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Prediction of residual stresses and strains in cold-formed steel members
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Thin-Walled Structures 46 (2008) 1274–1289 www.elsevier.com/locate/tws
Prediction of residual stresses and strains in cold-formed steel members Cristopher D. Moen, Takeru Igusa, B.W. Schafer Department of Civil Engineering, Johns Hopkins University, Latrobe Hall 210, Baltimore, MD 21218, USA Received 22 October 2007; received in revised form 13 February 2008; accepted 13 February 2008 Available online 9 April 2008
Abstract The objective of this paper is to provide an unambiguous mechanics-based prediction method for determination of initial residual stresses and effective plastic strains in cold-formed steel members. The method is founded on basic physical assumptions regarding plastic deformations and common industry practice in manufacturing. Sheet steel coiling and cross-section roll-forming are the manufacturing processes considered. The structural mechanics employed in the method are defined for each manufacturing stage and the end result is a series of closed-form algebraic equations for the prediction of residual stresses and strains. Prediction validity is evaluated with measured residual strains from existing experiments, and good agreement is shown. The primary motivation for the development of this method is to define the initial state of a cold-formed steel member for use in a subsequent nonlinear finite element analysis. The work also has impact on our present understanding of cold-work of forming effects in cold-formed steel members. r 2008 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel; Residual stresses; Residual strains; Finite element analysis
1. Introduction Thin cold-formed steel members begin as thick, molten, hot steel slabs. Each slab is typically hot-rolled, coldreduced, and annealed before coiling and shipping the thin steel sheet to roll-forming producers [1]. Once at a plant, the sheet is unwound through a production line and plastically folded to form the final shape of a structural member, as shown in Fig. 1. This manufacturing process imparts residual stresses and plastic strains through the sheet thickness. These residual stresses and strains influence the load–displacement response and ultimate strength of cold-formed steel members. In previous work, a statistical approach was employed to draw conclusions on the magnitude and distribution of longitudinal residual stresses using a data set of surface strain measurements collected by researchers between 1975 and 1997 [2]. The measured surface strains are converted to residual stresses using Hooke’s Law and then distributed through the thickness as membrane (constant) and bending (linear variation) components. These residual stress disCorresponding author.
E-mail address: [email protected] (C.D. Moen). 0263-8231/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.02.002
tributions are a convenient way to express the measured residual surface strains, and are convenient as well for use in nonlinear finite element analyses, but they are not necessarily consistent with the underlying mechanics. Plastic bending, followed by elastic springback, creates a nonlinear through-thickness residual stress distribution, in the direction of bending, as shown in Fig. 2 [3]. The presence of nonlinear residual stress distributions in coldformed steel members has been confirmed in experiments [4] and in nonlinear finite element modeling of pressbraking steel sheets [5]. A closed-form analytical prediction method for residual stresses and equivalent plastic strains from coiling, uncoiling, and mechanical flattening of sheet steel is presented in [6]. The same plastic bending that creates these residual stresses also initiates the cold-work of forming effect, where plastic strains increase the apparent yield stress in the steel sheet (and ultimate strength in some cases) as discussed in Ref. [7]. Together, these residual stresses and plastic strains comprise the initial material state of a cold-formed steel member. A general method for predicting the manufacturing residual stresses and plastic strains in cold-formed steel members is proposed here. The procedure is founded on common industry manufacturing practices and basic
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Fig. 1. Cold-formed steel roll-forming: (left) sheet coil enters roll-forming line, (right) steel sheet is cold-formed into C-shape cross-section (photos courtesy of Bradbury Group).
Elastic springback σ
Plastic bending
Plastic bending
Elastic springback
ε
compression
tension plastic bending Elastic springback
Nonlinear residual stress distribution
Fig. 2. Forming a bend: plastic bending and elastic springback of thin sheets results in a nonlinear through-thickness residual stress distribution.
physical assumptions. The primary motivation for the development of this method is to define the initial state of a cold-formed steel member for use in a subsequent nonlinear finite element analysis. The derivation of the prediction method is provided for each manufacturing step, and the predictions are evaluated with measured residual strains from existing experiments. The end result of the method is intended to be accessible to a wide audience including manufacturers, design engineers, and the academic community. 2. Stress–strain coordinate system and notation The stress–strain coordinate system and geometric notation used in the forthcoming derivations are defined in Fig. 3. The x-axis is referred to as the transverse direction and the z-axis as the longitudinal direction of a structural member. Cross-section elements are referred to as either ‘corners’ or ‘flats’. The sign convention for stress and strain is positive for tension and negative for compression. 3. Prediction method assumptions The following assumptions are employed to develop this prediction method:
(a) Plane sections remain plane before and after coldforming of the sheet steel. This assumption permits the use of beam mechanics to derive prediction equations. (b) The sheet thickness t remains constant before and after cold-forming of the sheet steel. A constant sheet thickness is expected after cold-bending if the bending is performed without applied tension [8]. Cross-section measurements demonstrate modest sheet thinning at the corners, where t in the corners is typically 5% less than in the flange and web [9]. This thinning is ignored here to simplify the derivations, although a reduced thickness based on the plastic strain calculations in Section 5 could be used if a higher level of accuracy is required. (c) The sheet neutral axis remains constant before and after cross-section cold-forming. Theoretical models used in metal forming theory do predict a small shift in the through-thickness neutral axis towards the inside of the corner as the sheet plastifies [8]. This shift is calculated as 6% of the sheet thickness, t, when assuming a centerline corner radius, rz, of 2.5t. A neutral axis shift of similar magnitude has been observed in the nonlinear finite element model results for thin press-braked steel sheets [5]. This small shift is ignored here to simplify the derivations.
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roller dies
rx
A
A
Forming direction
sheet steel coil Elevation view
x
t y
z
rz
y x z Section A-A
Fig. 3. Stress–strain coordinate system as related to the coiling and cold-forming processes.
(d) The steel stress–strain curve is assumed as elastic– perfectly plastic when calculating residual stresses. More detailed stress–strain models that include hardening are obviously possible, but a basic model is chosen to simplify the derivations. The implication of this assumption is that the residual stresses may be underestimated, especially in corner regions where the sheet has yielded completely through the thickness. (e) Plane strain behavior is assumed to exist during coiling, uncoiling, and flattening (ex ¼ 0) and during crosssection cold-forming (ez ¼ 0). (f) The steel sheet is fed from the top of the coil into the roll-forming bed as shown in Fig. 4a. This assumption is consistent with measured bending residual stress data (see Section 7) and manufacturing setups suggested by roll-forming equipment suppliers (Fig. 1). The first author did observe the alternative setup in Fig. 4b (sheet steel unrolling from the bottom of the coil) at a roll-forming plant, suggesting that the direction of uncoiling is a source of variability in measured residual stress data. (g) Membrane residual stresses are zero. Membrane residual stresses have been measured by several researchers (Refs. [4,9–14]), although the magnitudes are small relative to bending residual stresses (see Table 1). Membrane residual stresses are experimentally determined by averaging the measured surface strains on the two faces of a thin steel sheet. Given the variability inherent in these measurements it is difficult to know if the resulting membrane stresses (strains) are real or simply unavoidable experimental error.
4. Derivation of the residual stress prediction method The prediction method proposed here assumes that two manufacturing processes contribute to the through-thickness residual stresses in cold-formed steel members: (1) sheet coiling, uncoiling, and flattening, and (2) crosssection roll-forming. Algebraic equations for predicting the through-thickness residual stress and effective plastic strains in corners and flats are derived here and then summarized in flowcharts in Appendices A and B. 4.1. Residual stresses from sheet coiling, uncoiling, and flattening Coiling the sheet steel after annealing and galvanizing, but prior to shipment, may yield the steel if the virgin yield strain, eyield, is exceeded. If plastic deformation does occur, a residual curvature will exist in the sheet as it is uncoiled. This residual curvature is locked into a structural member resulting in longitudinal residual stresses as the sheet is flattened by the roll-formers. This process of coiling, uncoiling with residual curvature, and flattening is described in Fig. 5. 4.1.1. Coiling The through-thickness strain induced from coiling is related to the radial location of the sheet in the coil rx, with the well-known relationship from beam mechanics: z 1 ¼ . y rx
(1)
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Roll-forming bed
Sheet has residual CONCAVE curvature coming off the coil
Sheet has residual CONVEX curvature coming off the coil Fig. 4. Roll-forming setup with sheet coil fed from the (a) top of the coil and (b) bottom of coil. The orientation of the coil with reference to the rollforming bed influences the direction of the coiling residual stresses.
Table 1 Statistics of the residual stresses in roll-formed members Element
No. of samples
Residual stress as % syield Membrane
Corners Flats
Bending
Mean
S.D.
Mean
S.D.
5.7 1.8
10.1 10.7
32.0 25.2
23.8 20.7
resulting in a change in the through-thickness stress. This stress distribution is determined by first calculating the plastic coiling moment 2 t 1 2 coil ðrx yield Þ , (4) M x ¼ syield 2 3 and then applying an opposing moment elastically to simulate the removal of the imposed radial displacement
23 120
suncoil ¼ z ez is the engineering strain through the thickness y in the coiling (longitudinal) direction z. y varies from t/2 to t/2, where t is the sheet thickness. The radius associated with the elastic–plastic threshold initiating through-thickness yielding from coiling, rep, is derived by substituting ez ¼ eyield and y ¼ t/2 (outer fiber strain) into Eq. (2): rep ¼
t 2yield
.
(2)
When the coil radius rx is greater than rep the sheet steel experiences only elastic deformation on the coil. For sheet steel rolled to a coil radius rx less than rep, through-thickness yielding will occur as shown in Fig. 6. When rxorep, the depth of the elastic core c is defined as: c ¼ 2rx yield pt.
(3)
4.1.2. Uncoiling As the yielded sheet is uncoiled in preparation for the roll-forming line, the sheet steel springs back elastically
12M coil x y . 3 t
(5)
4.1.3. Flattening After the sheet has been unrolled, a permanent radius of curvature will still exist if rx was less than rep on the coil. This permanent radius is runcoil ¼ x
1 . ð1=rx Þ ðM coil x =EIÞ
(6)
Steel sheet with permanent curvature from coiling is pressed flat as the sheet enters the roll-forming line. The longitudinal stresses resulting from flattening the sheet are simply y sflatten ¼ E uncoil . (7) z rx 4.1.4. Residual stress distribution The total through-thickness longitudinal residual stress distribution due to coiling, uncoiling, and flattening is presented in Fig. 7. The resulting residual stress, sz, is selfequilibrating for axial force through the thickness but causes a residual longitudinal moment. Section 7 compares
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DETAIL A
Coiling
Uncoiled with residual curvature Flattened as sheet enters the roll-formers
Change in curvature locks in bending residual stresses in final member
Fig. 5. Coiling of the steel sheet may result in residual curvature which results in bending residual stresses as the sheet is flattened.
y +σyield
c
z
-σyield Fig. 6. Longitudinal residual stress distribution from coiling.
the stresses caused by this moment with surface strains (stresses) measured in experiments. The longitudinal residual stresses also will create transverse stresses across the width of the coil, assuming plane strain conditions for an infinitely wide sheet. Supporting the plane strain assumption is the observation that while the actual width of the sheet is finite, it remains several orders of magnitude greater than the sheet thickness. Under this assumption, and further assuming only elastic stresses, the transverse stresses are: uncoil sx ¼ n scoil . (8) þ sflatten z þ sz z Poisson’s ratio, n, is assumed here as 0.30 for steel deformed elastically. The through-thickness deformation from the uncoiling and flattening components will occur elastically, and the coiling component will be at least partially elastic through the thickness for the range of sheet thicknesses common in industry.
residual stresses are assumed to exist only at the location of the formed corners, between the roller die reactions, as shown in Fig. 8. Some yielding is expected to occur outside of the roller reactions as the stress distribution transitions from fully plastic to fully elastic; however, this transition area is not considered here to simplify the derivation. The engineering strain in the steel sheet, ex, and the bend radius, rz, are related for both small and large deformations with the strain–curvature relationship 1 x ¼ . rz y
This geometric relationship is valid for elastic and plastic bending of the steel sheet. For the small bend radii common in the cold-formed steel industry (rz ¼ 2t to 8t), the steel sheet yields through its thickness during the coldforming process. The steel sheet will reach the fully plastic stress state shown in Fig. 9 as the corner approaches its final manufactured radius. After the sheet becomes fully plastic through its thickness, the engineering strain continues to increase as the radius decreases. When the final bend radius is reached and the imposed radial displacement is removed, an elastic springback occurs that elastically unloads the corner (see Fig. 2). The change in stress through the thickness from this elastic rebound is derived with the plastic moment force couple shown in Fig. 10. The plastic moment is calculated with the equation t syield 1 t t syield t2 M bend ¼ , (10) ¼ FP ¼ z 2 2 2 4 which is then applied elastically through the thickness to simulate the stress distribution from elastic rebound of the sheet steel:
4.2. Residual stresses from cross-section roll-forming srebound ¼ x A set of algebraic equations is derived here to predict the transverse and longitudinal residual stresses created by rollforming a cross-section. Roll-forming residual stresses are cumulative with the coiling residual stresses derived in Section 4.1 and provide a complete prediction of the initial stress state of the member cross-section. The roll-forming
(9)
ðsyield t2 =4Þy 3syield y M bend y z . ¼ ¼ ð1=12Þ 1 t3 t I
(11)
The final transverse stress state is the summation of the fully plastic stress distribution through the thickness and the unloading stress from the elastic springback of the corner as shown in Fig. 11, where sx is the transverse residual stress through the thickness from the
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y
y
c
y
zuncoil
+σyield
z
+
1279
y
flatten z
+
z
z
= z z
-σyield Coil
Uncoil
Flatten
Residual Stress
Fig. 7. Predicted longitudinal residual stress distribution from coiling, uncoiling, and flattening of a steel sheet.
Assume bend is fully plastic between roller dies
Roller Die (Typ.)
Fig. 8. Cold-forming of a steel sheet.
y
y
-σyield
+σyield Fp t/2
x
+σyield
Fp
x
-σyield
Fig. 9. Fully plastic transverse stress state from cold-forming.
Fig. 10. Force couple applied to simulate the elastic springback of the steel sheet after the imposed radial deformation is removed.
cold-forming of the corner. This stress is nonlinear through the thickness and is self-equilibrating, meaning that axial and bending sectional forces are absent in the x-direction after forming. The transverse residual stresses will create stress in the longitudinal direction due to the assumed plane strain conditions (see Section 3):
The Poisson’s ratio, n, is assumed as 0.30 for steel deformed elastically and 0.50 for fully plastic deformation. The longitudinal residual stresses through the thickness, sz, are determined based on these assumptions as shown in Fig. 12. Longitudinal residual stress, sz, is selfequilibrating for axial force through the thickness but causes a residual longitudinal moment. This moment is hypothesized to contribute to the observed longitudinal residual strains measured in experiments (refer to
sz ¼ nsx .
(12)
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y -σyield
+1.5σyield
+0.5σyield
-σyield
-1.5σyield
+σyield
-0.5σyield Transverse Residual Stress x
Elastic Springback xrebound
Plastic Bending xbend
x
+σyield
Fig. 11. Cold-forming of a steel sheet occurs as plastic bending and elastic springback, resulting in a self-equilibrating transverse residual stress.
-σyield
-0.05σyield
+1.5σyield
0.50
-0.50σyield
0.30
-1.5σyield
x
+0.50σyield z
+σyield Plastic Bending plastic bend
y
Elastic Springback elasticxrebound
+0.05σyield Longitudinal Residual Stress z
Fig. 12. Plastic bending and elastic springback from cold-forming in the transverse direction result in longitudinal residual stresses because of the plane strain conditions.
Section 7for a comparison of this prediction to actual measurements). A flowchart summarizing the proposed prediction method for residual stresses in roll-formed members is provided in Appendix A. Appendix A explicitly demonstrates how coiling, uncoiling, flattening, and roll-forming contribute to the residual stresses locked into the crosssection during manufacturing.
5. Derivation of effective plastic strain prediction method In the method proposed here, plastic strains occur from sheet coiling and cold-forming, and together with residual stresses describe the initial material state of the member. The general state of plastic strain at a point can be quantified by using the von Mises yield criterion extended to plastic deformations (see Ref. [15]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ¼ 23 21 þ 22 þ 23 , (13) where ep is the effective plastic strain, and e1, e2, and e3 are the principal strains. All of the strains are ‘true’ strains, which may be calculated from the engineering
y zp
c
z
Fig. 13. Plastic strain distribution from sheet coiling with a radius less than elastic–plastic threshold rep.
strains via: 1 ¼ lnð1 þ x Þ;
2 ¼ lnð1 þ y Þ;
3 ¼ lnð1 þ z Þ,
(14)
where ex, ey, ez are in the Cartesian coordinate system (Fig. 3) and x, y, z are coincident with the principal directions. True strains are employed instead of engineering strains to accommodate the large deformations from plastic bending. Also, from a practical standpoint, nonlinear FE codes such as ABAQUS [16] require the engineer
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to provide true stress, true strain information (as large deformation theory is employed). The steel sheet is assumed to remain incompressible while experiencing plastic deformations, therefore when calculating ep 1 þ 2 þ 3 ¼ 0.
1281
y
(15)
5.1. Effective plastic strain from sheet coiling Engineering plastic strains, as shown in Fig. 13, accumulate during the coiling of sheet steel if the coiling radius rx is less than the elastic–plastic threshold rep. The engineering plastic strain distribution from coiling is: y c pz ¼ yield ; yX , rx 2 y c pz ¼ yield ; yp , rx 2 pz ¼ 0;
otherwise,
(16)
where the elastic core, c, is defined in Eq. (3). Plane strain conditions result in e1 ¼ 0, and e2 ¼ e3 via the incompressibility assumption of Eq. (15). Further, the Cartesian coordinate system is coincident with the principal axes, resulting in the following true principal plastic strains: 1 ¼ 0;
2 ¼ lnð1 þ pz Þ;
3 ¼ lnð1 þ pz Þ.
(17)
Substituting the principal strains into Eq. (13) and simplifying leads to the through-thickness effective plastic strain from coiling 2 (18) coiling ¼ pffiffiffi ln 1 þ pz . p 3 This plastic strain distribution, depicted in Fig. 14, will exist at all locations in the cross-section (corners and flats) when rx is less than the elastic–plastic threshold rep. The plastic strain from coiling, coiling , will generally be p much smaller in magnitude than the plastic strain from cross-section cold-forming, bend , as discussed in following p section.
Fig. 15. Effective von Mises true plastic strain at the location of coldforming of a steel sheet.
bent. The engineering plastic strain distribution from coldforming is described via y (19) px ¼ , rz which assumes that the elastic core at the center of the sheet is infinitesimally small. This assumption is consistent with the small bend radii common in industry (see Section 4.2). Plane strain conditions and Eq. (15) result in e3 ¼ 0, e2 ¼ e1. Physically these conditions imply that the sheet will experience some thinning at the location of coldforming (see Section 3), but the tendency to plastically shorten longitudinally will be resisted by the adjacent undeformed portion of the cross-section. As before, the Cartesian coordinate system is coincident with the principal axes, resulting in the following plastic principal strains: 1 ¼ lnð1 þ px Þ;
2 ¼ lnð1 þ px Þ;
3 ¼ 0.
(20)
Substituting for the principal strains and simplifying, the effective plastic strain at a cold-formed corner is: 2 bend ¼ pffiffiffi ln 1 þ px . p 3
(21)
5.2. Effective plastic strain from cross-section cold-forming
This effective plastic strain distribution is shown in Fig. 15. The distribution exists only at the cold-bent locations in a cross-section and should be added to the coiling plastic strain distribution in Fig. 14. A flowchart summarizing the prediction method for effective plastic strains in rollformed members is provided in Appendix B.
Large transverse plastic strains occur through the thickness of a thin steel sheet when the sheet is permanently
6. Employing the prediction method in practice: quantifying the coil radius influence
coiling p y
c
Fig. 14. Effective plastic strain in a cold-formed steel member from sheet coiling when the radius rx is less than the elastic–plastic threshold rep.
The residual stress and plastic strain distributions derived for cross-section cold-forming (Sections 4.2 and 5.2) are straight-forward to calculate if the yield stress, syield, and thickness, t, of the sheet steel are known. The coiling residual stresses and plastic strains are more difficult to calculate because the coil radius coinciding with the as-formed member, i.e., the radial location of the sheet, rx, is almost always unknown in practice. However, rx can be derived in an average sense though, since the range of inner and outer coil radii are known and the probability that a structural member will be manufactured from a certain rx can be quantified.
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The relationship between coil radius, rx, and corresponding linear location S of the sheet within the coil can be described using Archimedes spiral [17] p S ¼ r2x r2inner . (22) t The spiral maintains a constant pitch with varying radii, where the pitch is the thickness of the steel sheet, t, as shown in Fig. 16, L is the total length of sheet in the coil,
and rinner and router are the inside and outside coil radii, respectively. As-shipped outer coil radii range from 610 to 915 mm and inner coil radii range from 255 to 305 mm. These ranges were determined by the first author during a visit to a local roll-forming plant. Archimedes spiral is used to describe the probability that the steel sheet will come from a certain range of radial locations in the coil. The cumulative distribution function (CDF), FR(rx) ¼ probability that the radius is less than rx, is obtained by normalizing S by L S r2 r2inner ¼ F R ðrx Þ ¼ 2 x . L router r2inner
S
(23)
The probability density function (PDF) of rx is calculated by taking the derivative of FR(rx) rx t
f R ðrx Þ ¼
End S=L
Start S=0
dF R ðrx Þ 2rx ¼ 2 . drx router r2inner
(24)
The mean value of the radial location for a given inner and outer coil radii is Z
router
r¯x ¼ rinner
2 rinner router rinner þ router f R ðrx Þrx drx ¼ . 3 rinner þ router
Fig. 16. Coil coordinate system and notation.
(25)
Longitudinal Residual Stresses σyield = 210 MPa t/2
flat
corner
σyield = 345 MPa flat
corner
σyield = 550 MPa flat
corner y
t = 0.9 mm
-t/2 -σyield 0
σyield
t/2
z y
t = 1.8 mm
-t/2 -σyield 0
z
σyield
t/2
y
t = 2.6 mm -t/2 -σyield 0
σyield
z
Positive stress is tension, negative stress is compression Fig. 17. Influence of sheet thickness and yield stress on through-thickness longitudinal residual stresses (z-direction, solid lines are predictions for mean coil radius, dashed lines for mean radius7one standard deviation).
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7.1. Measurement statistics
The variance of the radial location is s2R ¼ ¼
Z
router
f R ðrx Þðrx r¯x Þ2 dr
rinner
ðrouter rinner Þ2 1 2 router þ 4router rinner þ r2inner . 18 ðrouter þ rinner Þ2
1283
(26)
These statistics for rx can then be used with the prediction method for coiling, uncoiling, and flattening residual stresses and plastic strains described in Sections 4 and 5. Fig. 17 summarizes the influence of sheet thickness and virgin yield stress on the longitudinal residual stress distributions in flats and corners. (The method proposed in this paper provides residual stresses and strains for the entire member, only the longitudinal residual stresses are shown in Fig. 17.) The solid lines in Fig. 17 are calculated using the mean value, r¯x ¼ 474 mm, from Eq. (25) assuming rinner ¼ 305 mm and router ¼ 610 mm. The distributions with the dashed lines are calculated with r¯x sR , where sR ¼ 86 mm is calculated with Eq. (26). The residual stresses are nonlinear through the thickness and have different shapes for flats and corners. The stress magnitudes at the outer fibers increase for thicker sheets and lower yield stresses. The accuracy of the linear bending residual stress model commonly employed in finite element analyses is perhaps sufficient when yield stress is low and thickness is high (relatively), but for typical thicknesses (0.9–1.8 mm) and yield stress (345 MPa) the assumption of a linear longitudinal stress distribution is not consistent with the mechanics-based predictions in Fig. 17.
The mean and standard deviation of the residual stresses for the 18 roll-formed specimens used in this comparison are provided in Table 1. Positive membrane stresses are tensile stresses and positive bending stresses cause tension at y ¼ t/2 (see Fig. 3 for coordinate system). The statistics demonstrate that both membrane and bending residual stress measurements are highly variable and that the membrane stresses are small relative to the steel yield stresses. Details on the residual stress measurements for each of the 18 specimens can be found in Ref. [18]. 7.2. Mean-squared error (MSE) estimate of radial location To explore the validity of the prediction method, the flat and corner residual stress measurements from the 18 specimens are used to estimate the radial location rx from which each specimen originated. These estimated radial locations are then used to calculate the difference between the predicted and measured longitudinal residual stresses. 7.2.1. MSE minimization The location of the specimen in the coil, rx, is estimated by minimizing the sum of the MSE for the p ¼ 1, 2, y, nq measurements taken around the cross-section of the q ¼ 1, 2, y, 18 specimens !2 nq X smeasured spredicted pq pq . (27) r^x;q ¼ arg min rx syield;pq p¼1 Both corner and flat measurements are included in the minimization.
7. Comparison of prediction method to measured residual stresses 8 7 6 5 MSE
The flat and corner residual surface strain measurements from 18 roll-formed specimens are used to evaluate the proposed residual stress prediction method. The prediction method provides the complete throughthickness longitudinal strain (stress) distribution if the radial location in the coil from which the specimen originated in the coil, rx, is known. Since the radial coil location of the 18 specimens is unknown, rx is statistically estimated for each specimen using the coil radius that best fits the predicted surface strains to the measured surface strains from a specimen cross-section (for both corners and flats). Once the best-fit radial locations have been calculated, they are examined to determine if their magnitude is rational when compared to typical inner and outer dimensions of a sheet coil. Although this comparison only provides a partial evaluation of the prediction method, it is as far as one can go with the available data. Qualitatively the prediction method is consistent with the more detailed through-thickness findings in Refs. [4,5].
4 3 2 1 0 1
1.5
2
2.5
3
3.5
rx/rinner Fig. 18. The mean-squared error of the predicted and measured bending residual stresses for de M. Batista and Rodrigues [12], Specimen CP1 is minimized when rx ¼ 1.60rinner.
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7.2.2. Bending component of longitudinal residual stress distribution The bending component of the predicted residual stress distribution must be isolated to compare with the measured values. The total predicted longitudinal residual stress distribution in the flats and corners of each cross-section is integrated to calculate the sectional moment through the thickness Z t=2 Mx ¼ sz y dy. (28)
7.2.3. Estimated coil radii using MSE Fig. 18 demonstrates the MSE results for de M. Batista and Rodrigues [12] Specimen CP1. The radial location that minimizes the prediction error is 1.60rinner in this case, and is summarized in Table 2 for all 18 roll-formed specimens considered. The estimated radial locations fall within the range of inner and outer coil radii assumed in the prediction (rinner to 2.40rinner) except for Dat RFC13 which is slightly outside the range at 2.45rinner. The MSE radial location cannot be determined in the three Bernard specimens [14] since the bending residual stresses in the flats are predicted to be zero. These three specimens are cold-formed steel decking with a thin sheet thickness t ranging from 0.75 to 1 mm and a relatively high yield stress syield ranging from 600 to 650 MPa. In this case, the coiling and uncoiling of the steel sheet will occur elastically as demonstrated in Fig. 17. Measured bending residual stress magnitudes in the flats of the Bernard specimens are on average 0.03syield which is consistent with the prediction method.
ðt=2Þ
Mx is then converted into a predicted outer fiber bending residual stress which can then be directly compared to the measurements ¼ spredicted pq
M x ðt=2Þ . I
(29)
Table 2 Radial location in the coil that minimizes the sum of the mean square prediction error for roll-formed members Researcher
Specimen
rx/rinner
de M. Batista and Rodrigues [12] de M. Batista and Rodrigues [12] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Dat [9] Dat [9] Bernard [14] Bernard [14] Bernard [14] Abdel-Rahman and Siva [19] Abdel-Rahman and Siva [19] Abdel-Rahman and Siva [19]
CP2 CP1 RFC13 RFC14 R13 R14 P3300 P4100 DC-12 DC-14 RFC14 RFC13 Bondek 1 Bondek 2 Condeck HP Type A—Spec 1 Type A—Spec 2 Type B—Spec 1
1.20 1.60 1.80 1.10 1.45 1.30 1.95 1.50 2.30 1.60 2.00 2.45 N/A N/A N/A 1.55 1.55 1.25
7.3. Statistical variations between measurements and predictions The predicted radial locations in Table 2 are now used to calculate the statistical variations between the experiments and predictions. The bending residual stresses in the 18 roll-formed members are calculated using the MSEpredicted radial location rx in Eq. (27) with the residual stress prediction method summarized in Section 4. The bending component of the residual stress prediction is then obtained with Eq. (29). The difference between the predicted and measured residual bending stresses, epq, for the p ¼ 1, 2, y, n measurements taken around the cross-section of the q ¼ 1, 2, y, 18 specimens is calculated as
epq ¼
rinner ¼ 254 mm, router ¼ 610 mm ¼ 2.40 rinner. N/A coiling residual stresses are predicted as zero.
40 e = 0.03yield e = 0.15yield
30
σmeasured/σyield
Observations
35 25 20 15 10 5 0
-1
-0.5
0
0.5
(σmeasured-σpredicted)/σyield
1
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2
smeasured spredicted pq pq syield;pq
0
.
(30)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 σpredicted/σyield
Fig. 19. (a) Histogram and (b) scattergram of bending residual stress prediction error (flat cross-sectional elements) for 18 roll-formed specimens.
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40 30
e = −0.16σyield
25
e = 0.19yield
σ measured /σyield
Observations
35
20 15 10 5 0 -1
-0.5 0 0.5 (σ measured-σ predicted)/σyield
1
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2
0
1285
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 σpredicted/σyield
Fig. 20. (a) Histogram and (b) scattergram of bending residual stress prediction error (corner cross-sectional elements) for 18 roll-formed specimens.
The error histogram for the flat cross-sectional elements in Fig. 19a demonstrates that the mean difference me is near zero with a standard deviation se ¼ 0.15syield. The scattergram in Fig. 19b demonstrates the strength of the correlation between the measurements and predictions in the flats; the solid regression line passes nearly through zero (y-intercept ¼ 0.05syield) and has nearly a unit slope (m ¼ 0.92). Also, the majority of the data lies within 7one standard deviation of the estimate, denoted as the dashed lines in the figure. It is concluded that the prediction method is consistent with the measured data in the flats. The corner element error histogram in Fig. 20a shows a negative bias of me ¼ 0.16syield meaning that the predicted residual stresses are generally higher than the measured values. The standard deviation of the error is large (se ¼ 0.19syield) but is less than the standard deviation of the corner residual stress measurements in Table 1 (sm ¼ 0.24syield). This demonstrates a greater match between the measurements and predictions, although more corner residual stress measurements are needed to improve the strength of this comparison. The scattergram in Fig. 20b highlights the variability in the measured corner data, especially in the region corresponding to spredicted ¼ 0.4syield , where bending residual stresses (strains) vary from 0 to 0.7syield. 8. Discussion The residual stresses and strains predicted with this method (Section 4 for stress, Section 5 for strain) form the initial material state in the cross-section. In design, this initial material state is sometimes considered through the so-called cold-work of forming effect, where the yield stress of the material is increased above the virgin yield stress, syield, to account for the ‘working of the corners’. For onedimensional stress–strain this concept is expressed as shown in Fig. 21, where ‘working the corners’ results in a residual plastic strain, ep, such that when the section is reloaded the stress at which yielding re-initiates, sey, is
ey yield
Virgin steel
Coldformed steel ers
Apparent yield stress
Effective residual stress yield + p
Fig. 21. Definition of apparent yield stress, effective residual stress, and effective plastic strain as related to a uniaxial tensile coupon test.
greater than the virgin yield stress, syield. If no residual stresses existed the apparent increase in the yield stress from syield to sey can be significant. However, as Fig. 16 illustrates, ‘working the corners’ also contributes to residual stresses, srs e , and these residual stresses may decrease the apparent yield stress. The prediction method presented herein provides a more nuanced understanding of the cold-work of forming effects. The residual plastic strains may increase the apparent yield stress, but those strains vary through the thickness and have contributions from both transverse and longitudinal strains. Further, residual stresses follow their own relatively complicated distribution through the thickness. In a multi-axial stress state using the von Mises yield criterion, Fig. 21 is enforced for the effective
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stress–effective strain pair for every point in the crosssection. As a result, the apparent yield stress upon loading varies through the thickness and is influenced by both the residual stresses and strains. Even under simple loading conditions (e.g., compression) a cold-formed member undergoes plate bending well in advance of collapse, so the strains demanded of the material also vary through the thickness and around the cross-section. While it is indeed possible to model such effects in a finite element analysis, assuming these effects can be collapsed into a generic increase in the yield stress for the entire sections as is currently done in design would seem to be an oversimplification. Implementation of the residual stresses and initial plastic strains into a commercial finite element program such as ABAQUS [16], where the member is modeled using shell elements, is relatively straightforward. The number of through-thickness section (integration) points must be increased to resolve the nonlinear through-thickness residual stress and strain distributions. The residual stresses and strains predicted herein can be relatively large. Further, conventional loading (e.g., compression, majoraxis bending) may cause loading or unloading of these initial stresses at a given point in the cross-section. As a result, the hardening rule: isotropic, kinematic, or mixed can have practical differences in the observed response even when the applied loads themselves are not reversing. For this situation, kinematic hardening, which approximates the Bauschinger effect, provides a more conservative model of the anticipated material behavior than isotropic hardening. However, to model kinematic hardening the location of the center of the yield surface in stress space (also known as the backstress) must be determined for each point in the cross-section at the end of the manufacturing process. This location is a function of the extent of yielding, in the example of Fig. 21, the backstress would be the Ds1, Ds2, Ds3 triad that results in the effective stress increasing from syield to sey. Unfortunately, the elastic–perfectly plastic assumption used to predict residual stresses herein does not directly allow for the calculation of the backstress. However, the effective plastic strain may be used to approximate the backstress as provided in Appendix C to this paper. Further examination of the predicted residual stress and strains and their impact on the peak strength and collapse response of cold-formed steel members in nonlinear finite element analysis is currently underway.
9. Conclusions A general mechanics-based prediction method for residual stresses and effective plastic strains in cold-formed
steel members is developed and presented. Throughthickness residual stress and strain distributions caused by coiling, uncoiling, and flattening of the steel sheet and cross-section cold-forming are considered. A cold-formed steel member is predicted to have both transverse and longitudinal residual stress distributions and the shapes of these distributions are nonlinear through the thickness. The stress and strain magnitudes are shown to increase with decreasing steel yield stress and increasing sheet thickness. The location in the coil from which a structural member originates is identified as a source of variability, and a prediction equation is derived to assist in the implementation of the prediction method in practice. The residual stress prediction method is evaluated against existing measurements from 18 roll-formed members. The measurement statistics demonstrate that the membrane residual stress component is small relative to the steel yield stress and that generally measurement variability is high. The radial coil locations for each member are estimated using the MSE of the predictions and measurements and are observed to be consistent with the physical dimensions of the coils. These estimated radial locations are used to evaluate the differences between the predictions and measurements in flats and corners. The prediction method is consistent with measurements in the flat cross-section elements but appears to moderately overpredict the longitudinal residual stresses at the corners. Additional work is needed to experimentally evaluate the relationship between coil radius and residual stresses. A study evaluating the influence of the proposed residual stress and plastic strain distributions in nonlinear finite element models when compared to more conventional model techniques is underway.
Acknowledgments The development of this residual stress prediction method would not have been possible without accurate information about the manufacturing process of sheet steel coils and cold-formed steel members. Thanks to Clark Western Building Systems, Mittal Steel USA, and the Cold-Formed Steel Engineers Institute (CFSEI) for their important contributions to this research, especially Bill Craig, Ken Curtis, Tom Lemler, Joe Wellinghoff, Ezio Defrancesco, Jean Frasier, Narayan Pottore, and Don Allen. The authors are also grateful for the support provided by the American Iron and Steel Institute (AISI) in this effort.
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Appendix A The following flowchart summarizing the prediction method for residual stresses in roll-formed members.
Start
Flat
Corner Flat or Corner?
End
No
t 2yield Yes, yields on coil t rx > 2yield No, remains elastic rx ≤
Yielding on the Coil?
No residual stresses!
c t ≤ y≤ 2 2 c c − < y< 2 2 t c − ≤ y≤− 2 2
+ yield
Yes
y E rx
zcoil = Sheet Coiling
− yield
Yielding on the Coil?
No
Yes c = 2rxεyield Sheet Coiling
xcoil = elastic zcoil elastic = 0.30 zuncoil =
− M xcoil y I
M xcoil =yield
Sheet Uncoiling
t 2
xuncoil = elastic zuncoil zflatten = − E
End
Sheet Flattening
rxuncoil =
2
1 2 1 − (rxyield ) I = ⋅ 1⋅ t 3 3 12
Sheet Uncoiling
elastic = 0.30
y rxuncoil
1 1 M xcoil − rx EI
Sheet Flattening
xflatten = elastic zflatten elastic = 0.30
xbend
=
− yield 0 ≤ y ≤
t 2
t + yield − ≤ y ≤ 0 2
zbend = plastic xbend
(For rz<8t)
plastic = 0.50
t
t t ≤ y≤ 2 2
zrebound = elastic xrebound
elastic = 0.30
xrebound =
Corner Bending
3yield y
−
Corner Rebound
End
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Appendix B The following flowchart summarizing the prediction method for effective plastic strains in roll-formed members.
Start
Flat
Corner Flat or Corner?
End
No
rx ≤ Yielding on the Coil? rx >
No equivalent plastic strains!
Yielding on the Coil?
t
No
No, remains elastic
2yield
Yes 2
pcoiling =
3 y
End
t 2yield Yes, yields on coil
Sheet Coiling zp =
rx y rx
Yes ln 1 + zp
− εyield
y≤−
− εyield
2 3
Sheet Coiling
2 c
c = 2rxyield
2
otherwise
0
bend = p
c
y≥
p ln 1 + x
xp = −
−
t 2
≤ y≤
t 2
Corner Bending
y rz End
Appendix C. Backstress for kinematic hardening implementation Implementation of a kinematic hardening rule requires that the center of the yield surface, in stress space, be known for any material which has been yielded prior to the loading of interest. The coordinates of the center of the yield surface (Ds1, Ds2, Ds3), known as the backstress, cannot be directly calculated from the stresses derived herein because work hardening was ignored in the residual stress derivations. However, the plastic strains developed in the manufacturing process provide a means by which the
backstress may be approximated, as provided in this appendix. The general equation for effective stress is defined as 1 se ¼ pffiffiffi 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs3 s1 Þ2 .
(C.1)
Given that the through-thickness sheet stresses are zero (s2 ¼ 0), Eq. (C.1) reduces to se ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s21 s1 s3 þ s23 .
(C.2)
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Consider the contribution to the backstress that develops due to coiling. From Eq. (18) we know the plastic strain, coiling . With coiling and knowing the material stress–strain p p relation (i.e., Fig. 21) the effective stress at that plastic strain, scoiling maybe determined. Consistent with the ey residual stress derivation of Eq. (8), we assume n ¼ 0.3 and scoiling ¼ nscoiling . 1 3
(C.3)
Finally, substituting the preceding into Eq. (C.2) results in scoiling ey ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi. p scoiling ¼ 3 n2 n þ 1
(C.4)
Similarly for cold bending the corners, from Eq. (21) we . With bend and knowing the know the plastic strain, bend p p material stress–strain relation (i.e., Fig. 21), we determine the effective stress at that plastic strain, sbend ey . Consistent with the residual stress derivation of Eq. (12), we assume n ¼ 0.5 and sbend ¼ nsbend , 3 1
(C.5)
and thus find sbend ey sbend ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 1 2 n nþ1 The backstress is then determined as:
(C.6)
Ds1 ¼ scoiling þ sbend syield , 1 1 Ds2 ¼ 0, Ds3 ¼ scoiling þ sbend syield . 3 3
(C.7)
where syield is the virgin yield stress of the steel. This estimate assumes that the changes in material properties from coiling, uncoiling, and flattening and cold-forming do not influence one another. References [1] US Steel. The making, shaping, and treating of steel. 10th ed. Pittsburgh, PA: Herbick & Held; 1985.
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[2] Schafer BW, Peko¨z T. Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses. J Constr Res 1997;47:193–210. [3] Shanley FR. Strength of materials. New York, NY: McGraw-Hill Book Company, Inc.; 1957. [4] Key PW, Hancock GJ. A theoretical investigation of the column behavior of cold-formed square hollow sections. Thin-Walled Struct 1993;16:31–64. [5] Quach WM, Teng JG, Chung KF. Finite element predictions of residual stresses in press-braked thin-walled steel sections. Eng Struct 2006;26:1609–19. [6] Quach WM, Teng JG, Chung KF. Residual stresses in steel sheets due to coiling and uncoiling: a closed-form analytical solution. Engineering Structures 2004;26:1249–59. [7] Yu WW. Cold-formed steel design. 3rd ed. New York: Wiley; 2000. [8] Hill R. The mathematical theory of plasticity. New York: Oxford University Press; 1983. [9] Dat DT. The strength of cold-formed steel columns. Cornell University Department of Structural Engineering report no. 80-4, Ithaca, NY, 1980. [10] Ingvarsson L. Cold-forming residual stresses, effect of buckling. In: Proceedings of the third international specialty conference on cold-formed steel structures, University of Missouri-Rolla; 1975. p. 85–119. [11] Weng CC, Peko¨z T. Residual stresses in cold-formed steel members. ASCE J Struct Eng 1990;116(6):1611–25. [12] de M. Batista E, Rodrigues FC. Residual stress measurements on cold-formed profiles. Exp Tech 1992;16(5):25–9. [13] Kwon YB. Post-buckling behaviour of thin-walled sections. Ph.D. thesis, University of Sydney, Australia, 1992. [14] Bernard ES. Flexural behaviour of cold-formed profiled steel decking. Ph.D. thesis, University of Sydney, Australia, 1993. [15] Chen WF, Han DJ. Plasticity for structural engineers. New York, NY: Springer; 1988. [16] ABAQUS. ABAQUS/Standard user’s manual version 6.5. ABAQUS, Inc., Providence, RI, /www.abaqus.comS; 2004. [17] CRC. Standard mathematical tables and formulae. New York, NY: CRC Press; 2003. [18] Moen C, Schafer BW. Direct strength design for cold-formed steel members with perforations, Progress report #4. Washington, DC: American Iron and Steel Institute; 2006. [19] Abdel-Rahman N, Sivakumaran KS. Material properties models for analysis of cold-formed steel members. ASCE J Struct Eng 1997; 123(9):1135–43.
Thin-Walled Structures 46 (2008) 1274–1289 www.elsevier.com/locate/tws
Prediction of residual stresses and strains in cold-formed steel members Cristopher D. Moen, Takeru Igusa, B.W. Schafer Department of Civil Engineering, Johns Hopkins University, Latrobe Hall 210, Baltimore, MD 21218, USA Received 22 October 2007; received in revised form 13 February 2008; accepted 13 February 2008 Available online 9 April 2008
Abstract The objective of this paper is to provide an unambiguous mechanics-based prediction method for determination of initial residual stresses and effective plastic strains in cold-formed steel members. The method is founded on basic physical assumptions regarding plastic deformations and common industry practice in manufacturing. Sheet steel coiling and cross-section roll-forming are the manufacturing processes considered. The structural mechanics employed in the method are defined for each manufacturing stage and the end result is a series of closed-form algebraic equations for the prediction of residual stresses and strains. Prediction validity is evaluated with measured residual strains from existing experiments, and good agreement is shown. The primary motivation for the development of this method is to define the initial state of a cold-formed steel member for use in a subsequent nonlinear finite element analysis. The work also has impact on our present understanding of cold-work of forming effects in cold-formed steel members. r 2008 Elsevier Ltd. All rights reserved. Keywords: Cold-formed steel; Residual stresses; Residual strains; Finite element analysis
1. Introduction Thin cold-formed steel members begin as thick, molten, hot steel slabs. Each slab is typically hot-rolled, coldreduced, and annealed before coiling and shipping the thin steel sheet to roll-forming producers [1]. Once at a plant, the sheet is unwound through a production line and plastically folded to form the final shape of a structural member, as shown in Fig. 1. This manufacturing process imparts residual stresses and plastic strains through the sheet thickness. These residual stresses and strains influence the load–displacement response and ultimate strength of cold-formed steel members. In previous work, a statistical approach was employed to draw conclusions on the magnitude and distribution of longitudinal residual stresses using a data set of surface strain measurements collected by researchers between 1975 and 1997 [2]. The measured surface strains are converted to residual stresses using Hooke’s Law and then distributed through the thickness as membrane (constant) and bending (linear variation) components. These residual stress disCorresponding author.
E-mail address: [email protected] (C.D. Moen). 0263-8231/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2008.02.002
tributions are a convenient way to express the measured residual surface strains, and are convenient as well for use in nonlinear finite element analyses, but they are not necessarily consistent with the underlying mechanics. Plastic bending, followed by elastic springback, creates a nonlinear through-thickness residual stress distribution, in the direction of bending, as shown in Fig. 2 [3]. The presence of nonlinear residual stress distributions in coldformed steel members has been confirmed in experiments [4] and in nonlinear finite element modeling of pressbraking steel sheets [5]. A closed-form analytical prediction method for residual stresses and equivalent plastic strains from coiling, uncoiling, and mechanical flattening of sheet steel is presented in [6]. The same plastic bending that creates these residual stresses also initiates the cold-work of forming effect, where plastic strains increase the apparent yield stress in the steel sheet (and ultimate strength in some cases) as discussed in Ref. [7]. Together, these residual stresses and plastic strains comprise the initial material state of a cold-formed steel member. A general method for predicting the manufacturing residual stresses and plastic strains in cold-formed steel members is proposed here. The procedure is founded on common industry manufacturing practices and basic
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Fig. 1. Cold-formed steel roll-forming: (left) sheet coil enters roll-forming line, (right) steel sheet is cold-formed into C-shape cross-section (photos courtesy of Bradbury Group).
Elastic springback σ
Plastic bending
Plastic bending
Elastic springback
ε
compression
tension plastic bending Elastic springback
Nonlinear residual stress distribution
Fig. 2. Forming a bend: plastic bending and elastic springback of thin sheets results in a nonlinear through-thickness residual stress distribution.
physical assumptions. The primary motivation for the development of this method is to define the initial state of a cold-formed steel member for use in a subsequent nonlinear finite element analysis. The derivation of the prediction method is provided for each manufacturing step, and the predictions are evaluated with measured residual strains from existing experiments. The end result of the method is intended to be accessible to a wide audience including manufacturers, design engineers, and the academic community. 2. Stress–strain coordinate system and notation The stress–strain coordinate system and geometric notation used in the forthcoming derivations are defined in Fig. 3. The x-axis is referred to as the transverse direction and the z-axis as the longitudinal direction of a structural member. Cross-section elements are referred to as either ‘corners’ or ‘flats’. The sign convention for stress and strain is positive for tension and negative for compression. 3. Prediction method assumptions The following assumptions are employed to develop this prediction method:
(a) Plane sections remain plane before and after coldforming of the sheet steel. This assumption permits the use of beam mechanics to derive prediction equations. (b) The sheet thickness t remains constant before and after cold-forming of the sheet steel. A constant sheet thickness is expected after cold-bending if the bending is performed without applied tension [8]. Cross-section measurements demonstrate modest sheet thinning at the corners, where t in the corners is typically 5% less than in the flange and web [9]. This thinning is ignored here to simplify the derivations, although a reduced thickness based on the plastic strain calculations in Section 5 could be used if a higher level of accuracy is required. (c) The sheet neutral axis remains constant before and after cross-section cold-forming. Theoretical models used in metal forming theory do predict a small shift in the through-thickness neutral axis towards the inside of the corner as the sheet plastifies [8]. This shift is calculated as 6% of the sheet thickness, t, when assuming a centerline corner radius, rz, of 2.5t. A neutral axis shift of similar magnitude has been observed in the nonlinear finite element model results for thin press-braked steel sheets [5]. This small shift is ignored here to simplify the derivations.
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roller dies
rx
A
A
Forming direction
sheet steel coil Elevation view
x
t y
z
rz
y x z Section A-A
Fig. 3. Stress–strain coordinate system as related to the coiling and cold-forming processes.
(d) The steel stress–strain curve is assumed as elastic– perfectly plastic when calculating residual stresses. More detailed stress–strain models that include hardening are obviously possible, but a basic model is chosen to simplify the derivations. The implication of this assumption is that the residual stresses may be underestimated, especially in corner regions where the sheet has yielded completely through the thickness. (e) Plane strain behavior is assumed to exist during coiling, uncoiling, and flattening (ex ¼ 0) and during crosssection cold-forming (ez ¼ 0). (f) The steel sheet is fed from the top of the coil into the roll-forming bed as shown in Fig. 4a. This assumption is consistent with measured bending residual stress data (see Section 7) and manufacturing setups suggested by roll-forming equipment suppliers (Fig. 1). The first author did observe the alternative setup in Fig. 4b (sheet steel unrolling from the bottom of the coil) at a roll-forming plant, suggesting that the direction of uncoiling is a source of variability in measured residual stress data. (g) Membrane residual stresses are zero. Membrane residual stresses have been measured by several researchers (Refs. [4,9–14]), although the magnitudes are small relative to bending residual stresses (see Table 1). Membrane residual stresses are experimentally determined by averaging the measured surface strains on the two faces of a thin steel sheet. Given the variability inherent in these measurements it is difficult to know if the resulting membrane stresses (strains) are real or simply unavoidable experimental error.
4. Derivation of the residual stress prediction method The prediction method proposed here assumes that two manufacturing processes contribute to the through-thickness residual stresses in cold-formed steel members: (1) sheet coiling, uncoiling, and flattening, and (2) crosssection roll-forming. Algebraic equations for predicting the through-thickness residual stress and effective plastic strains in corners and flats are derived here and then summarized in flowcharts in Appendices A and B. 4.1. Residual stresses from sheet coiling, uncoiling, and flattening Coiling the sheet steel after annealing and galvanizing, but prior to shipment, may yield the steel if the virgin yield strain, eyield, is exceeded. If plastic deformation does occur, a residual curvature will exist in the sheet as it is uncoiled. This residual curvature is locked into a structural member resulting in longitudinal residual stresses as the sheet is flattened by the roll-formers. This process of coiling, uncoiling with residual curvature, and flattening is described in Fig. 5. 4.1.1. Coiling The through-thickness strain induced from coiling is related to the radial location of the sheet in the coil rx, with the well-known relationship from beam mechanics: z 1 ¼ . y rx
(1)
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Roll-forming bed
Sheet has residual CONCAVE curvature coming off the coil
Sheet has residual CONVEX curvature coming off the coil Fig. 4. Roll-forming setup with sheet coil fed from the (a) top of the coil and (b) bottom of coil. The orientation of the coil with reference to the rollforming bed influences the direction of the coiling residual stresses.
Table 1 Statistics of the residual stresses in roll-formed members Element
No. of samples
Residual stress as % syield Membrane
Corners Flats
Bending
Mean
S.D.
Mean
S.D.
5.7 1.8
10.1 10.7
32.0 25.2
23.8 20.7
resulting in a change in the through-thickness stress. This stress distribution is determined by first calculating the plastic coiling moment 2 t 1 2 coil ðrx yield Þ , (4) M x ¼ syield 2 3 and then applying an opposing moment elastically to simulate the removal of the imposed radial displacement
23 120
suncoil ¼ z ez is the engineering strain through the thickness y in the coiling (longitudinal) direction z. y varies from t/2 to t/2, where t is the sheet thickness. The radius associated with the elastic–plastic threshold initiating through-thickness yielding from coiling, rep, is derived by substituting ez ¼ eyield and y ¼ t/2 (outer fiber strain) into Eq. (2): rep ¼
t 2yield
.
(2)
When the coil radius rx is greater than rep the sheet steel experiences only elastic deformation on the coil. For sheet steel rolled to a coil radius rx less than rep, through-thickness yielding will occur as shown in Fig. 6. When rxorep, the depth of the elastic core c is defined as: c ¼ 2rx yield pt.
(3)
4.1.2. Uncoiling As the yielded sheet is uncoiled in preparation for the roll-forming line, the sheet steel springs back elastically
12M coil x y . 3 t
(5)
4.1.3. Flattening After the sheet has been unrolled, a permanent radius of curvature will still exist if rx was less than rep on the coil. This permanent radius is runcoil ¼ x
1 . ð1=rx Þ ðM coil x =EIÞ
(6)
Steel sheet with permanent curvature from coiling is pressed flat as the sheet enters the roll-forming line. The longitudinal stresses resulting from flattening the sheet are simply y sflatten ¼ E uncoil . (7) z rx 4.1.4. Residual stress distribution The total through-thickness longitudinal residual stress distribution due to coiling, uncoiling, and flattening is presented in Fig. 7. The resulting residual stress, sz, is selfequilibrating for axial force through the thickness but causes a residual longitudinal moment. Section 7 compares
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DETAIL A
Coiling
Uncoiled with residual curvature Flattened as sheet enters the roll-formers
Change in curvature locks in bending residual stresses in final member
Fig. 5. Coiling of the steel sheet may result in residual curvature which results in bending residual stresses as the sheet is flattened.
y +σyield
c
z
-σyield Fig. 6. Longitudinal residual stress distribution from coiling.
the stresses caused by this moment with surface strains (stresses) measured in experiments. The longitudinal residual stresses also will create transverse stresses across the width of the coil, assuming plane strain conditions for an infinitely wide sheet. Supporting the plane strain assumption is the observation that while the actual width of the sheet is finite, it remains several orders of magnitude greater than the sheet thickness. Under this assumption, and further assuming only elastic stresses, the transverse stresses are: uncoil sx ¼ n scoil . (8) þ sflatten z þ sz z Poisson’s ratio, n, is assumed here as 0.30 for steel deformed elastically. The through-thickness deformation from the uncoiling and flattening components will occur elastically, and the coiling component will be at least partially elastic through the thickness for the range of sheet thicknesses common in industry.
residual stresses are assumed to exist only at the location of the formed corners, between the roller die reactions, as shown in Fig. 8. Some yielding is expected to occur outside of the roller reactions as the stress distribution transitions from fully plastic to fully elastic; however, this transition area is not considered here to simplify the derivation. The engineering strain in the steel sheet, ex, and the bend radius, rz, are related for both small and large deformations with the strain–curvature relationship 1 x ¼ . rz y
This geometric relationship is valid for elastic and plastic bending of the steel sheet. For the small bend radii common in the cold-formed steel industry (rz ¼ 2t to 8t), the steel sheet yields through its thickness during the coldforming process. The steel sheet will reach the fully plastic stress state shown in Fig. 9 as the corner approaches its final manufactured radius. After the sheet becomes fully plastic through its thickness, the engineering strain continues to increase as the radius decreases. When the final bend radius is reached and the imposed radial displacement is removed, an elastic springback occurs that elastically unloads the corner (see Fig. 2). The change in stress through the thickness from this elastic rebound is derived with the plastic moment force couple shown in Fig. 10. The plastic moment is calculated with the equation t syield 1 t t syield t2 M bend ¼ , (10) ¼ FP ¼ z 2 2 2 4 which is then applied elastically through the thickness to simulate the stress distribution from elastic rebound of the sheet steel:
4.2. Residual stresses from cross-section roll-forming srebound ¼ x A set of algebraic equations is derived here to predict the transverse and longitudinal residual stresses created by rollforming a cross-section. Roll-forming residual stresses are cumulative with the coiling residual stresses derived in Section 4.1 and provide a complete prediction of the initial stress state of the member cross-section. The roll-forming
(9)
ðsyield t2 =4Þy 3syield y M bend y z . ¼ ¼ ð1=12Þ 1 t3 t I
(11)
The final transverse stress state is the summation of the fully plastic stress distribution through the thickness and the unloading stress from the elastic springback of the corner as shown in Fig. 11, where sx is the transverse residual stress through the thickness from the
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y
y
c
y
zuncoil
+σyield
z
+
1279
y
flatten z
+
z
z
= z z
-σyield Coil
Uncoil
Flatten
Residual Stress
Fig. 7. Predicted longitudinal residual stress distribution from coiling, uncoiling, and flattening of a steel sheet.
Assume bend is fully plastic between roller dies
Roller Die (Typ.)
Fig. 8. Cold-forming of a steel sheet.
y
y
-σyield
+σyield Fp t/2
x
+σyield
Fp
x
-σyield
Fig. 9. Fully plastic transverse stress state from cold-forming.
Fig. 10. Force couple applied to simulate the elastic springback of the steel sheet after the imposed radial deformation is removed.
cold-forming of the corner. This stress is nonlinear through the thickness and is self-equilibrating, meaning that axial and bending sectional forces are absent in the x-direction after forming. The transverse residual stresses will create stress in the longitudinal direction due to the assumed plane strain conditions (see Section 3):
The Poisson’s ratio, n, is assumed as 0.30 for steel deformed elastically and 0.50 for fully plastic deformation. The longitudinal residual stresses through the thickness, sz, are determined based on these assumptions as shown in Fig. 12. Longitudinal residual stress, sz, is selfequilibrating for axial force through the thickness but causes a residual longitudinal moment. This moment is hypothesized to contribute to the observed longitudinal residual strains measured in experiments (refer to
sz ¼ nsx .
(12)
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1280
y -σyield
+1.5σyield
+0.5σyield
-σyield
-1.5σyield
+σyield
-0.5σyield Transverse Residual Stress x
Elastic Springback xrebound
Plastic Bending xbend
x
+σyield
Fig. 11. Cold-forming of a steel sheet occurs as plastic bending and elastic springback, resulting in a self-equilibrating transverse residual stress.
-σyield
-0.05σyield
+1.5σyield
0.50
-0.50σyield
0.30
-1.5σyield
x
+0.50σyield z
+σyield Plastic Bending plastic bend
y
Elastic Springback elasticxrebound
+0.05σyield Longitudinal Residual Stress z
Fig. 12. Plastic bending and elastic springback from cold-forming in the transverse direction result in longitudinal residual stresses because of the plane strain conditions.
Section 7for a comparison of this prediction to actual measurements). A flowchart summarizing the proposed prediction method for residual stresses in roll-formed members is provided in Appendix A. Appendix A explicitly demonstrates how coiling, uncoiling, flattening, and roll-forming contribute to the residual stresses locked into the crosssection during manufacturing.
5. Derivation of effective plastic strain prediction method In the method proposed here, plastic strains occur from sheet coiling and cold-forming, and together with residual stresses describe the initial material state of the member. The general state of plastic strain at a point can be quantified by using the von Mises yield criterion extended to plastic deformations (see Ref. [15]) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi p ¼ 23 21 þ 22 þ 23 , (13) where ep is the effective plastic strain, and e1, e2, and e3 are the principal strains. All of the strains are ‘true’ strains, which may be calculated from the engineering
y zp
c
z
Fig. 13. Plastic strain distribution from sheet coiling with a radius less than elastic–plastic threshold rep.
strains via: 1 ¼ lnð1 þ x Þ;
2 ¼ lnð1 þ y Þ;
3 ¼ lnð1 þ z Þ,
(14)
where ex, ey, ez are in the Cartesian coordinate system (Fig. 3) and x, y, z are coincident with the principal directions. True strains are employed instead of engineering strains to accommodate the large deformations from plastic bending. Also, from a practical standpoint, nonlinear FE codes such as ABAQUS [16] require the engineer
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to provide true stress, true strain information (as large deformation theory is employed). The steel sheet is assumed to remain incompressible while experiencing plastic deformations, therefore when calculating ep 1 þ 2 þ 3 ¼ 0.
1281
y
(15)
5.1. Effective plastic strain from sheet coiling Engineering plastic strains, as shown in Fig. 13, accumulate during the coiling of sheet steel if the coiling radius rx is less than the elastic–plastic threshold rep. The engineering plastic strain distribution from coiling is: y c pz ¼ yield ; yX , rx 2 y c pz ¼ yield ; yp , rx 2 pz ¼ 0;
otherwise,
(16)
where the elastic core, c, is defined in Eq. (3). Plane strain conditions result in e1 ¼ 0, and e2 ¼ e3 via the incompressibility assumption of Eq. (15). Further, the Cartesian coordinate system is coincident with the principal axes, resulting in the following true principal plastic strains: 1 ¼ 0;
2 ¼ lnð1 þ pz Þ;
3 ¼ lnð1 þ pz Þ.
(17)
Substituting the principal strains into Eq. (13) and simplifying leads to the through-thickness effective plastic strain from coiling 2 (18) coiling ¼ pffiffiffi ln 1 þ pz . p 3 This plastic strain distribution, depicted in Fig. 14, will exist at all locations in the cross-section (corners and flats) when rx is less than the elastic–plastic threshold rep. The plastic strain from coiling, coiling , will generally be p much smaller in magnitude than the plastic strain from cross-section cold-forming, bend , as discussed in following p section.
Fig. 15. Effective von Mises true plastic strain at the location of coldforming of a steel sheet.
bent. The engineering plastic strain distribution from coldforming is described via y (19) px ¼ , rz which assumes that the elastic core at the center of the sheet is infinitesimally small. This assumption is consistent with the small bend radii common in industry (see Section 4.2). Plane strain conditions and Eq. (15) result in e3 ¼ 0, e2 ¼ e1. Physically these conditions imply that the sheet will experience some thinning at the location of coldforming (see Section 3), but the tendency to plastically shorten longitudinally will be resisted by the adjacent undeformed portion of the cross-section. As before, the Cartesian coordinate system is coincident with the principal axes, resulting in the following plastic principal strains: 1 ¼ lnð1 þ px Þ;
2 ¼ lnð1 þ px Þ;
3 ¼ 0.
(20)
Substituting for the principal strains and simplifying, the effective plastic strain at a cold-formed corner is: 2 bend ¼ pffiffiffi ln 1 þ px . p 3
(21)
5.2. Effective plastic strain from cross-section cold-forming
This effective plastic strain distribution is shown in Fig. 15. The distribution exists only at the cold-bent locations in a cross-section and should be added to the coiling plastic strain distribution in Fig. 14. A flowchart summarizing the prediction method for effective plastic strains in rollformed members is provided in Appendix B.
Large transverse plastic strains occur through the thickness of a thin steel sheet when the sheet is permanently
6. Employing the prediction method in practice: quantifying the coil radius influence
coiling p y
c
Fig. 14. Effective plastic strain in a cold-formed steel member from sheet coiling when the radius rx is less than the elastic–plastic threshold rep.
The residual stress and plastic strain distributions derived for cross-section cold-forming (Sections 4.2 and 5.2) are straight-forward to calculate if the yield stress, syield, and thickness, t, of the sheet steel are known. The coiling residual stresses and plastic strains are more difficult to calculate because the coil radius coinciding with the as-formed member, i.e., the radial location of the sheet, rx, is almost always unknown in practice. However, rx can be derived in an average sense though, since the range of inner and outer coil radii are known and the probability that a structural member will be manufactured from a certain rx can be quantified.
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The relationship between coil radius, rx, and corresponding linear location S of the sheet within the coil can be described using Archimedes spiral [17] p S ¼ r2x r2inner . (22) t The spiral maintains a constant pitch with varying radii, where the pitch is the thickness of the steel sheet, t, as shown in Fig. 16, L is the total length of sheet in the coil,
and rinner and router are the inside and outside coil radii, respectively. As-shipped outer coil radii range from 610 to 915 mm and inner coil radii range from 255 to 305 mm. These ranges were determined by the first author during a visit to a local roll-forming plant. Archimedes spiral is used to describe the probability that the steel sheet will come from a certain range of radial locations in the coil. The cumulative distribution function (CDF), FR(rx) ¼ probability that the radius is less than rx, is obtained by normalizing S by L S r2 r2inner ¼ F R ðrx Þ ¼ 2 x . L router r2inner
S
(23)
The probability density function (PDF) of rx is calculated by taking the derivative of FR(rx) rx t
f R ðrx Þ ¼
End S=L
Start S=0
dF R ðrx Þ 2rx ¼ 2 . drx router r2inner
(24)
The mean value of the radial location for a given inner and outer coil radii is Z
router
r¯x ¼ rinner
2 rinner router rinner þ router f R ðrx Þrx drx ¼ . 3 rinner þ router
Fig. 16. Coil coordinate system and notation.
(25)
Longitudinal Residual Stresses σyield = 210 MPa t/2
flat
corner
σyield = 345 MPa flat
corner
σyield = 550 MPa flat
corner y
t = 0.9 mm
-t/2 -σyield 0
σyield
t/2
z y
t = 1.8 mm
-t/2 -σyield 0
z
σyield
t/2
y
t = 2.6 mm -t/2 -σyield 0
σyield
z
Positive stress is tension, negative stress is compression Fig. 17. Influence of sheet thickness and yield stress on through-thickness longitudinal residual stresses (z-direction, solid lines are predictions for mean coil radius, dashed lines for mean radius7one standard deviation).
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7.1. Measurement statistics
The variance of the radial location is s2R ¼ ¼
Z
router
f R ðrx Þðrx r¯x Þ2 dr
rinner
ðrouter rinner Þ2 1 2 router þ 4router rinner þ r2inner . 18 ðrouter þ rinner Þ2
1283
(26)
These statistics for rx can then be used with the prediction method for coiling, uncoiling, and flattening residual stresses and plastic strains described in Sections 4 and 5. Fig. 17 summarizes the influence of sheet thickness and virgin yield stress on the longitudinal residual stress distributions in flats and corners. (The method proposed in this paper provides residual stresses and strains for the entire member, only the longitudinal residual stresses are shown in Fig. 17.) The solid lines in Fig. 17 are calculated using the mean value, r¯x ¼ 474 mm, from Eq. (25) assuming rinner ¼ 305 mm and router ¼ 610 mm. The distributions with the dashed lines are calculated with r¯x sR , where sR ¼ 86 mm is calculated with Eq. (26). The residual stresses are nonlinear through the thickness and have different shapes for flats and corners. The stress magnitudes at the outer fibers increase for thicker sheets and lower yield stresses. The accuracy of the linear bending residual stress model commonly employed in finite element analyses is perhaps sufficient when yield stress is low and thickness is high (relatively), but for typical thicknesses (0.9–1.8 mm) and yield stress (345 MPa) the assumption of a linear longitudinal stress distribution is not consistent with the mechanics-based predictions in Fig. 17.
The mean and standard deviation of the residual stresses for the 18 roll-formed specimens used in this comparison are provided in Table 1. Positive membrane stresses are tensile stresses and positive bending stresses cause tension at y ¼ t/2 (see Fig. 3 for coordinate system). The statistics demonstrate that both membrane and bending residual stress measurements are highly variable and that the membrane stresses are small relative to the steel yield stresses. Details on the residual stress measurements for each of the 18 specimens can be found in Ref. [18]. 7.2. Mean-squared error (MSE) estimate of radial location To explore the validity of the prediction method, the flat and corner residual stress measurements from the 18 specimens are used to estimate the radial location rx from which each specimen originated. These estimated radial locations are then used to calculate the difference between the predicted and measured longitudinal residual stresses. 7.2.1. MSE minimization The location of the specimen in the coil, rx, is estimated by minimizing the sum of the MSE for the p ¼ 1, 2, y, nq measurements taken around the cross-section of the q ¼ 1, 2, y, 18 specimens !2 nq X smeasured spredicted pq pq . (27) r^x;q ¼ arg min rx syield;pq p¼1 Both corner and flat measurements are included in the minimization.
7. Comparison of prediction method to measured residual stresses 8 7 6 5 MSE
The flat and corner residual surface strain measurements from 18 roll-formed specimens are used to evaluate the proposed residual stress prediction method. The prediction method provides the complete throughthickness longitudinal strain (stress) distribution if the radial location in the coil from which the specimen originated in the coil, rx, is known. Since the radial coil location of the 18 specimens is unknown, rx is statistically estimated for each specimen using the coil radius that best fits the predicted surface strains to the measured surface strains from a specimen cross-section (for both corners and flats). Once the best-fit radial locations have been calculated, they are examined to determine if their magnitude is rational when compared to typical inner and outer dimensions of a sheet coil. Although this comparison only provides a partial evaluation of the prediction method, it is as far as one can go with the available data. Qualitatively the prediction method is consistent with the more detailed through-thickness findings in Refs. [4,5].
4 3 2 1 0 1
1.5
2
2.5
3
3.5
rx/rinner Fig. 18. The mean-squared error of the predicted and measured bending residual stresses for de M. Batista and Rodrigues [12], Specimen CP1 is minimized when rx ¼ 1.60rinner.
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7.2.2. Bending component of longitudinal residual stress distribution The bending component of the predicted residual stress distribution must be isolated to compare with the measured values. The total predicted longitudinal residual stress distribution in the flats and corners of each cross-section is integrated to calculate the sectional moment through the thickness Z t=2 Mx ¼ sz y dy. (28)
7.2.3. Estimated coil radii using MSE Fig. 18 demonstrates the MSE results for de M. Batista and Rodrigues [12] Specimen CP1. The radial location that minimizes the prediction error is 1.60rinner in this case, and is summarized in Table 2 for all 18 roll-formed specimens considered. The estimated radial locations fall within the range of inner and outer coil radii assumed in the prediction (rinner to 2.40rinner) except for Dat RFC13 which is slightly outside the range at 2.45rinner. The MSE radial location cannot be determined in the three Bernard specimens [14] since the bending residual stresses in the flats are predicted to be zero. These three specimens are cold-formed steel decking with a thin sheet thickness t ranging from 0.75 to 1 mm and a relatively high yield stress syield ranging from 600 to 650 MPa. In this case, the coiling and uncoiling of the steel sheet will occur elastically as demonstrated in Fig. 17. Measured bending residual stress magnitudes in the flats of the Bernard specimens are on average 0.03syield which is consistent with the prediction method.
ðt=2Þ
Mx is then converted into a predicted outer fiber bending residual stress which can then be directly compared to the measurements ¼ spredicted pq
M x ðt=2Þ . I
(29)
Table 2 Radial location in the coil that minimizes the sum of the mean square prediction error for roll-formed members Researcher
Specimen
rx/rinner
de M. Batista and Rodrigues [12] de M. Batista and Rodrigues [12] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Weng and Peko¨z [11] Dat [9] Dat [9] Bernard [14] Bernard [14] Bernard [14] Abdel-Rahman and Siva [19] Abdel-Rahman and Siva [19] Abdel-Rahman and Siva [19]
CP2 CP1 RFC13 RFC14 R13 R14 P3300 P4100 DC-12 DC-14 RFC14 RFC13 Bondek 1 Bondek 2 Condeck HP Type A—Spec 1 Type A—Spec 2 Type B—Spec 1
1.20 1.60 1.80 1.10 1.45 1.30 1.95 1.50 2.30 1.60 2.00 2.45 N/A N/A N/A 1.55 1.55 1.25
7.3. Statistical variations between measurements and predictions The predicted radial locations in Table 2 are now used to calculate the statistical variations between the experiments and predictions. The bending residual stresses in the 18 roll-formed members are calculated using the MSEpredicted radial location rx in Eq. (27) with the residual stress prediction method summarized in Section 4. The bending component of the residual stress prediction is then obtained with Eq. (29). The difference between the predicted and measured residual bending stresses, epq, for the p ¼ 1, 2, y, n measurements taken around the cross-section of the q ¼ 1, 2, y, 18 specimens is calculated as
epq ¼
rinner ¼ 254 mm, router ¼ 610 mm ¼ 2.40 rinner. N/A coiling residual stresses are predicted as zero.
40 e = 0.03yield e = 0.15yield
30
σmeasured/σyield
Observations
35 25 20 15 10 5 0
-1
-0.5
0
0.5
(σmeasured-σpredicted)/σyield
1
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2
smeasured spredicted pq pq syield;pq
0
.
(30)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 σpredicted/σyield
Fig. 19. (a) Histogram and (b) scattergram of bending residual stress prediction error (flat cross-sectional elements) for 18 roll-formed specimens.
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40 30
e = −0.16σyield
25
e = 0.19yield
σ measured /σyield
Observations
35
20 15 10 5 0 -1
-0.5 0 0.5 (σ measured-σ predicted)/σyield
1
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2
0
1285
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 σpredicted/σyield
Fig. 20. (a) Histogram and (b) scattergram of bending residual stress prediction error (corner cross-sectional elements) for 18 roll-formed specimens.
The error histogram for the flat cross-sectional elements in Fig. 19a demonstrates that the mean difference me is near zero with a standard deviation se ¼ 0.15syield. The scattergram in Fig. 19b demonstrates the strength of the correlation between the measurements and predictions in the flats; the solid regression line passes nearly through zero (y-intercept ¼ 0.05syield) and has nearly a unit slope (m ¼ 0.92). Also, the majority of the data lies within 7one standard deviation of the estimate, denoted as the dashed lines in the figure. It is concluded that the prediction method is consistent with the measured data in the flats. The corner element error histogram in Fig. 20a shows a negative bias of me ¼ 0.16syield meaning that the predicted residual stresses are generally higher than the measured values. The standard deviation of the error is large (se ¼ 0.19syield) but is less than the standard deviation of the corner residual stress measurements in Table 1 (sm ¼ 0.24syield). This demonstrates a greater match between the measurements and predictions, although more corner residual stress measurements are needed to improve the strength of this comparison. The scattergram in Fig. 20b highlights the variability in the measured corner data, especially in the region corresponding to spredicted ¼ 0.4syield , where bending residual stresses (strains) vary from 0 to 0.7syield. 8. Discussion The residual stresses and strains predicted with this method (Section 4 for stress, Section 5 for strain) form the initial material state in the cross-section. In design, this initial material state is sometimes considered through the so-called cold-work of forming effect, where the yield stress of the material is increased above the virgin yield stress, syield, to account for the ‘working of the corners’. For onedimensional stress–strain this concept is expressed as shown in Fig. 21, where ‘working the corners’ results in a residual plastic strain, ep, such that when the section is reloaded the stress at which yielding re-initiates, sey, is
ey yield
Virgin steel
Coldformed steel ers
Apparent yield stress
Effective residual stress yield + p
Fig. 21. Definition of apparent yield stress, effective residual stress, and effective plastic strain as related to a uniaxial tensile coupon test.
greater than the virgin yield stress, syield. If no residual stresses existed the apparent increase in the yield stress from syield to sey can be significant. However, as Fig. 16 illustrates, ‘working the corners’ also contributes to residual stresses, srs e , and these residual stresses may decrease the apparent yield stress. The prediction method presented herein provides a more nuanced understanding of the cold-work of forming effects. The residual plastic strains may increase the apparent yield stress, but those strains vary through the thickness and have contributions from both transverse and longitudinal strains. Further, residual stresses follow their own relatively complicated distribution through the thickness. In a multi-axial stress state using the von Mises yield criterion, Fig. 21 is enforced for the effective
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stress–effective strain pair for every point in the crosssection. As a result, the apparent yield stress upon loading varies through the thickness and is influenced by both the residual stresses and strains. Even under simple loading conditions (e.g., compression) a cold-formed member undergoes plate bending well in advance of collapse, so the strains demanded of the material also vary through the thickness and around the cross-section. While it is indeed possible to model such effects in a finite element analysis, assuming these effects can be collapsed into a generic increase in the yield stress for the entire sections as is currently done in design would seem to be an oversimplification. Implementation of the residual stresses and initial plastic strains into a commercial finite element program such as ABAQUS [16], where the member is modeled using shell elements, is relatively straightforward. The number of through-thickness section (integration) points must be increased to resolve the nonlinear through-thickness residual stress and strain distributions. The residual stresses and strains predicted herein can be relatively large. Further, conventional loading (e.g., compression, majoraxis bending) may cause loading or unloading of these initial stresses at a given point in the cross-section. As a result, the hardening rule: isotropic, kinematic, or mixed can have practical differences in the observed response even when the applied loads themselves are not reversing. For this situation, kinematic hardening, which approximates the Bauschinger effect, provides a more conservative model of the anticipated material behavior than isotropic hardening. However, to model kinematic hardening the location of the center of the yield surface in stress space (also known as the backstress) must be determined for each point in the cross-section at the end of the manufacturing process. This location is a function of the extent of yielding, in the example of Fig. 21, the backstress would be the Ds1, Ds2, Ds3 triad that results in the effective stress increasing from syield to sey. Unfortunately, the elastic–perfectly plastic assumption used to predict residual stresses herein does not directly allow for the calculation of the backstress. However, the effective plastic strain may be used to approximate the backstress as provided in Appendix C to this paper. Further examination of the predicted residual stress and strains and their impact on the peak strength and collapse response of cold-formed steel members in nonlinear finite element analysis is currently underway.
9. Conclusions A general mechanics-based prediction method for residual stresses and effective plastic strains in cold-formed
steel members is developed and presented. Throughthickness residual stress and strain distributions caused by coiling, uncoiling, and flattening of the steel sheet and cross-section cold-forming are considered. A cold-formed steel member is predicted to have both transverse and longitudinal residual stress distributions and the shapes of these distributions are nonlinear through the thickness. The stress and strain magnitudes are shown to increase with decreasing steel yield stress and increasing sheet thickness. The location in the coil from which a structural member originates is identified as a source of variability, and a prediction equation is derived to assist in the implementation of the prediction method in practice. The residual stress prediction method is evaluated against existing measurements from 18 roll-formed members. The measurement statistics demonstrate that the membrane residual stress component is small relative to the steel yield stress and that generally measurement variability is high. The radial coil locations for each member are estimated using the MSE of the predictions and measurements and are observed to be consistent with the physical dimensions of the coils. These estimated radial locations are used to evaluate the differences between the predictions and measurements in flats and corners. The prediction method is consistent with measurements in the flat cross-section elements but appears to moderately overpredict the longitudinal residual stresses at the corners. Additional work is needed to experimentally evaluate the relationship between coil radius and residual stresses. A study evaluating the influence of the proposed residual stress and plastic strain distributions in nonlinear finite element models when compared to more conventional model techniques is underway.
Acknowledgments The development of this residual stress prediction method would not have been possible without accurate information about the manufacturing process of sheet steel coils and cold-formed steel members. Thanks to Clark Western Building Systems, Mittal Steel USA, and the Cold-Formed Steel Engineers Institute (CFSEI) for their important contributions to this research, especially Bill Craig, Ken Curtis, Tom Lemler, Joe Wellinghoff, Ezio Defrancesco, Jean Frasier, Narayan Pottore, and Don Allen. The authors are also grateful for the support provided by the American Iron and Steel Institute (AISI) in this effort.
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Appendix A The following flowchart summarizing the prediction method for residual stresses in roll-formed members.
Start
Flat
Corner Flat or Corner?
End
No
t 2yield Yes, yields on coil t rx > 2yield No, remains elastic rx ≤
Yielding on the Coil?
No residual stresses!
c t ≤ y≤ 2 2 c c − < y< 2 2 t c − ≤ y≤− 2 2
+ yield
Yes
y E rx
zcoil = Sheet Coiling
− yield
Yielding on the Coil?
No
Yes c = 2rxεyield Sheet Coiling
xcoil = elastic zcoil elastic = 0.30 zuncoil =
− M xcoil y I
M xcoil =yield
Sheet Uncoiling
t 2
xuncoil = elastic zuncoil zflatten = − E
End
Sheet Flattening
rxuncoil =
2
1 2 1 − (rxyield ) I = ⋅ 1⋅ t 3 3 12
Sheet Uncoiling
elastic = 0.30
y rxuncoil
1 1 M xcoil − rx EI
Sheet Flattening
xflatten = elastic zflatten elastic = 0.30
xbend
=
− yield 0 ≤ y ≤
t 2
t + yield − ≤ y ≤ 0 2
zbend = plastic xbend
(For rz<8t)
plastic = 0.50
t
t t ≤ y≤ 2 2
zrebound = elastic xrebound
elastic = 0.30
xrebound =
Corner Bending
3yield y
−
Corner Rebound
End
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Appendix B The following flowchart summarizing the prediction method for effective plastic strains in roll-formed members.
Start
Flat
Corner Flat or Corner?
End
No
rx ≤ Yielding on the Coil? rx >
No equivalent plastic strains!
Yielding on the Coil?
t
No
No, remains elastic
2yield
Yes 2
pcoiling =
3 y
End
t 2yield Yes, yields on coil
Sheet Coiling zp =
rx y rx
Yes ln 1 + zp
− εyield
y≤−
− εyield
2 3
Sheet Coiling
2 c
c = 2rxyield
2
otherwise
0
bend = p
c
y≥
p ln 1 + x
xp = −
−
t 2
≤ y≤
t 2
Corner Bending
y rz End
Appendix C. Backstress for kinematic hardening implementation Implementation of a kinematic hardening rule requires that the center of the yield surface, in stress space, be known for any material which has been yielded prior to the loading of interest. The coordinates of the center of the yield surface (Ds1, Ds2, Ds3), known as the backstress, cannot be directly calculated from the stresses derived herein because work hardening was ignored in the residual stress derivations. However, the plastic strains developed in the manufacturing process provide a means by which the
backstress may be approximated, as provided in this appendix. The general equation for effective stress is defined as 1 se ¼ pffiffiffi 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðs1 s2 Þ2 þ ðs2 s3 Þ2 þ ðs3 s1 Þ2 .
(C.1)
Given that the through-thickness sheet stresses are zero (s2 ¼ 0), Eq. (C.1) reduces to se ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s21 s1 s3 þ s23 .
(C.2)
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Consider the contribution to the backstress that develops due to coiling. From Eq. (18) we know the plastic strain, coiling . With coiling and knowing the material stress–strain p p relation (i.e., Fig. 21) the effective stress at that plastic strain, scoiling maybe determined. Consistent with the ey residual stress derivation of Eq. (8), we assume n ¼ 0.3 and scoiling ¼ nscoiling . 1 3
(C.3)
Finally, substituting the preceding into Eq. (C.2) results in scoiling ey ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi. p scoiling ¼ 3 n2 n þ 1
(C.4)
Similarly for cold bending the corners, from Eq. (21) we . With bend and knowing the know the plastic strain, bend p p material stress–strain relation (i.e., Fig. 21), we determine the effective stress at that plastic strain, sbend ey . Consistent with the residual stress derivation of Eq. (12), we assume n ¼ 0.5 and sbend ¼ nsbend , 3 1
(C.5)
and thus find sbend ey sbend ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . 1 2 n nþ1 The backstress is then determined as:
(C.6)
Ds1 ¼ scoiling þ sbend syield , 1 1 Ds2 ¼ 0, Ds3 ¼ scoiling þ sbend syield . 3 3
(C.7)
where syield is the virgin yield stress of the steel. This estimate assumes that the changes in material properties from coiling, uncoiling, and flattening and cold-forming do not influence one another. References [1] US Steel. The making, shaping, and treating of steel. 10th ed. Pittsburgh, PA: Herbick & Held; 1985.
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[2] Schafer BW, Peko¨z T. Computational modeling of cold-formed steel: characterizing geometric imperfections and residual stresses. J Constr Res 1997;47:193–210. [3] Shanley FR. Strength of materials. New York, NY: McGraw-Hill Book Company, Inc.; 1957. [4] Key PW, Hancock GJ. A theoretical investigation of the column behavior of cold-formed square hollow sections. Thin-Walled Struct 1993;16:31–64. [5] Quach WM, Teng JG, Chung KF. Finite element predictions of residual stresses in press-braked thin-walled steel sections. Eng Struct 2006;26:1609–19. [6] Quach WM, Teng JG, Chung KF. Residual stresses in steel sheets due to coiling and uncoiling: a closed-form analytical solution. Engineering Structures 2004;26:1249–59. [7] Yu WW. Cold-formed steel design. 3rd ed. New York: Wiley; 2000. [8] Hill R. The mathematical theory of plasticity. New York: Oxford University Press; 1983. [9] Dat DT. The strength of cold-formed steel columns. Cornell University Department of Structural Engineering report no. 80-4, Ithaca, NY, 1980. [10] Ingvarsson L. Cold-forming residual stresses, effect of buckling. In: Proceedings of the third international specialty conference on cold-formed steel structures, University of Missouri-Rolla; 1975. p. 85–119. [11] Weng CC, Peko¨z T. Residual stresses in cold-formed steel members. ASCE J Struct Eng 1990;116(6):1611–25. [12] de M. Batista E, Rodrigues FC. Residual stress measurements on cold-formed profiles. Exp Tech 1992;16(5):25–9. [13] Kwon YB. Post-buckling behaviour of thin-walled sections. Ph.D. thesis, University of Sydney, Australia, 1992. [14] Bernard ES. Flexural behaviour of cold-formed profiled steel decking. Ph.D. thesis, University of Sydney, Australia, 1993. [15] Chen WF, Han DJ. Plasticity for structural engineers. New York, NY: Springer; 1988. [16] ABAQUS. ABAQUS/Standard user’s manual version 6.5. ABAQUS, Inc., Providence, RI, /www.abaqus.comS; 2004. [17] CRC. Standard mathematical tables and formulae. New York, NY: CRC Press; 2003. [18] Moen C, Schafer BW. Direct strength design for cold-formed steel members with perforations, Progress report #4. Washington, DC: American Iron and Steel Institute; 2006. [19] Abdel-Rahman N, Sivakumaran KS. Material properties models for analysis of cold-formed steel members. ASCE J Struct Eng 1997; 123(9):1135–43.