eq. (39) from Poisson’s ratio m to the stress ratio xc results in " ! ry ð1 xc þ x2c Þ1=2 ep;c ¼ ln E 2xc 1 !#xc pffiffiffi pffiffiffi 3 3 ð2x 1Þ c tan1 þð1 2mÞ ð40Þ 3 3 m
Similarly, the equivalent plastic strain increment dep;u during the uncoiling stage can be obtained as dep;u ¼
ry ½ð2 xu Þ þ mð2xu 1Þ dxu Eð1 xu þ x2u Þð1 2xu Þ
ð41Þ
Integrating the left-hand side of Eq. (41) from the equivalent plastic strain ep;c at the end of coiling to the final equivalent plastic strain ep;r after flattening, and the right-hand side from the stress ratio xuy at the onset of reverse yielding to the stress ratio xu
corresponding to ez,r or ep;r , results in " ! ry ð1 xu þ x2u Þ1=2 ep;r ¼ ep;c þ ln E 2xu 1 !#xu pffiffiffi pffiffiffi 3 3ð2xu 1Þ 1 tan þð1 2mÞ 3 3
ð42Þ
xuy
With the stress ratios xc and xu determined at each location y from Eqs. (21) and (37), the equivalent plastic strain during coiling and uncoiling can be calculated from Eqs. (41) and (42), respectively. 3. Finite element simulation In order to verify the analytical model presented in the preceding section, the coiling and uncoiling process of steel sheets was also simulated using the finite element package ABAQUS [17]. A flat steel strip of 60 mm in length was modelled with one end fixed and the other end free (see Fig. 3). The steel was assumed to be elastic–perfectly plastic with the following properties: yield stress ry ¼ 250 MPa, Young’s modulus E ¼ 200 GPa and Poisson’s ratio m ¼ 0:3. The sheet had a thickness of 2 mm. Twenty four layers of CPE4R elements, which are 2-D plane strain four-node elements with reduced integration and hourglass control, were employed to capture the through-thickness variations of stresses (Fig. 3). Two steps were required to simulate the whole coiling–uncoiling process: coiling was simulated as pure bending of the cantilever steel strip to a coiling radius r (¼D=2) of 250 mm, and uncoiling including flattening was simulated as reverse bending of the strip to the initial zero curvature. The steps involved are summarized in Fig. 4. At the fixed end, all nodes were constrained in the longitudinal direction. In addition, the node at the mid-depth was constrained also in the through-thickness direction. By means of kinematic coupling, the longitudinal displacements of all nodes at the free end were constrained to the rigid body motion of the reference node located at the mid-depth on the free end, to ensure that the plane section remained plane. Both coiling and uncoiling were achieved by specifying the desired amounts of translation and rotation of the reference node at the free end, corresponding to the desired coiling curvature and final zero curvature, respectively. Both geometrical and material non-linearities were considered. The ABAQUS metal plasticity model is characterized by the von Mises yield criterion and the Prandtl–Reuss flow rule with isotropic hardening. The finite element model predicted residual stresses which are uniform along the whole length, so only the stress
W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259
Fig. 3.
Mesh and boundary conditions.
distributions at the fixed end are compared in Figs. 5 and 6 with the predictions of the analytical solution. The longitudinal and transverse residual stresses as well as the equivalent plastic strains predicted by both the analytical solution and the finite element simulation are seen to be in very close agreement, which demonstrates the validity and accuracy of both approaches. The results in Figs. 5 and 6 also show that the residual stresses are not linearly distributed across the thickness. Therefore, the assumption of linear residual stress distribution adopted in existing experimentally-based residual stress studies is not appropriate. 4. Stress path of the coiling–uncoiling process Using the predictions from the analytical solution presented and verified above, the steel sheet described in the preceding section is examined here to illustrate the development of stresses at different stages of the coiling-flattening process. Fig. 2 shows the stress path of the extreme tensile fibre of the 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 path O–E represents elastic coiling, with point E being reached (attainment
Fig. 4.
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of yielding) when the applied curvature jcy reaches 0.00128 mm1. During this stage, the through-thickness distributions of both the longitudinal and transverse stresses are linear (Fig. 7a). With further loading, the stress path of the extreme fibre is represented by EP in Fig. 2, with point P denoting the end of the coiling stage. In this stage, the through-thickness stress distributions become non-linear and the equivalent plastic strain ep;c starts to accumulate (Figs. 7b and 8). At the end of the coiling stage, two plastic zones are developed near the top and bottom surfaces of the sheet with an elastic core in the middle (Fig. 7b). The maximum longitudinal and transverse coiling stresses are 1.154ry and 0.557ry, respectively. Elastic uncoiling 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. 7c. The next stage, involving reverse yielding, is represented by stress path UE–UP in Fig. 2. During this stage, plastic strains are induced. At the end of this stage, the residual stresses are as shown in Fig. 7d, while the distribution of equivalent plastic strain is shown in Fig. 8. The maximum longitudinal and transverse residual stresses are 1.145ry and 0.444ry, respectively. It is seen that, due to the zero curvature after flattening,
Pure bending of a flat strip.
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Fig. 5. Comparison of residual stresses between the analytical solution and finite element analysis, (a) longitudinal coiling stress; (b) transverse coiling stress; (c) final longitudinal residual stress; (d) final transverse residual stress.
no residual stresses are found in the elastic core. Two zones of high tensile and compressive longitudinal residual stresses, with magnitudes greater than ry, are located near the inner and outer surfaces of the flattened sheet, respectively, and each has a size larger than 0.25t. On the surfaces of the steel sheet, the stress ratio xc after coiling is 0.482 while the stress ratio xu after flattening is 0.387. Because elastic strains are considered in the present analysis, the stress ratios xc and xu will never reach the value of 0.5 for rigid-plastic bending. However, from Eqs. (20) and (36), it can be observed that, as the longitudinal coiling strain ez,c and the final strain after flattening ez,r approach infinity, the stress ratios xc and xu will approach 0.5. This means that the point UL on the yield envelope (Fig. 2) can only be approached but can never be reached. 5. Effect of coiling curvature
Fig. 6. Comparison of equivalent plastic strains between the analytical solution and finite element analysis, (a) after coiling; (b) after flattening.
Obviously, the residual stresses resulting from the coiling–uncoiling process depend on the coiling curvature (or coiling radius), so in this section, the effect of coiling curvature on residual stresses is explored for coiling radii ranging from 200 to 700 mm for steel sheets of 2 mm in thickness. The steel was assumed to have the same properties as given earlier in the paper except that three different yield stresses are considered in this section (ry ¼ 250, 350 and 450 MPa). The range
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Fig. 7. Residual stresses over thickness during the coiling–uncoiling process, (a) onset of surface yielding; (b) end of coiling; (c) onset of reverse surface yielding; (d) after flattening.
of radii was chosen to cover the lower values of a practical range as a coiling radius greater than 781 mm will not lead to any residual stresses. Fig. 9 shows that the magnitudes of residual stresses and the sizes of affected zones are quite sensitive to the coiling curvature. It is of interest to note that the coiling curvature limit jcy, which signifies the onset of yielding on sheet surfaces, depends on the material properties and the sheet thickness (refer to Eq. (22)). The uncoiling curvature limit juy, which signifies the onset of reverse yielding, depends not only on the
Fig. 8.
Equivalent plastic strain distributions.
material properties, but also on the bending history and the through-thickness location concerned (refer to Eq. (27)). Although the value of juy depends on the bending history, which is related to the applied coiling curvature jc, a change of the r=t ratio from 100 to 350 leads only to a change of juy value on the sheet surfaces from 0.00250 to 0.00256 mm1. This change in juy, which is greater than changes elsewhere across the thickness, is less than 3%. This indicates that the uncoiling curvature limit juy at which the steel starts to experience reverse yielding is not sensitive to the applied coiling curvature jc. As shown in Fig. 9 (also see Table 1), as the r=t ratio decreases, the maximum surface residual stresses initially increase rapidly until the r=t ratio is small enough for reverse yielding to occur during uncoiling. With further decreases in the r=t ratio, the maximum surface residual stresses become stable as required by the von Mises yield criterion, but the residual stress zones continuously expand. The total thickness of the two residual stress zones referred to in Table 1 is given by (t 2ycy ), where ycy is half the thickness of the central elastic core. Table 1 summarizes the magnitudes of maximum surface residual stresses and the total thickness of the residual stress zones. In addition to the r=t ratio, the effect of steel yield strength is also illustrated in Table 1,
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W.M. Quach et al. / Engineering Structures 26 (2004) 1249–1259 Table 1 Effect of r=t ratio and yield strength on the magnitudes of maximum surface residual stresses and the size of residual stress zones r=t
ry (MPa)
Total thickness of residual stress zones over sheet thickness ðt 2ycy Þ=t
Maximum rz;r =ry
Maximum rx;r =ry
100
250 350 450 250 350 450 250 350 450 250 350 450 250 350 450 250 350 450
0.74 0.64 0.54 0.62 0.46 0.31 0.49 0.28 0.08 0.36 0.10 N.A.a 0.23 N.A. N.A. 0.10 N.A. N.A.
1.153 1.136 1.095 1.127 0.943 0.485 1.047 0.428 0.091 0.612 0.124 N.A. 0.326 N.A. N.A. 0.124 N.A. N.A.
0.522 0.386 0.230 0.347 0.141 0.056 0.163 0.048 0.007 0.077 0.010 N.A. 0.033 N.A. N.A. 0.010 N.A. N.A.
150
200
250
300
350
a
Fig. 9. Effect of r=t ratio on residual stress distributions. (a) final longitudinal residual stress; (b) final transverse residual stress.
where results for three different yield stresses (ry ¼ 250, 350 and 450 MPa) are shown. As expected, as the r=t ratio and the yield stress increase, the residual stresses reduce in both value and extent. At r=t ¼ 100, more than half of the sheet thickness experiences plastic bending and develops residual stresses for all three yield strengths. On the other hand, at r=t ¼ 350, residual stresses are only induced when the yield stress is 250 MPa, and these residual stresses are small. Obviously, at high values of both the r=t ratio and the yield strength ry, the coiling curvature jc < jcy and no plastic bending is involved. 6. Conclusions Residual stresses in cold-formed sections are in general due to both the cold-forming process (press-braking or cold rolling) and the prior coiling–uncoiling process. In both processes, residual stresses are induced as a result of plastic bending. This paper has been concerned with the accurate prediction of residual stresses resulting from the coiling–uncoiling process, as such predictions are a necessary starting point for the prediction of residual stresses from the process of cold forming.
N.A.: No value is available, since no plastic bending is involved.
In this paper, an analytical solution for the residual stresses from the coiling–uncoiling process has been presented in which coiling, uncoiling and flattening are all taken into account in a plane strain pure elastic– plastic bending model. A finite element simulation of the same problem has also been presented. The analytical predictions have been shown to agree closely with finite element results, demonstrating the validity of both approaches. Results from both methods showed that through-thickness variations of residual stresses are non-linear. The analytical solution was employed to generate numerical results to illustrate the development process of residual stresses and the effects of coiling curvature and steel yield stress on residual stresses. The magnitude and extent of residual stresses were found to be sensitive to the coiling radius and the yield stress of steel. The analytical solution provides accurate residual stresses which can be specified as initial stresses in a finite element simulation of the cold-forming process. Such exploitation of the present analytical solution will be reported in a forthcoming paper [18]. It should be noted that the analytical solution presented in this paper is based on the plane strain assumption which is invalid for a narrow zone along each longitudinal edge of a wide plate of finite width. As a result, the solution is unlikely to give satisfactory predictions if applied to the coiling of a rather narrow strip (e.g. a width-to-thickness ratio below 50). It should also be noted that steel sheets may be coiled at an elevated temperature which may affect the mechanical
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properties of the steel and hence the resulting residual stresses. Both the effects of elevated temperature and the plane strain assumption warrant attention in future research. Finally, it is worth noting that although the present solution has been presented in the context of cold-formed steel sections, it can be applied to any other situations where a wide steel plate is bent into a cylindrical surface. Acknowledgements The authors would like to thank The Hong Kong Polytechnic University (Project No. G-V864), the University of Macau (Ref. No. SD009(3)/99-00S/ QWM/FST) and the Research Grants Council of the Hong Kong S.A.R. (Project No. PolyU5056/02E) for their financial support. References [1] Rowlands RE. Residual stress. In: Kobayashi AS, editor. Handbook on experimental mechanics. 2nd revised ed. New York: VCH Publ; 1993, p. 785–828. [2] Weng CC, Peko¨z T. Residual stresses in cold-formed steel members. Journal of Structural Engineering, ASCE 1990;116(6): 1611–1625. [3] Batista EM, Rodrigues FC. Residual stresses measurements on cold-formed profiles. Experimental Techniques 1992;September/ October:25–9. [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. [5] Weng CC, White RN. Cold-bending of thick high-strength steel plates. Journal of Structural Engineering, ASCE 1990;116(1):40–54.
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