- Email: [email protected]
Proposed residual stress model for roller bent steel wide flange sections
Journal of Constructional Steel Research 67 (2011) 992–1000
Contents lists available at ScienceDirect
Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Proposed residual stress model for roller bent steel wide flange sections R.C. Spoorenberg a,b,∗ , H.H. Snijder b , J.C.D. Hoenderkamp b a
Materials Innovation Institute M2i, P.O. Box 5008, 2600 GA Delft, The Netherlands
b
Eindhoven University of Technology, Faculty of Architecture, Building and Planning, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
article
info
Article history: Received 28 July 2010 Accepted 13 January 2011 Keywords: Residual stress model Curved steel Roller bending Wide flange section
abstract The manufacturing process of structural wide flange steel sections introduces residual stresses in the material. These stresses due to hot-rolling or welding influence the inelastic buckling response of structural steel members and need to be taken into account in the design. Based on experimental data standardized residual stress models have been proposed for inclusion in inelastic buckling analyses. By incorporating these residual stress models their effect on the resistance of beams and columns can be obtained. Residual stress models for roller bent steel sections are currently not available. Roller bent wide flange sections are manufactured by curving straight members at ambient temperature. This manufacturing technique, which is also known as roller bending, stresses the material beyond its yield stress, thereby overriding the initial residual stresses prior to bending and generating an entirely new pattern. This paper proposes a residual stress model for roller bent wide flange sections, based on earlier conducted numerical investigations which were validated by experimental research performed at Eindhoven University of Technology. The proposed residual stress model can serve as an initial state of a roller bent steel section in fully non-linear finite element analyses to accurately predict its influence on the inelastic buckling response. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Residual stresses in straight steel members
The manufacturing process of structural steel members induces residual stresses into the material. The residual stresses influence the structural behavior of steel structures since their presence causes early yielding of the material in specific locations of the member. Especially for steel structures prone to inelastic buckling, residual stresses are of major importance due to their detrimental influence on the maximum resistance. The availability of a large experimental database of residual stress measurements has led to the development of residual stress models for different section types, manufacturing procedures (e.g. hot-rolling, welding) and material properties (mild steel, high strength steel). A residual stress model comprises a simplified pattern and magnitude of the residual stresses over the cross section of the profile. The influence of residual stresses on the maximum resistance has been investigated by employing residual stress models as an initial state in a finite element model. Extensive computational work has led to design rules which make allowance for the effect of residual stresses on inelastic buckling.
1.1.1. Hot-rolled sections The residual stresses in hot-rolled steel sections are caused by differential cooling after roll forming. For a large number of wide flange sections with different height-to-width ratios and steel grades these stresses have been measured and published by Beedle and Tall [1], Mas and Massonet [2], Lay and Ward [3], Daddi and Mazzolani [4] and Young [5]. The results were summarized and several residual stress models were proposed by Lay and Ward [3], Young [5], Mazzolani [6] and ECCS [7], These models are featured by compressive stresses at the flange tips and tension stresses at the web-to-flange junctions. The webs of hot-rolled sections are subject to either tensile or compressive stresses (Fig. 1(a)). Throughout the paper tensile and compressive residual stresses will be annotated by (+) and (−) respectively. Although the proposed models suggest rather straightforward residual stress patterns which are not significantly influenced by the size of the section, the variability of the hot-rolled residual stress patterns was emphasized by Young [5] and Tall and Alpsten [8]. Section dimensions and cooling conditions have an influence on hot-rolled residual stresses.
∗ Corresponding author at: Eindhoven University of Technology, Faculty of Architecture, Building and Planning, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Tel.: +31 40 247 2948; fax: +31 40 245 0328. E-mail address: [email protected] (R.C. Spoorenberg). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.01.009
1.1.2. Welded sections Welded sections are manufactured by welding different plates together. Due to the large amount of heat-input for the welding
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
(a) Hot-rolled.
(b) Welded.
993
(c) Roller straightened.
Fig. 1. Indicative residual stress patterns for hot-rolled, welded and roller straightened sections.
process, large residual stresses emerge as a result of unequal cooling of the sections. The residual stresses in welded sections have been measured in mild steel members by Tebedge et al. [9] and in high strength steel by Beg and Hladnik [10]. The experimental results are characterized by large tensile stresses, with a magnitude of nearly the yield stress of the material in the vicinity of the weld and smaller compressive stresses in the remaining areas to equilibrate the tensile stresses. Models for residual stresses in welded sections were presented by Young and Robinson [11], see Fig. 1(b), and Chernenko and Kennedy [12].
and hence a lower inelastic buckling resistance. In addition, the resistance reduction due to residual stresses depends on the type of buckling, reflecting the complex interaction between residual stresses and inelastic buckling. Current design codes take into account the detrimental influence of residual stresses for beams and columns which buckle in the inelastic range. Design codes distinguish between welded and rolled sections for both minor and major axis flexural buckling and lateral torsional buckling.
1.1.3. Roller straightened sections After rolling and cooling in the mill, hot-rolled members may possess an out-of-straightness in excess of the specified tolerances. These members can be roller straightened or rotorized to meet the fabrication tolerances for hot-rolled wide flange steel sections. During this process the member is fed through a series of rolls, thereby reducing its initial deformations. It has been found by Alpsten [13] that the initial stresses due to differential cooling are altered due to roller straightening. A new residual stress pattern emerges with smaller values for the stresses when compared to hot-rolled residual stresses, especially in the flanges. The residual stress pattern depends to a large extent on the degree of straightening forced on the section. It was shown by Alpsten [13] that roller straightened wide flange sections have a higher resistance than their non-rotorized counterparts as a result of their lower residual stresses and improved straightness. The influence of the roller-straightening process is, however, not implemented in current design rules. A distinctive residual stress pattern proposed by ECCS [7] (Fig. 1(c)) can be used for numerical computations for rotorized wide flange sections, irrespective of their height-to-width ratio.
In recent years curved steel and arches have experienced an expanding area of application. Curved steel has been frequently used for roofs and bridges, combining architectural demands with structural merits. Curved steel is often fabricated by bending straight hot-rolled sections at ambient temperatures, Bjorhovde [14]. The member is placed between three rolls, which are positioned in a triangular arrangement, and a combination of movement of the rolls and feeding the member between these rolls induces a permanent curvature. This manufacturing technique is better known as the roller bending process. Wide flange sections can either be curved about the weak or strong axis into different shapes, e.g. ellipse, parabola or circle. This paper is limited to circular arches curved about the strong axis. When bent about the strong axis the top flange and bottom flange experience plastic elongation and shortening in longitudinal direction respectively. Since the roller bending process involves plastic bending, the initial residual stresses prior to roller bending are altered due to the curving process [15,16].
1.1.4. Influence on resistance The proposed indicative stress models shown in Fig. 1 have been employed as initial states of stress in structural steel members for computer analyses to assess their influence on the resistance and load–deflection characteristics. All these residual stress models are based on the membrane component (averaged residual stress over web thickness and flange thickness) without bending (residual stress gradient over web thickness and flange thickness). Nonlinear finite element analyses have shown that the maximum resistance is significantly influenced by the magnitude and pattern of the residual stresses, as summarized in ECCS [7]. In general it was observed that especially large compressive residual stresses in the flange tips are detrimental to the inelastic buckling resistance of beams and columns. For these members premature yielding takes place at the flange tips, resulting in a stiffness reduction
1.2. Roller bent steel
1.3. Residual stresses and inelastic arch buckling Due to their slenderness and in-plane force distribution which is characterized by large compressive actions, arches are susceptible to either in-plane (Fig. 2(a)) or out-of-plane buckling (Fig. 2(b)). The inelastic buckling behavior of steel arches has mainly been investigated by means of finite element analyses, incorporating geometrical imperfections, material non-linearities and residual stresses. The residual stress models were based on either typical welding residual stresses (Fig. 3(a) and (b) for which fy = the yield stress) or hot-rolled residual stresses (Fig. 3(c)). Studies on in-plane buckling have been presented by Pi and Trahair [17] and Pi et al. [18]. Finite element analyses on inelastic out-of-plane buckling were carried out by Komatsu and Sakimoto [19], Sakimoto and Komatsu [20], Pi and Trahair [21], Pi and Bradford [22] and Pi and Bradford [23]. The finite element computations led to the development of design rules to obtain the resistance for steel arches.
994
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
(a) In-plane instability.
(b) Out-of-plane instability.
Fig. 2. Instability phenomena of arches.
(a) Komatsu and Sakimoto [19].
(b) Sakimoto and Komatsu [20].
(c) Pi and Bradford [23].
(d) Suggested roller bending.
Fig. 3. Adopted residual stress patterns for studying inelastic buckling behavior of steel arches.
The influence of the roller bending process on the pattern and magnitude of the residual stresses was not implemented in the finite element models. A classical cold bent residual stress pattern as initially proposed by Timoshenko [24] for solid rectangular sections and suggested by King and Brown [25] (Fig. 3(d)) to be applicable to roller bent wide flange sections, was mentioned by Pi and Bradford [23] and Pi et al. [18]. In Fig. 3(d) Mpl is the full plastic moment and Mel is the elastic moment of the cross section. The study of the structural behavior of steel arches subject to in-plane loading necessitates a detailed knowledge of residual stresses in roller bent steel sections bent about the major axis in view of their influence on the inelastic buckling response of steel arches. In the research program on the behavior of steel arch structures at Eindhoven University of Technology, residual stresses were first measured in a number of roller bent and straight wide flange sections of different steel grade, section type and bending radius. The results have been published earlier by Spoorenberg et al. [15]. The experimental phase was followed by the development of a finite element model to estimate the residual stresses in roller bent sections, [16]. This finite element model was used to simulate the complete roller bending process. Experimental values were used to corroborate numerical results. Good agreement in averaged residual stresses over the flange thickness and web thickness was observed, demonstrating the validity of the finite element model. Subsequently, it was found that residual stresses in roller bent wide flange sections display a pattern that is significantly different from a typical hot-rolled pattern (Fig. 3(c)) or cold bent pattern based on the proposal by Timoshenko (Fig. 3(d)). Considering the observations from the first research phases, the final step is an examination of all numerical data to arrive at a residual stress model for circular roller bent wide flange steel sections bent around the major axis. The proposed residual stress model will be compared against the experimental and numerical data as a final check on the accuracy.
thickness and the flange thickness. The finite element model was not developed to simulate the residual stress gradients over the web thickness and the flange thickness. However, it yields averaged residual stresses over the web and flange thickness for the entire cross section. An experimental approach for obtaining residual stresses cannot yield data for the entire cross section due to placement requirements of the strain gauges. It was therefore decided to employ finite element analyses to generate residual stresses which can be used in the development of a proposal for a roller bent residual stress model, thereby ignoring residual stress gradients over the web thickness and the flange thickness. In view of earlier developed residual stress models [4,10] for straight steel members, which also ignored the measured residual stress gradient over the web thickness and the flange thickness, this simplification was considered appropriate. Residual stresses were obtained for a total of 18 wide flange steel sections by simulation in the finite element environment. Two steel grades, four different sections and five bending radii were used which yielded 8 different bending ratios (Table 1). The bending ratio equals the radius of the circular arch divided by the nominal height of the cross section. An increase of the bending ratio means a decrease of the degree of cold working, when the height of the section is kept constant. As part of the experimental procedure to measure residual stresses, tensile tests were performed on coupons taken from the flanges of straight reference sections to obtain the yield stress and ultimate tensile stress, according to NEN-EN 10002-1 [26] and EN 10025 [27]. These straight sections were originally attached to the roller bent specimens and give information on the state of the material prior to roller bending. Dividing the numerically obtained residual stresses by the measured yield stresses of the straight material gives normalized residual stress values.
2. Residual stress analyses based on finite element computations
The roller bent residual stresses obtained by finite element analysis of all 18 wide flange sections are presented in Fig. 4 at normalized locations. The residual stresses in the top flange range from 0fy to 0.2fy at the tips and at the web-to-flange junction they range from −0.2fy to 0.2fy . The bottom flange is featured by compressive stresses at the flange tips that range from −0.1fy to
It has been shown by Spoorenberg et al. [16] that close agreement can be attained between finite element analyses and experimental results for the averaged residual stresses over the web
2.1. Numerically obtained roller bent residual stresses
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
995
Table 1 Mechanical properties and bending ratio of steel members. Section type
Steel grade
S235 HE 100A S355
S235 HE 100B S355
HE 360B
S235 S355 S235
IPE 360 S355
Bending radius R (mm)
Mechanical properties straight reference sections (N/mm2 )
Bending ratio R/h (–)
Yield stress fy
Tensile stress ft
1910 2546 3820 1910 2546 3820
322 279
433 418
364
566
1910 2546 3820 1910 2546 3820
248 285
411 412
386 390
492 495
8000 8000
269 357
389 534
22.22 22.22
4500 8000 4500 8000
297
414
361
528
12.5 22.22 12.5 22.22
19.89 26.52 39.79 19.89 26.52 39.79 19.1 25.46 38.2 19.1 25.46 38.2
analyses are consistently located in the bottom flange at the webto-flange junction and flange tips respectively. It was therefore decided that the residual stress patterns across the bottom flange, which was subjected to compressive actions during roller bending, would be used to investigate the influence of bending ratio and original yield stress. 2.2. Influence of bending radius The amount of cold work or plastic straining applied during roller bending may affect the magnitude of the residual stresses. For steel exhibiting a clear hardening stage, a decrease of the bending ratio is expected to induce higher residual stresses. For each steel section as presented in Table 1, the numerically obtained maximum tensile and maximum compressive normalized residual stresses in the bottom flange are displayed as a function of the bending ratio in Fig. 5(left). It can bee seen that the bending ratio has no clear influence on the extreme residual stresses within the examined range of 12.5 ≤ R/h ≤ 39.79. Consequently a residual stress model can be developed which is independent of the bending radius, and applicable to a bending ratio range of approximately 10 ≤ R/h ≤ 40. Fig. 4. Normalized residual stresses in 18 roller bent wide flange sections.
−0.3fy and tensile stresses at the web-to-flange junction ranging from 0.4fy to 0.7fy . The web shows a large scatter when compared to the stress characteristics in the flanges. Although consistent trends from the web stresses cannot readily be observed, it can be concluded that the upper part of the web is in tension with an average maximum stress of approximately 0.3fy . The lower part of the web is mainly subject to compression with an average maximum stress of approximately −0.4fy . Steep stress gradients are observed at the web-to-flange junctions, as reflected by the large difference between the stress values in the top and bottom of the web and in the middle of the top flange and bottom flange respectively. In Table 1 it can be seen that the numerical calculations comprise a wide range of bending radii and steel grades. For the development of a single residual stress model the influence of the bending ratio and the yield stress of the original material on the numerical residual stresses were studied first. It was observed that the maximum tensile and compressive residual stresses from all 18
2.3. Influence of original yield stress The yield stress of the original material may also affect the magnitude of the residual stresses. Structural steel with a higher yield stress prior to roller bending may have higher residual stresses after bending when compared to steel with low yield stress levels. Residual stress measurements published by Gardner and Cruise [28] on press braked angles and cold-rolled box sections have shown a clear relationship between material yield stress and magnitude of residual stresses. Fig. 5(right) shows the maximum tensile and compressive residual stresses in the bottom flange versus the measured yield stress of the material before roller bending. The residual stress values are not normalized, in order to display the influence of the yield stress on the magnitude of the residual stress. 3. Roller bent residual stress model for wide flange sections Based on the numerically obtained roller bent residual stresses, a model of these stresses for wide flange steel sections is suggested. This proposal is intended to be generally applicable, independent
996
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
Fig. 5. Normalized maximum tensile and compressive residual stresses in the bottom flange versus bending ratio (left) and maximum tensile and compressive residual stresses in the bottom flange versus yield stress of original material (right).
(a) Dimensions.
(b) Residual stresses.
(c) Equilibrium of stress blocks in web.
Fig. 6. Proposed residual stress model for roller bent wide flange steel sections.
of the bending ratio for the range of 10 ≤ R/h ≤ 40 (Section 2.2) and linearly related to the magnitude of the original yield stress for S235 and S355 steel sections (Section 2.3). The magnitudes and pattern for the residual stress model are determined to best fit the finite element results. 3.1. Residual stress pattern and magnitudes The residual stress model is based on the numerically obtained residual stresses as shown in Fig. 4 and the trend lines as shown in Fig. 5 and is shown in Fig. 6. A symmetric bi-linear stress pattern along the half bottom flange is suggested with a maximum tensile stress value of 0.7fy at the web-to-flange junction and 0.35fy compression at the flange tip. These stress values are based on the gradient of the trend lines as shown in Fig. 5. A somewhat larger compressive stress value for the flange tips that shown in the trend line of Fig. 5 has been proposed, since numerical data could not be obtained at the flange tips but at a distance away from the ends (integration point location). Using the gradient of the trend line would therefore result in an underestimation of the compressive stresses at the flange tips. For the top flange a linear stress gradient is suggested based on a qualitative fit with numerical data, featured by 0.2fy tensile stresses and 0.2fy compressive stresses at the flange tips and flange center, respectively (Fig. 6(c)). For the web two triangular stress blocks are suggested. The tensile and compressive stress peak of the triangles are located at a distance of 1/4h0 and 3/4h0 from the web-to-flange junction of the top flange, respectively. The stress values are equal to zero at the top and bottom of the web. The maximum tensile and compressive residual stress values in the web are annotated by σwrt and σwrc
respectively and their magnitudes are governed by the internal equilibrium requirement. Equilibrium conditions for residual stresses consist of the axial force, major moment and minor moment equilibrium requirement as stated earlier by Lay and Ward [3] and Szalai and Papp [29]. Equilibrium about the minor bending axis is automatically satisfied due to the symmetric pattern of the suggested residual stress model about this axis. The bi-linear residual stress pattern in the bottom flange yields a net tensile force, which is balanced by residual stresses in the web, according to the equilibrium requirements. The two unknown web stresses, σwrt and σwrc , can be obtained from the two remaining equilibrium equations: the normal forces and the major bending moment requirement Fig. 6(c). The equilibrium equations are simplified by neglecting the fillets in the wide flange section. Axial force (N ) and major moment (M ) equilibrium equations can be set up by summing forces for all stress blocks and summing the product of these normal forces and their distances to the center of the top flange, as depicted in Fig. 6(c):
−
7
1
1
+ σwrt 8 4 1 1 + + σwrc fy = 0 8 4 − 7 1 1 1 5 M = btf h0 + tw h0 + h0 σwrt 80 8 6 4 12 1 7 1 5 + h0 + h0 σwrt fy = 0. N =
80
btf + tw h0
4
12
8
6
(1)
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
Eq. (1) can be simplified into
−
7
3
3
σwrt + σwrc fy = 0 8 − 1 7 1 h0 fy = 0, M = btf h0 + h0 tw σwrt + σwrc N =
80
btf + tw h0
8
80
8
(2)
4
and written in matrix notations:
N M
3 8 = fy h0 tw 1 h0
8
3 8 σwrt 1 σwrc
h0
− fy
4
7 btf 80
80
0 . 0
(3)
Solving (3) for σwrt and σwrc yields
σwrt =
7btf 30h0 tw
fy ,
σwrc = −
14btf 30h0 tw
fy .
flange width b or section height h. Due to placement requirements of the strain gauges, average values could not be obtained close to the junction of the flanges and web. Similarly to the numerical stress values, the experimental results were normalized with respect to the measured yield stress of the straight material (Table 1). The proposed residual stress model agrees well with the measured residual stress value for the HE 100A, HE 100B and HE 360B sections, although less coherence is found in the web of the IPE 360 series. 4.2. Comparison to numerical analyses
7 btf h0
=
997
(4)
With these values the residual stress model for roller bent wide flange steel sections is complete. 3.2. Residual stress model features From Eq. (4) it is clear that the requirement of internal equilibrium renders the proposed model to be different per section geometry. The residual stress values in the web are governed by the ratio between the area of the flange and area of the web. The proposed model is qualitatively identical for all wide flange sections but the magnitude of the residual stresses in the web is dependent on the geometry of the cross section. Fig. 4 shows that the numerical residual stress pattern is featured by a sharp stress gradient at the web-to-flange junctions of the roller bent wide flange sections. This sharp stress gradient has been approximated in the proposed residual stress model by a so-called ‘stress jump’ at the web-to-flange junctions (i.e. the stress value at the top of the web or bottom of the web is different from the stress value in the middle of the top flange or middle of the bottom flange, respectively). A stress jump is a stress change over an infinitesimal distance and can also be found in residual stress models for welded sections, [20] (Fig. 3(b)). Modeling the large stress gradients instead of the stress-jumps in the web-to-flange junctions in a residual stress model would significantly complicate the stress pattern in the web and improve its accuracy only marginally as the pertaining section areas are extremely small. The stress jump provides therefore a simplification to the residual stress model and enhances the simplicity of the equilibrium equations and ease of applicability when employed in numerical models. 4. Discussion 4.1. Comparison to experiments Residual stress measurements were performed on both straight and roller bent sections using the sectioning method. Measurements taken from both sides of the flanges and the web were used to obtain the average residual stress values. Residual stresses in HE 100A, HE 360B and IPE 360 sections were published earlier by Spoorenberg et al. [15]. A full overview of the averaged experimental results compared with the suggested residual stress model is shown in Fig. 7. The averaged experimental results for roller bent sections and the suggested residual stress model are plotted against the location over the section normalized by the
The proposed residual stress model is based on the finite element patterns and magnitudes from all wide flange sections as summarized in Fig. 4 and the trend lines shown in Fig. 5. However, in order to meet the internal equilibrium requirements, the suggested model will be different for all wide flange steel sections. The proposed residual stress model is therefore compared to the finite element residual stresses in all 4 different section types; see Fig. 8. A good correlation of results can be observed for the HE 100A, HE 100B and HE 360B series but larger discrepancies are found in the top flange of the IPE 360 series. 4.3. Comparison between existing hot-rolled and proposed roller bent residual stress models It was mentioned in Section 1.1.4 that compressive residual stresses in the flange tips are detrimental to the resistance of members susceptible to either flexural or lateral torsional buckling. Compressive residual stresses in the flange tips reduce the flexural stiffness during inelastic arch buckling and subsequently will cause early collapse of the arch. A hot-rolled residual stress model is featured by compressive stresses in all four flange tips, whereas the proposed model displays only compressive stresses in the tips of the bottom flange. From a qualitative comparison between the hot-rolled model and the proposed roller bent model it can be concluded that the proposed residual stress model for roller bent wide flange steel sections will be more favorable to the resistance of steel arches, when either failing by in-plane or out-of-plane inelastic buckling. 4.4. Range of applicability residual stress model Good coherence between the residual stress model and experimental results for various roller bent wide flange sections can be observed in Fig. 7. This allows the application of the residual stress model to other wide flange sections that did not make part of the experimental program. In addition, since the residual stress model can be expressed as a function of the yield stress of the straight material, it can be applied to other steel grades (e.g. steel grade S275). Under the assumption that for bending ratios R/h > 40 the required amount of cold work on the section will be marginally smaller, the reduction in residual stresses will be of minor influence. Application of the residual stress model to roller bent arches with larger bending ratios than currently investigated seems appropriate but is probably conservative. 5. Conclusions After manufacturing wide flange steel sections by the hotrolling process, additional cold working such as roller bending has a significant influence on the residual stress patterns. Earlier publications by Spoorenberg et al. [15,16] demonstrate a significant difference between residual stresses in hot-rolled and roller bent wide flange sections. In this paper residual stresses obtained from finite element analyses by Spoorenberg et al. [16] have been summarized and normalized with respect to the yield stress of the material before roller bending to develop and present
998
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
Fig. 7. Normalized measured averaged residual stresses and proposed residual stress model.
a residual stress model. It was found that the bending ratio does not influence the magnitudes of the residual stresses, allowing for a residual stress model representing all examined bending radii for 10 ≤ R/h ≤ 40. The residual stresses were found to be linearly dependent on the yield stress of the original material and a single normalized residual stress model could therefore represent all roller bent sections, for steel grades S235 and S355. Based on a best fit of the finite element data, the proposed linear stress gradient along the width of the top flange (the flange that is plastically elongated in longitudinal direction during roller bending) is featured by stress magnitudes of 0.2fy in tension and 0.2fy in compression at the flange tips and flange center respectively. The residual stress in the bottom flange (the flange that is plastically shortened in longitudinal direction during
roller bending) can be represented by a bi-linear pattern with a maximum compressive stress of 0.35fy at the flange tip, zero stress at the quarter points of the flange width and a maximum tension of 0.70fy at the web-to-flange junction. The residual stress pattern over the height of the web can be represented by two triangular stress blocks: tensile stress in the upper region of the web near the top flange and compressive stress in the lower region. The maximum value for the two zones can be determined from internal equilibrium which results in 0.2–0.4fy for tension and 0.4–0.8fy for compression residual stresses. The residual stress model was compared to experimental and numerical residual stress measurements and good agreement was observed. The suggested residual stress model can be implemented in finite element models with beam or shell elements to assess its influence
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
999
Fig. 8. Normalized computed residual stresses and proposed residual stress model.
on the inelastic buckling response of steel arches. In view of the detrimental influence of compressive residual stress at the flange tips it is expected that the proposed residual stress model will yield higher resistances compared to finite element analyses using models for hot-rolled residual stresses when investigating inelastic arch buckling. Acknowledgements This research was carried out under the project number MC1.06262 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl).
References [1] Beedle LS, Tall L. Basic column strength. Transactions of the ASCE 1962;127: 138–79. [2] Mas E, Massonet Ch. Part prise par la belgique dans les recherches experimentales de la convention europeenne des associations de la construction metallique sur le flambement centriques des barres en acier doux. Acier-StahlSteel 1966;9:393–400. [3] Lay MG, Ward R. Residual stresses in steel sections. Journal of the Australian Institute of Steel Construction 1969;3(3):2–21. [4] Daddi I, Mazzolani FM. Determinazione sperimentale delle imperfezioni strutturali nei profilati di acciaio: Universita degli studi di Napoli istituto di tecnica delle costruzioni; 1971. [5] Young BW. Residual stresses in hot rolled members. IABSE reports of the working commissions 1975;23:25–38. [6] Mazzolani FM. Buckling curves of hot-rolled steel shapes with structural imperfections. IABSE reports of the working commissions 1975;23:152–61.
1000
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
[7] ECCS. Manual on stability of steel structures - ECCS Committee 8 Stability: 1976. [8] Tall L, Alpsten G. On the scatter in yield strength and residual stresses in steel members. IABSE reports of the working commissions 1969;4:151–63. [9] Tebedge N, Alpsten G, Tall L. Residual-stress measurement by the sectioning method. Experimental Mechanics 1973;13:88–96. [10] Beg D, Hladnik L. Eigenspannungen bei geschweissten I-profilen aus hochfesten Stahlen. Stahlbau 1994;63(5):134–9. [11] Young BW, Robinson KW. Buckling of axially loaded welded steel columns. The Structural Engineer 1975;53(5):203–7. [12] Chernenko DE, Kennedy DJL. An analysis of the performance of welded wide flange columns. Canadian Journal of Civil Engineering 1991;18(14):537–54. [13] Alpsten G. Residual stresses, yield stress, and the column strength of hotrolled and roller-straightened steel shapes. IABSE reports of the working commissions 1975;23:39–59. [14] Bjorhovde R. Cold bending of wide-flange shapes for construction. Engineering Journal 2006;43(4):271–86. [15] Spoorenberg RC, Snijder HH, Hoenderkamp JCD. Experimental investigation of residual stresses in roller bent wide flange steel sections. Journal of Constructional Steel Research 2010;66(6):737–47. [16] Spoorenberg RC, Snijder HH, Hoenderkamp JCD. Finite element simulations of residual stresses in roller bent wide flange sections. Journal of Constructional Steel Research 2011;67(1):39–50. [17] Pi Y-L, Trahair NS. In-plane inelastic buckling and strengths of steel arches. Journal of Structural Engineering ASCE 1996;122(7):734–47.
[18] Pi Y-L, Bradford MA, Tin-Loi F. In-plane strength of steel arches. Advanced Steel Construction 2008;4(4):306–22. [19] Komatsu S, Sakimoto T. Ultimate load carrying capacity of steel arches. Journal of the Structural Division 1977;103(12):2323–36. [20] Sakimoto T, Komatsu S. Ultimate strength formula for central-arch-girder bridges. Proc Japan Soc Civ Engrs 1983;333:183–6. [21] Pi Y-L, Trahair NS. Out-of-plane inelastic buckling and strength of steel arches. Journal of Structural Engineering ASCE 1998;124(2):174–83. [22] Pi Y-L, Bradford MA. Elasto-plastic buckling and postbuckling of arches subjected to a central load. Computers & Structures 2003;81(8):1811–25. [23] Pi Y-L, Bradford MA. Out-of-plane strength design of fixed steel I-section arches. Journal of Structural Engineering ASCE 2005;131(4):560–8. [24] Timoshenko SP. Strength of materials. Part II advanced theory and problems. D. Van Nostrand Company, Inc.; 1940. [25] King C, Brown D. Design of Curved Steel: The Steel Construction Institute; 2001. [26] NEN-EN 10002-1. Metalen - Trekproef - Deel 1: Beproevingsmethoden bij omgevingstemperatuur: Nederlands Normalisatie Instituut; 2001. [27] EN 10025. Hot rolled products of non-alloy structural steels. Technical delivery conditions.: CEN European Committee Standardization; 1993. [28] Gardner L, Cruise RB. Modeling of residual stresses in structural stainless steel sections. Journal of Structural Engineering ASCE 2009;135(1):42–53. [29] Szalai J, Papp F. A new residual stress distribution for hot-rolled I-shaped sections. Journal of Constructional Steel Research 2005;61(6):845–61.
Contents lists available at ScienceDirect
Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr
Proposed residual stress model for roller bent steel wide flange sections R.C. Spoorenberg a,b,∗ , H.H. Snijder b , J.C.D. Hoenderkamp b a
Materials Innovation Institute M2i, P.O. Box 5008, 2600 GA Delft, The Netherlands
b
Eindhoven University of Technology, Faculty of Architecture, Building and Planning, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands
article
info
Article history: Received 28 July 2010 Accepted 13 January 2011 Keywords: Residual stress model Curved steel Roller bending Wide flange section
abstract The manufacturing process of structural wide flange steel sections introduces residual stresses in the material. These stresses due to hot-rolling or welding influence the inelastic buckling response of structural steel members and need to be taken into account in the design. Based on experimental data standardized residual stress models have been proposed for inclusion in inelastic buckling analyses. By incorporating these residual stress models their effect on the resistance of beams and columns can be obtained. Residual stress models for roller bent steel sections are currently not available. Roller bent wide flange sections are manufactured by curving straight members at ambient temperature. This manufacturing technique, which is also known as roller bending, stresses the material beyond its yield stress, thereby overriding the initial residual stresses prior to bending and generating an entirely new pattern. This paper proposes a residual stress model for roller bent wide flange sections, based on earlier conducted numerical investigations which were validated by experimental research performed at Eindhoven University of Technology. The proposed residual stress model can serve as an initial state of a roller bent steel section in fully non-linear finite element analyses to accurately predict its influence on the inelastic buckling response. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction
1.1. Residual stresses in straight steel members
The manufacturing process of structural steel members induces residual stresses into the material. The residual stresses influence the structural behavior of steel structures since their presence causes early yielding of the material in specific locations of the member. Especially for steel structures prone to inelastic buckling, residual stresses are of major importance due to their detrimental influence on the maximum resistance. The availability of a large experimental database of residual stress measurements has led to the development of residual stress models for different section types, manufacturing procedures (e.g. hot-rolling, welding) and material properties (mild steel, high strength steel). A residual stress model comprises a simplified pattern and magnitude of the residual stresses over the cross section of the profile. The influence of residual stresses on the maximum resistance has been investigated by employing residual stress models as an initial state in a finite element model. Extensive computational work has led to design rules which make allowance for the effect of residual stresses on inelastic buckling.
1.1.1. Hot-rolled sections The residual stresses in hot-rolled steel sections are caused by differential cooling after roll forming. For a large number of wide flange sections with different height-to-width ratios and steel grades these stresses have been measured and published by Beedle and Tall [1], Mas and Massonet [2], Lay and Ward [3], Daddi and Mazzolani [4] and Young [5]. The results were summarized and several residual stress models were proposed by Lay and Ward [3], Young [5], Mazzolani [6] and ECCS [7], These models are featured by compressive stresses at the flange tips and tension stresses at the web-to-flange junctions. The webs of hot-rolled sections are subject to either tensile or compressive stresses (Fig. 1(a)). Throughout the paper tensile and compressive residual stresses will be annotated by (+) and (−) respectively. Although the proposed models suggest rather straightforward residual stress patterns which are not significantly influenced by the size of the section, the variability of the hot-rolled residual stress patterns was emphasized by Young [5] and Tall and Alpsten [8]. Section dimensions and cooling conditions have an influence on hot-rolled residual stresses.
∗ Corresponding author at: Eindhoven University of Technology, Faculty of Architecture, Building and Planning, Den Dolech 2, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. Tel.: +31 40 247 2948; fax: +31 40 245 0328. E-mail address: [email protected] (R.C. Spoorenberg). 0143-974X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jcsr.2011.01.009
1.1.2. Welded sections Welded sections are manufactured by welding different plates together. Due to the large amount of heat-input for the welding
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
(a) Hot-rolled.
(b) Welded.
993
(c) Roller straightened.
Fig. 1. Indicative residual stress patterns for hot-rolled, welded and roller straightened sections.
process, large residual stresses emerge as a result of unequal cooling of the sections. The residual stresses in welded sections have been measured in mild steel members by Tebedge et al. [9] and in high strength steel by Beg and Hladnik [10]. The experimental results are characterized by large tensile stresses, with a magnitude of nearly the yield stress of the material in the vicinity of the weld and smaller compressive stresses in the remaining areas to equilibrate the tensile stresses. Models for residual stresses in welded sections were presented by Young and Robinson [11], see Fig. 1(b), and Chernenko and Kennedy [12].
and hence a lower inelastic buckling resistance. In addition, the resistance reduction due to residual stresses depends on the type of buckling, reflecting the complex interaction between residual stresses and inelastic buckling. Current design codes take into account the detrimental influence of residual stresses for beams and columns which buckle in the inelastic range. Design codes distinguish between welded and rolled sections for both minor and major axis flexural buckling and lateral torsional buckling.
1.1.3. Roller straightened sections After rolling and cooling in the mill, hot-rolled members may possess an out-of-straightness in excess of the specified tolerances. These members can be roller straightened or rotorized to meet the fabrication tolerances for hot-rolled wide flange steel sections. During this process the member is fed through a series of rolls, thereby reducing its initial deformations. It has been found by Alpsten [13] that the initial stresses due to differential cooling are altered due to roller straightening. A new residual stress pattern emerges with smaller values for the stresses when compared to hot-rolled residual stresses, especially in the flanges. The residual stress pattern depends to a large extent on the degree of straightening forced on the section. It was shown by Alpsten [13] that roller straightened wide flange sections have a higher resistance than their non-rotorized counterparts as a result of their lower residual stresses and improved straightness. The influence of the roller-straightening process is, however, not implemented in current design rules. A distinctive residual stress pattern proposed by ECCS [7] (Fig. 1(c)) can be used for numerical computations for rotorized wide flange sections, irrespective of their height-to-width ratio.
In recent years curved steel and arches have experienced an expanding area of application. Curved steel has been frequently used for roofs and bridges, combining architectural demands with structural merits. Curved steel is often fabricated by bending straight hot-rolled sections at ambient temperatures, Bjorhovde [14]. The member is placed between three rolls, which are positioned in a triangular arrangement, and a combination of movement of the rolls and feeding the member between these rolls induces a permanent curvature. This manufacturing technique is better known as the roller bending process. Wide flange sections can either be curved about the weak or strong axis into different shapes, e.g. ellipse, parabola or circle. This paper is limited to circular arches curved about the strong axis. When bent about the strong axis the top flange and bottom flange experience plastic elongation and shortening in longitudinal direction respectively. Since the roller bending process involves plastic bending, the initial residual stresses prior to roller bending are altered due to the curving process [15,16].
1.1.4. Influence on resistance The proposed indicative stress models shown in Fig. 1 have been employed as initial states of stress in structural steel members for computer analyses to assess their influence on the resistance and load–deflection characteristics. All these residual stress models are based on the membrane component (averaged residual stress over web thickness and flange thickness) without bending (residual stress gradient over web thickness and flange thickness). Nonlinear finite element analyses have shown that the maximum resistance is significantly influenced by the magnitude and pattern of the residual stresses, as summarized in ECCS [7]. In general it was observed that especially large compressive residual stresses in the flange tips are detrimental to the inelastic buckling resistance of beams and columns. For these members premature yielding takes place at the flange tips, resulting in a stiffness reduction
1.2. Roller bent steel
1.3. Residual stresses and inelastic arch buckling Due to their slenderness and in-plane force distribution which is characterized by large compressive actions, arches are susceptible to either in-plane (Fig. 2(a)) or out-of-plane buckling (Fig. 2(b)). The inelastic buckling behavior of steel arches has mainly been investigated by means of finite element analyses, incorporating geometrical imperfections, material non-linearities and residual stresses. The residual stress models were based on either typical welding residual stresses (Fig. 3(a) and (b) for which fy = the yield stress) or hot-rolled residual stresses (Fig. 3(c)). Studies on in-plane buckling have been presented by Pi and Trahair [17] and Pi et al. [18]. Finite element analyses on inelastic out-of-plane buckling were carried out by Komatsu and Sakimoto [19], Sakimoto and Komatsu [20], Pi and Trahair [21], Pi and Bradford [22] and Pi and Bradford [23]. The finite element computations led to the development of design rules to obtain the resistance for steel arches.
994
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
(a) In-plane instability.
(b) Out-of-plane instability.
Fig. 2. Instability phenomena of arches.
(a) Komatsu and Sakimoto [19].
(b) Sakimoto and Komatsu [20].
(c) Pi and Bradford [23].
(d) Suggested roller bending.
Fig. 3. Adopted residual stress patterns for studying inelastic buckling behavior of steel arches.
The influence of the roller bending process on the pattern and magnitude of the residual stresses was not implemented in the finite element models. A classical cold bent residual stress pattern as initially proposed by Timoshenko [24] for solid rectangular sections and suggested by King and Brown [25] (Fig. 3(d)) to be applicable to roller bent wide flange sections, was mentioned by Pi and Bradford [23] and Pi et al. [18]. In Fig. 3(d) Mpl is the full plastic moment and Mel is the elastic moment of the cross section. The study of the structural behavior of steel arches subject to in-plane loading necessitates a detailed knowledge of residual stresses in roller bent steel sections bent about the major axis in view of their influence on the inelastic buckling response of steel arches. In the research program on the behavior of steel arch structures at Eindhoven University of Technology, residual stresses were first measured in a number of roller bent and straight wide flange sections of different steel grade, section type and bending radius. The results have been published earlier by Spoorenberg et al. [15]. The experimental phase was followed by the development of a finite element model to estimate the residual stresses in roller bent sections, [16]. This finite element model was used to simulate the complete roller bending process. Experimental values were used to corroborate numerical results. Good agreement in averaged residual stresses over the flange thickness and web thickness was observed, demonstrating the validity of the finite element model. Subsequently, it was found that residual stresses in roller bent wide flange sections display a pattern that is significantly different from a typical hot-rolled pattern (Fig. 3(c)) or cold bent pattern based on the proposal by Timoshenko (Fig. 3(d)). Considering the observations from the first research phases, the final step is an examination of all numerical data to arrive at a residual stress model for circular roller bent wide flange steel sections bent around the major axis. The proposed residual stress model will be compared against the experimental and numerical data as a final check on the accuracy.
thickness and the flange thickness. The finite element model was not developed to simulate the residual stress gradients over the web thickness and the flange thickness. However, it yields averaged residual stresses over the web and flange thickness for the entire cross section. An experimental approach for obtaining residual stresses cannot yield data for the entire cross section due to placement requirements of the strain gauges. It was therefore decided to employ finite element analyses to generate residual stresses which can be used in the development of a proposal for a roller bent residual stress model, thereby ignoring residual stress gradients over the web thickness and the flange thickness. In view of earlier developed residual stress models [4,10] for straight steel members, which also ignored the measured residual stress gradient over the web thickness and the flange thickness, this simplification was considered appropriate. Residual stresses were obtained for a total of 18 wide flange steel sections by simulation in the finite element environment. Two steel grades, four different sections and five bending radii were used which yielded 8 different bending ratios (Table 1). The bending ratio equals the radius of the circular arch divided by the nominal height of the cross section. An increase of the bending ratio means a decrease of the degree of cold working, when the height of the section is kept constant. As part of the experimental procedure to measure residual stresses, tensile tests were performed on coupons taken from the flanges of straight reference sections to obtain the yield stress and ultimate tensile stress, according to NEN-EN 10002-1 [26] and EN 10025 [27]. These straight sections were originally attached to the roller bent specimens and give information on the state of the material prior to roller bending. Dividing the numerically obtained residual stresses by the measured yield stresses of the straight material gives normalized residual stress values.
2. Residual stress analyses based on finite element computations
The roller bent residual stresses obtained by finite element analysis of all 18 wide flange sections are presented in Fig. 4 at normalized locations. The residual stresses in the top flange range from 0fy to 0.2fy at the tips and at the web-to-flange junction they range from −0.2fy to 0.2fy . The bottom flange is featured by compressive stresses at the flange tips that range from −0.1fy to
It has been shown by Spoorenberg et al. [16] that close agreement can be attained between finite element analyses and experimental results for the averaged residual stresses over the web
2.1. Numerically obtained roller bent residual stresses
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
995
Table 1 Mechanical properties and bending ratio of steel members. Section type
Steel grade
S235 HE 100A S355
S235 HE 100B S355
HE 360B
S235 S355 S235
IPE 360 S355
Bending radius R (mm)
Mechanical properties straight reference sections (N/mm2 )
Bending ratio R/h (–)
Yield stress fy
Tensile stress ft
1910 2546 3820 1910 2546 3820
322 279
433 418
364
566
1910 2546 3820 1910 2546 3820
248 285
411 412
386 390
492 495
8000 8000
269 357
389 534
22.22 22.22
4500 8000 4500 8000
297
414
361
528
12.5 22.22 12.5 22.22
19.89 26.52 39.79 19.89 26.52 39.79 19.1 25.46 38.2 19.1 25.46 38.2
analyses are consistently located in the bottom flange at the webto-flange junction and flange tips respectively. It was therefore decided that the residual stress patterns across the bottom flange, which was subjected to compressive actions during roller bending, would be used to investigate the influence of bending ratio and original yield stress. 2.2. Influence of bending radius The amount of cold work or plastic straining applied during roller bending may affect the magnitude of the residual stresses. For steel exhibiting a clear hardening stage, a decrease of the bending ratio is expected to induce higher residual stresses. For each steel section as presented in Table 1, the numerically obtained maximum tensile and maximum compressive normalized residual stresses in the bottom flange are displayed as a function of the bending ratio in Fig. 5(left). It can bee seen that the bending ratio has no clear influence on the extreme residual stresses within the examined range of 12.5 ≤ R/h ≤ 39.79. Consequently a residual stress model can be developed which is independent of the bending radius, and applicable to a bending ratio range of approximately 10 ≤ R/h ≤ 40. Fig. 4. Normalized residual stresses in 18 roller bent wide flange sections.
−0.3fy and tensile stresses at the web-to-flange junction ranging from 0.4fy to 0.7fy . The web shows a large scatter when compared to the stress characteristics in the flanges. Although consistent trends from the web stresses cannot readily be observed, it can be concluded that the upper part of the web is in tension with an average maximum stress of approximately 0.3fy . The lower part of the web is mainly subject to compression with an average maximum stress of approximately −0.4fy . Steep stress gradients are observed at the web-to-flange junctions, as reflected by the large difference between the stress values in the top and bottom of the web and in the middle of the top flange and bottom flange respectively. In Table 1 it can be seen that the numerical calculations comprise a wide range of bending radii and steel grades. For the development of a single residual stress model the influence of the bending ratio and the yield stress of the original material on the numerical residual stresses were studied first. It was observed that the maximum tensile and compressive residual stresses from all 18
2.3. Influence of original yield stress The yield stress of the original material may also affect the magnitude of the residual stresses. Structural steel with a higher yield stress prior to roller bending may have higher residual stresses after bending when compared to steel with low yield stress levels. Residual stress measurements published by Gardner and Cruise [28] on press braked angles and cold-rolled box sections have shown a clear relationship between material yield stress and magnitude of residual stresses. Fig. 5(right) shows the maximum tensile and compressive residual stresses in the bottom flange versus the measured yield stress of the material before roller bending. The residual stress values are not normalized, in order to display the influence of the yield stress on the magnitude of the residual stress. 3. Roller bent residual stress model for wide flange sections Based on the numerically obtained roller bent residual stresses, a model of these stresses for wide flange steel sections is suggested. This proposal is intended to be generally applicable, independent
996
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
Fig. 5. Normalized maximum tensile and compressive residual stresses in the bottom flange versus bending ratio (left) and maximum tensile and compressive residual stresses in the bottom flange versus yield stress of original material (right).
(a) Dimensions.
(b) Residual stresses.
(c) Equilibrium of stress blocks in web.
Fig. 6. Proposed residual stress model for roller bent wide flange steel sections.
of the bending ratio for the range of 10 ≤ R/h ≤ 40 (Section 2.2) and linearly related to the magnitude of the original yield stress for S235 and S355 steel sections (Section 2.3). The magnitudes and pattern for the residual stress model are determined to best fit the finite element results. 3.1. Residual stress pattern and magnitudes The residual stress model is based on the numerically obtained residual stresses as shown in Fig. 4 and the trend lines as shown in Fig. 5 and is shown in Fig. 6. A symmetric bi-linear stress pattern along the half bottom flange is suggested with a maximum tensile stress value of 0.7fy at the web-to-flange junction and 0.35fy compression at the flange tip. These stress values are based on the gradient of the trend lines as shown in Fig. 5. A somewhat larger compressive stress value for the flange tips that shown in the trend line of Fig. 5 has been proposed, since numerical data could not be obtained at the flange tips but at a distance away from the ends (integration point location). Using the gradient of the trend line would therefore result in an underestimation of the compressive stresses at the flange tips. For the top flange a linear stress gradient is suggested based on a qualitative fit with numerical data, featured by 0.2fy tensile stresses and 0.2fy compressive stresses at the flange tips and flange center, respectively (Fig. 6(c)). For the web two triangular stress blocks are suggested. The tensile and compressive stress peak of the triangles are located at a distance of 1/4h0 and 3/4h0 from the web-to-flange junction of the top flange, respectively. The stress values are equal to zero at the top and bottom of the web. The maximum tensile and compressive residual stress values in the web are annotated by σwrt and σwrc
respectively and their magnitudes are governed by the internal equilibrium requirement. Equilibrium conditions for residual stresses consist of the axial force, major moment and minor moment equilibrium requirement as stated earlier by Lay and Ward [3] and Szalai and Papp [29]. Equilibrium about the minor bending axis is automatically satisfied due to the symmetric pattern of the suggested residual stress model about this axis. The bi-linear residual stress pattern in the bottom flange yields a net tensile force, which is balanced by residual stresses in the web, according to the equilibrium requirements. The two unknown web stresses, σwrt and σwrc , can be obtained from the two remaining equilibrium equations: the normal forces and the major bending moment requirement Fig. 6(c). The equilibrium equations are simplified by neglecting the fillets in the wide flange section. Axial force (N ) and major moment (M ) equilibrium equations can be set up by summing forces for all stress blocks and summing the product of these normal forces and their distances to the center of the top flange, as depicted in Fig. 6(c):
−
7
1
1
+ σwrt 8 4 1 1 + + σwrc fy = 0 8 4 − 7 1 1 1 5 M = btf h0 + tw h0 + h0 σwrt 80 8 6 4 12 1 7 1 5 + h0 + h0 σwrt fy = 0. N =
80
btf + tw h0
4
12
8
6
(1)
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
Eq. (1) can be simplified into
−
7
3
3
σwrt + σwrc fy = 0 8 − 1 7 1 h0 fy = 0, M = btf h0 + h0 tw σwrt + σwrc N =
80
btf + tw h0
8
80
8
(2)
4
and written in matrix notations:
N M
3 8 = fy h0 tw 1 h0
8
3 8 σwrt 1 σwrc
h0
− fy
4
7 btf 80
80
0 . 0
(3)
Solving (3) for σwrt and σwrc yields
σwrt =
7btf 30h0 tw
fy ,
σwrc = −
14btf 30h0 tw
fy .
flange width b or section height h. Due to placement requirements of the strain gauges, average values could not be obtained close to the junction of the flanges and web. Similarly to the numerical stress values, the experimental results were normalized with respect to the measured yield stress of the straight material (Table 1). The proposed residual stress model agrees well with the measured residual stress value for the HE 100A, HE 100B and HE 360B sections, although less coherence is found in the web of the IPE 360 series. 4.2. Comparison to numerical analyses
7 btf h0
=
997
(4)
With these values the residual stress model for roller bent wide flange steel sections is complete. 3.2. Residual stress model features From Eq. (4) it is clear that the requirement of internal equilibrium renders the proposed model to be different per section geometry. The residual stress values in the web are governed by the ratio between the area of the flange and area of the web. The proposed model is qualitatively identical for all wide flange sections but the magnitude of the residual stresses in the web is dependent on the geometry of the cross section. Fig. 4 shows that the numerical residual stress pattern is featured by a sharp stress gradient at the web-to-flange junctions of the roller bent wide flange sections. This sharp stress gradient has been approximated in the proposed residual stress model by a so-called ‘stress jump’ at the web-to-flange junctions (i.e. the stress value at the top of the web or bottom of the web is different from the stress value in the middle of the top flange or middle of the bottom flange, respectively). A stress jump is a stress change over an infinitesimal distance and can also be found in residual stress models for welded sections, [20] (Fig. 3(b)). Modeling the large stress gradients instead of the stress-jumps in the web-to-flange junctions in a residual stress model would significantly complicate the stress pattern in the web and improve its accuracy only marginally as the pertaining section areas are extremely small. The stress jump provides therefore a simplification to the residual stress model and enhances the simplicity of the equilibrium equations and ease of applicability when employed in numerical models. 4. Discussion 4.1. Comparison to experiments Residual stress measurements were performed on both straight and roller bent sections using the sectioning method. Measurements taken from both sides of the flanges and the web were used to obtain the average residual stress values. Residual stresses in HE 100A, HE 360B and IPE 360 sections were published earlier by Spoorenberg et al. [15]. A full overview of the averaged experimental results compared with the suggested residual stress model is shown in Fig. 7. The averaged experimental results for roller bent sections and the suggested residual stress model are plotted against the location over the section normalized by the
The proposed residual stress model is based on the finite element patterns and magnitudes from all wide flange sections as summarized in Fig. 4 and the trend lines shown in Fig. 5. However, in order to meet the internal equilibrium requirements, the suggested model will be different for all wide flange steel sections. The proposed residual stress model is therefore compared to the finite element residual stresses in all 4 different section types; see Fig. 8. A good correlation of results can be observed for the HE 100A, HE 100B and HE 360B series but larger discrepancies are found in the top flange of the IPE 360 series. 4.3. Comparison between existing hot-rolled and proposed roller bent residual stress models It was mentioned in Section 1.1.4 that compressive residual stresses in the flange tips are detrimental to the resistance of members susceptible to either flexural or lateral torsional buckling. Compressive residual stresses in the flange tips reduce the flexural stiffness during inelastic arch buckling and subsequently will cause early collapse of the arch. A hot-rolled residual stress model is featured by compressive stresses in all four flange tips, whereas the proposed model displays only compressive stresses in the tips of the bottom flange. From a qualitative comparison between the hot-rolled model and the proposed roller bent model it can be concluded that the proposed residual stress model for roller bent wide flange steel sections will be more favorable to the resistance of steel arches, when either failing by in-plane or out-of-plane inelastic buckling. 4.4. Range of applicability residual stress model Good coherence between the residual stress model and experimental results for various roller bent wide flange sections can be observed in Fig. 7. This allows the application of the residual stress model to other wide flange sections that did not make part of the experimental program. In addition, since the residual stress model can be expressed as a function of the yield stress of the straight material, it can be applied to other steel grades (e.g. steel grade S275). Under the assumption that for bending ratios R/h > 40 the required amount of cold work on the section will be marginally smaller, the reduction in residual stresses will be of minor influence. Application of the residual stress model to roller bent arches with larger bending ratios than currently investigated seems appropriate but is probably conservative. 5. Conclusions After manufacturing wide flange steel sections by the hotrolling process, additional cold working such as roller bending has a significant influence on the residual stress patterns. Earlier publications by Spoorenberg et al. [15,16] demonstrate a significant difference between residual stresses in hot-rolled and roller bent wide flange sections. In this paper residual stresses obtained from finite element analyses by Spoorenberg et al. [16] have been summarized and normalized with respect to the yield stress of the material before roller bending to develop and present
998
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
Fig. 7. Normalized measured averaged residual stresses and proposed residual stress model.
a residual stress model. It was found that the bending ratio does not influence the magnitudes of the residual stresses, allowing for a residual stress model representing all examined bending radii for 10 ≤ R/h ≤ 40. The residual stresses were found to be linearly dependent on the yield stress of the original material and a single normalized residual stress model could therefore represent all roller bent sections, for steel grades S235 and S355. Based on a best fit of the finite element data, the proposed linear stress gradient along the width of the top flange (the flange that is plastically elongated in longitudinal direction during roller bending) is featured by stress magnitudes of 0.2fy in tension and 0.2fy in compression at the flange tips and flange center respectively. The residual stress in the bottom flange (the flange that is plastically shortened in longitudinal direction during
roller bending) can be represented by a bi-linear pattern with a maximum compressive stress of 0.35fy at the flange tip, zero stress at the quarter points of the flange width and a maximum tension of 0.70fy at the web-to-flange junction. The residual stress pattern over the height of the web can be represented by two triangular stress blocks: tensile stress in the upper region of the web near the top flange and compressive stress in the lower region. The maximum value for the two zones can be determined from internal equilibrium which results in 0.2–0.4fy for tension and 0.4–0.8fy for compression residual stresses. The residual stress model was compared to experimental and numerical residual stress measurements and good agreement was observed. The suggested residual stress model can be implemented in finite element models with beam or shell elements to assess its influence
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
999
Fig. 8. Normalized computed residual stresses and proposed residual stress model.
on the inelastic buckling response of steel arches. In view of the detrimental influence of compressive residual stress at the flange tips it is expected that the proposed residual stress model will yield higher resistances compared to finite element analyses using models for hot-rolled residual stresses when investigating inelastic arch buckling. Acknowledgements This research was carried out under the project number MC1.06262 in the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl).
References [1] Beedle LS, Tall L. Basic column strength. Transactions of the ASCE 1962;127: 138–79. [2] Mas E, Massonet Ch. Part prise par la belgique dans les recherches experimentales de la convention europeenne des associations de la construction metallique sur le flambement centriques des barres en acier doux. Acier-StahlSteel 1966;9:393–400. [3] Lay MG, Ward R. Residual stresses in steel sections. Journal of the Australian Institute of Steel Construction 1969;3(3):2–21. [4] Daddi I, Mazzolani FM. Determinazione sperimentale delle imperfezioni strutturali nei profilati di acciaio: Universita degli studi di Napoli istituto di tecnica delle costruzioni; 1971. [5] Young BW. Residual stresses in hot rolled members. IABSE reports of the working commissions 1975;23:25–38. [6] Mazzolani FM. Buckling curves of hot-rolled steel shapes with structural imperfections. IABSE reports of the working commissions 1975;23:152–61.
1000
R.C. Spoorenberg et al. / Journal of Constructional Steel Research 67 (2011) 992–1000
[7] ECCS. Manual on stability of steel structures - ECCS Committee 8 Stability: 1976. [8] Tall L, Alpsten G. On the scatter in yield strength and residual stresses in steel members. IABSE reports of the working commissions 1969;4:151–63. [9] Tebedge N, Alpsten G, Tall L. Residual-stress measurement by the sectioning method. Experimental Mechanics 1973;13:88–96. [10] Beg D, Hladnik L. Eigenspannungen bei geschweissten I-profilen aus hochfesten Stahlen. Stahlbau 1994;63(5):134–9. [11] Young BW, Robinson KW. Buckling of axially loaded welded steel columns. The Structural Engineer 1975;53(5):203–7. [12] Chernenko DE, Kennedy DJL. An analysis of the performance of welded wide flange columns. Canadian Journal of Civil Engineering 1991;18(14):537–54. [13] Alpsten G. Residual stresses, yield stress, and the column strength of hotrolled and roller-straightened steel shapes. IABSE reports of the working commissions 1975;23:39–59. [14] Bjorhovde R. Cold bending of wide-flange shapes for construction. Engineering Journal 2006;43(4):271–86. [15] Spoorenberg RC, Snijder HH, Hoenderkamp JCD. Experimental investigation of residual stresses in roller bent wide flange steel sections. Journal of Constructional Steel Research 2010;66(6):737–47. [16] Spoorenberg RC, Snijder HH, Hoenderkamp JCD. Finite element simulations of residual stresses in roller bent wide flange sections. Journal of Constructional Steel Research 2011;67(1):39–50. [17] Pi Y-L, Trahair NS. In-plane inelastic buckling and strengths of steel arches. Journal of Structural Engineering ASCE 1996;122(7):734–47.
[18] Pi Y-L, Bradford MA, Tin-Loi F. In-plane strength of steel arches. Advanced Steel Construction 2008;4(4):306–22. [19] Komatsu S, Sakimoto T. Ultimate load carrying capacity of steel arches. Journal of the Structural Division 1977;103(12):2323–36. [20] Sakimoto T, Komatsu S. Ultimate strength formula for central-arch-girder bridges. Proc Japan Soc Civ Engrs 1983;333:183–6. [21] Pi Y-L, Trahair NS. Out-of-plane inelastic buckling and strength of steel arches. Journal of Structural Engineering ASCE 1998;124(2):174–83. [22] Pi Y-L, Bradford MA. Elasto-plastic buckling and postbuckling of arches subjected to a central load. Computers & Structures 2003;81(8):1811–25. [23] Pi Y-L, Bradford MA. Out-of-plane strength design of fixed steel I-section arches. Journal of Structural Engineering ASCE 2005;131(4):560–8. [24] Timoshenko SP. Strength of materials. Part II advanced theory and problems. D. Van Nostrand Company, Inc.; 1940. [25] King C, Brown D. Design of Curved Steel: The Steel Construction Institute; 2001. [26] NEN-EN 10002-1. Metalen - Trekproef - Deel 1: Beproevingsmethoden bij omgevingstemperatuur: Nederlands Normalisatie Instituut; 2001. [27] EN 10025. Hot rolled products of non-alloy structural steels. Technical delivery conditions.: CEN European Committee Standardization; 1993. [28] Gardner L, Cruise RB. Modeling of residual stresses in structural stainless steel sections. Journal of Structural Engineering ASCE 2009;135(1):42–53. [29] Szalai J, Papp F. A new residual stress distribution for hot-rolled I-shaped sections. Journal of Constructional Steel Research 2005;61(6):845–61.