Residual Stress Pattern of Stainless Steel SHS

Residual Stress Pattern of Stainless Steel SHS M. Jandera & J. Machacek Czech Technical University in Prague, Czech Republic ABSTRACT: The investigation is focused on cold rolled square hollow sections made of stainless steel 1.4301. Current research at the Czech Technical University in Prague embraces experimental investigation of residual stresses induced by forming process of the sections determined by the sectioning method which is described in detail. Patterns of stress distribution along the section are generalized and suitable predictive formulas for general use developed. The patterns were finally used in a FEM model to show possible influence of each part of the residual stresses (especially of longitudinal and transversal bending ones) on compressive carrying capacity of the sections subjected to local and global buckling. The numerical model originates from ABAQUS software and includes the distribution of membrane and bending residual stresses in both longitudinal and transverse member directions, a non-linear stress-strain diagram and enhanced strength corner properties to obtain a relevant credibility of the solution. The parametric study of the influence of residual stresses on the stub column strengths is shown and assessed for all substantial parameters such as web plate slenderness, column slenderness and the nonlinearity parameter of Ramberg-Osgood formula. 1 INTRODUCTION 1.1 Material and sections Stainless steel is comparatively a new material in structural applications. Nevertheless, steadily expanding use may be seen in last two decades while an extensive research has shown notable beneficial structural properties (Gardner 2005). Up to recently the most common stainless steel is austenitic Grade EN 1.4301, which was also used in all tests and as a reference material for this study. However, great progress may be seen in other austenitic and duplex grades and also in development of new ones with low nickel content known as lean stainless steels. These steels may offer superior mechanical properties at lower cost and are currently being introduced into the design codes (Theofanous & Gardner 2009). 1.2 Residual stresses The forming process of thin-walled structures induces residual stresses which may have a significant effect on structural behaviour. Residual stresses in cold-formed sections are generally expected to have substantial bending part and comparatively low membrane part, and hereby being the opposite of thermally induced residual stresses in welded or hot-rolled sections (Rasmussen & Hancock 1988). This was also confirmed for residual stresses in stainless steel sections via experimental investigation of members in compression and bending (Rasmussen & Hancock 1993a, Rasmussen & Hancock 1993b) where large longitudinal bending residual stresses were found. For carbon square hollow sections (SHS) the residual stresses were measured by Key & Hancock (1993) and a model of their distribution in both directions was proposed. A recent experimental program concerning X-ray diffraction method measurements on SHS were published by Lia et al. (2009) showing through-thickness stress distribution in both directions. Nevertheless, no model was proposed and the longitudinal membrane residual stresses in tension found around the whole section were not qualified. Lately the residual stresses of stainless steel SHS were measured by Young & Lui (2005) using the sectioning method for longitudinal residual stresses investigation, and by Cruise & Gardner (2008), where similar procedure was used for extensive research and modelling of longitudinal bending stress distribution proposed also for press-braked and hotrolled angles. Nevertheless, the experimental methods are not the only possibility of residual stress determination and numerical simulations are also conveniently used for stainless steel cold-formed sections (Quach

et al. 2009a, Quach et al. 2009b, Rossi et al. 2009). But for hollow sections, where the forming process is more complicated and contains more stages no numerical model was published yet as far as authors are informed. 2 RESIDUAL STRESS DISTRIBUTION 2.1 Selected sections and material characteristics In the experimental part, two cross-sections were investigated, SHS 100x100x3 and SHS 120x120x4, all with a longitudinal weld. For each type of the sections three specimens were made for measurement of residual stresses on one web only. One of the investigated webs was always the one with the weld, the other one next to the web with the weld and the third one the one facing the web with the weld. So, all types of webs were considered. Coupon tensile tests were carried out for each type of section for flat part (taken always from the centre of the web) and corner part as well. Also, a stress relieved specimen (annealing at 650 C) for flat part was made for each type of section. Tensile material characteristics are summarised in Table 1, where E 0 is the initial tangent (Young s) modulus, σ 0.2 and σ 1.0 are the 0.2% and 1.0% proof strengths respectively, ε pu is the plastic ultimate tensile strength and n and n 1.0 are strain hardening exponents for the compound Ramberg Osgood material model described by Gardner & Ashraf (2006). Table 1. Measured material properties. Section E 0 σ 0.2 σ 1.0 σ u ε pu n n 1.0 MPa MPa MPa MPa - - -_ 100x100x3-F 205 750 417 457 753 0,60 7,1 2,3 100x100x3-FA 211 500 429 456 753 0,65 13,4 1,5 100x100x3-C 202 000 623 720 816 0,27 6,1 3,3 120x120x4-F 192 000 429 479 783 0,68 4,3 2,7 120x120x4-FA 205 500 405 441 762 0,74 8,1 2,1 120x120x4-C 212 500 522 617 745 0,37 5,4 3,0 F as-delivered flat part FA stress relieved flat part C as-delivered corner part 2.2 Measurement procedure For the strain record, resistance strain gauges were used on both surfaces of the web (with measuring grid 5x10 mm). To avoid potential damage of the strain-gauges, silicone coating was used. For 120 mm cross-section five web sections were cut out in the longitudinal direction and one in the transverse one, while for 100 mm cross-section only four in the longitudinal and one in the transverse direction (Figure 1). The width of the longitudinal section was at least 20 mm with length of 110 mm and that of transverse section was 30 mm with length equal to the whole web width. These conservatively large dimensions were taken to ensure the highest possible accuracy of results. At first, the strain gauges were attached to the outer surface of measured web followed by the initial readings (always five times for each strain gauge with two minutes brakes to avoid any mistake in readings), Figure 1. Subsequently, an opening was cut out in the web facing the measured one, strain gauges were attached to the inner surface of the measured web (Figure 2) and readings were taken for all strain gauges. The measured strain on the outer surface was attributed to the residual stress redistribution after the facing web opening. In the longitudinal direction the residual stresses were considered as pure membrane stress and in the transverse direction as pure bending one. This assumption is not so important in view of the fact that the magnitude of the redistributed stresses is very low, nearly negligible (typically less then 2 MPa). Figure 1. Specimen prepared for the sectioning method with strain gauges set on the outer surface and a strain-gauge for compensation of temperature on separate section. Figure 2. Cut out opening in the web facing the measured one and configuration of strain gauges on the inner surface. Figure 3. Sectioning of a specimen.

Finally the specimen was divided into the sections by a cooled thin cutting-disc (Figure 3). The depth of each turn of a cut didn t exceeded 0,4 mm. This procedure kept the temperature of the material safely under 100 C. Once the specimen was divided (Figure 4) and the temperature returned back to the room temperature, the final readings were taken on the strain gauges and residual stresses calculated in two parts membrane and bending in both longitudinal and transverse direction. For transverse membrane stresses approximately zero magnitude was obtained, which is correct with respect to the evident equilibrium condition. For calculation of stresses from the recorded strain the initial elastic modulus E 0 for the material of web (Table 1, 100x100x3-F and 120x120x4-F respectively) was used and for bending residual stresses plastic (block-like) distribution was assumed. This simplification was considered also by Cruise and Gardner (2008) with respect to large magnitudes of bending stresses and subsequently by Jandera et al. (2008) where measurement by X-ray diffraction method was carried out and the stresses seemed to be relatively constant through the half of the web thickness. σ m = (-0,253+1,483(x-x 2 )) σ 0.2 (1) σ b.pl = (0,833+1,866(x-x 2 )) σ 0.2 (2) where x means relative distance along the web width (x=0 and x=1 for the edges of the flat web, x=0,5 for the centre of the web). The models for longitudinal stresses are shown in Figures 5 and 6, where 95% predictive intervals and the measurements are also demonstrated. Figure 5. Membrane residual stress measurements and proposed predictive model (related to the web yield strength) across a web of SHS. Figure 4. A sectioned web of a specimen. 2.3 Proposed model According to these results models of residual stresses distribution were proposed. Both, the model and the results are always related to the 0.2 % proof strength σ 0.2 of material at the centre of the web. For membrane stresses the positive sign marks tension and negative ones compression. For bending residual stresses all values are related to the outer surface. Then positive values mark tension on the outer surface and compression on the inner one. For the distribution along the web a quadratic function was proposed according to the least squares. The procedure resulted into equation (1) for longitudinal membrane residual stresses (σ m ) and equation (2) for longitudinal bending residual stresses (σ pl.b ): Figure 6. Bending residual stress measurements and proposed predictive model (related to the web yield strength) across a web of SHS, where plastic distribution through thickness is assumed. The positive sign (tensile stresses) is related to the outer surface. The predictive formulas demonstrate good agreement with the experiments (coefficient of determination is R 2 = 0,893 for membrane stresses and R 2 = 0,828 for bending stresses). The same formula is used for webs with as well as without weld despite slight difference was monitored. For the transverse residual stresses, where only one strain gauge at the web center was used, uniform distribution along the web is assumed. The transverse bending residual stresses may be taken as: σ b.pl.t = -0,376 σ 0.2 (3)

3 FE MODEL 3.1 Introduction The proposed model of residual stress distribution was subsequently introduced into a FE model, which was successfully validated previously (Jandera et al. 2008) and where a detailed description of the model is available. The parametric studies of influence of residual stresses were divided into two parts: one focused on influence of residual stresses on global column buckling where a pinned column was modelled and the other focused on local web buckling, which was represented by a stub column model. The boundary conditions are shown in Figure 7. All: the longitudinal membrane and bending as well as transverse bending stresses, Max. all: the longitudinal membrane and bending stresses as well as transverse bending stresses, the longitudinal bending residual stresses were taken as the upper bound of the 95% predictive interval. Bending residual stresses in corners, where no measurement was carried out, were neglected in the model. As was shown previously by Cruise & Gardner (2008), the bending residual stresses in corners are low. This, together with increased strength in the corner area implies that their effect is insignificant. Longitudinal membrane residual stresses in corners were always calculated from condition of equilibrium over the all cross-section. Their magnitudes were very low, as was shown by Cruise & Gardner (2008) or in the patterns proposed by Key & Hancock (1993) for carbon steel SHS. 4 PARAMETRIC STUDY BASED ON SHS 120x120x4 Figure 7. FE models of stub and long column used for the parametric studies. 3.2 Initial geometric imperfections Initial deflections were considered in the shape of local buckling with the amplitudes measured on the specimens before tests (see Jandera et al. 2008). These web imperfections were introduced in models of stubs as well as long columns. Global initial column deflections were introduced in model of the long column in shape of the lowest global buckling eigenmode with amplitude of L/2000, where L is the column length. This value was proposed by Gardner (2002) as an average value, lately confirmed by Cruise & Gardner (2006). 3.3 Residual stresses The bending residual stresses were introduced in 6 integration points with quadratic integration trough the thickness. The proposed residual stresses pattern was introduced in five stages: Membrane: the longitudinal membrane stresses only, Longitudinal: the longitudinal membrane and bending stresses, Max. longitudinal: the longitudinal membrane and bending stresses, which were taken by the upper bound of the 95% predictive interval, 4.1 Introduction The study was based on sections SHS 120x120x4 tested at the Czech Technical University in Prague where the geometry and material characteristics were measured. The material characteristics for the corner were taken as measured. For flat parts the influence of bending residual stresses was removed with help of an analytical model. The resulting stress-strain diagram corresponded to the stressrelieved material diagram (Jandera 2009). The modelled corner area was assumed as sum of the own corner area extended for flat web parts considered as twice the web thickness (proposed for SHS by Gardner (2002) and Cruise (2007)). 4.2 Influence of residual stresses on global buckling By analysing column models of varying length, the influence of bending and membrane residual stresses on global buckling capacity over a range of slendernesses was assessed. The results of the study are presented in Figure 8. For non-dimensional slendernesses λ (defined as the square root of the ratio between yield load and elastic column buckling load) up to 1.3, the residual stresses may be seen to have a positive influence on load-carrying capacity. Beyond this slenderness, a negative influence is evident. Over the investigated range of slenderness, inclusion of residual stresses causes a variation in resistance between -16 % and +10 % and -20 % to + 14 % if the upper bound of the 95% predictive interval for the longitudinal bending residual stresses is considered.

The influence of membrane residual stresses was negligible for the whole employed slenderness range. Figure 8. Parametric study of residual stress influence on long column load-carrying capacity. Figure 9. Stress-strain relationship (left) and tangent modulus (right) of material with and without longitudinal bending residual stresses valid for the average web value of SHS 120x120x4. Figure 10. Load-strain diagrams of long columns. The variation of resistance results principally from the effect of the bending residual stresses on the non-linearity of the stress-strain curve. A positive influence of residual stresses arises when column failure strains coincide with a region of increased tangent modulus. This is illustrated for longitudinal bending residual stresses in Figure 9 where the material response with and without longitudinal bending residual stresses is depicted (the average magnitude of residual stresses for the web was employed). The material stress-strain curve containing residual stresses may be seen to be consistently below the residual stress free curve (i.e. the secant modulus is always lower). However, this is not valid for the tangent modulus, which is known to be fundamental in controlling column buckling resistance. Below approximately 0.12% strain, the tangent modulus of the stress free curve is higher than that of the residual stresses containing curve. Conversely, for higher strains, the reverse is true. From Figure 10, where load-strain curves of columns are printed (strain denotes the axial deformation to the column length ratio, similarly in Fig. 12), follows that the failure strains of columns of slendernesses up to 1.3 reach more than 0.12%. That s the range, where the longitudinal bending residual stresses were found to have positive influence on the tangential modulus. For higher column slenderness ( λ > 1.3), lower strains are reached at ultimate load. Therefore the tangent modulus for the material where bending residual stresses were included is lower than the tangent modulus of material without residual stresses (Figure 9), and thus longitudinal bending residual stresses were found to lead to a reduction in loadcarrying capacity. The magnitude of residual stresses clearly influences the variation in load-carrying capacity, and it was found that taking mean longitudinal bending residual stress values rather than the upper bound of the predictive interval, sensitivity of the column response was nearly halved for the most sensitive slendernesses. For the large slendernesses, the difference was lower. The transverse bending residual stress was found to emphasize the effect of the longitudinal one and having also significant effect. 4.3 Influence of residual stresses on local buckling The influence of residual stresses on local buckling capacity was assessed in a similar manner to the above studied global buckling. Stub column models of varying local plate slenderness λ p (defined as the square root of the yield load to the elastic local buckling load of the plate elements) with and without residual stresses were examined. The results are shown in Figure 11. The maximum influence of residual stresses in terms of load-carrying capacity was 9% and 11 % if the upper bound of the 95% predictive interval for the longitudinal bending residual stresses was considered. Although slightly less sensitive influence than in the column buckling results, similar conclusions can be drawn. The influence of membrane residual stresses was also found to be insignificant in comparison to the influence of the bending components.

However, the main difference is that for the local buckling no negative influence of residual stresses occurred. This is due to the post-buckling behaviour which increases the failure strain for very slender webs. Opposite to post-buckling behaviour of columns, the local buckling failure strain was always greater than 0.12%, where residual stresses have the positive influence on tangential modulus. The highest effect of bending residual stresses was monitored for the middle slendernesses ( λ p = 1.0 to 1.3) where the sensitivity of columns is large and the difference of the tangential modulus at failure strain between material with and without bending stresses is the highest. n σ σ ε = + 0.002 (4) E0 σ 0.2 where ε is strain and σ stress. The 0.2% proof strength was taken according to Eurocode 1993-1-4 where σ = 230 MPa for Grade 1.4301. The hardening exponents n was taken as: n = 4, which represents cold-formed austenitic steels, n = 6, which represents annealed austenitic steels, n = 16, which represents the lowest nonlinearity of stainless steels, n =, in fact bilinear stress-strain diagram which represents common carbon steels. The initial geometric imperfections and residual stresses were employed as above. 5.2 Influence of residual stresses on global buckling The parametric study was carried out for range of column slenderness λ = 0.8 to 1.8 (Jandera 2009) and only results of two the most representative slendernesses λ = 1.0 and 1.8 are shown here (Figure 13 and 14). Figure 11. Parametric study of residual stress influence on stub column load-carrying capacity. Figure 13. Influence of residual stresses on load-carrying capacity of long column with λ =1,0 according to Ramberg- Osgood factor n. Figure 12. Load-strain diagrams of stub columns. 5 PARAMETRIC STUDY ON INFLUENCE OF RAMBERG-OSGOOD STRAIN HARDENING EXPONENT 5.1 Introduction The second parametric study was made in order to explain the difference of influence of residual stresses in materials of various nonlinearities. For the simplicity, in all models only the one-stage Ramberg-Osgood diagram (4) was used for the whole area of cross-section: From the results follows that that the residual stresses for the nonlinear material (n = 4 to 6) may increase the load-carrying capacity for the middle slenderness ( λ = 1.0) but for high slenderness ( λ = 1.8) the opposite is true. For materials with lower nonlinearity of stressstrain diagram, especially for the bilinear stressstrain diagram (e.g. carbon steel), the inclusion of residual stresses cause always decrease in the load capacity as generally known and numerically confirmed for SHS by Key & Hancock (1993). This fact is due to a significant drop in stiffness of materials which offers none or low strain hardening beyond yield strength (e.g. Jandera 2009). The influence of membrane component of residual stresses is always low.

Figure 14. Influence of residual stresses on load-carrying capacity of long column with λ =1,8 according to Ramberg- Osgood factor n. The magnitude of the influence of residual stresses is, however, rather overestimated in this study due to the simplification of material characteristics. The increased strength of material in the corner area as well as slightly different stress-strain diagram (better represented by compound Ramberg-Osgood diagram, Gardner & Ashraf 2006) may suppress the residual stress influence significantly. 5.3 Influence of residual stresses on local buckling Similar study on local buckling showed similar results (for λ p = 1.0 and 1.8 see Figures 15 and 16). Figure 15. Influence of residual stresses on load-carrying capacity of stub column with λ p =1,0 according to Ramberg- Osgood factor n. As shown in the previous study based on real material characteristics, for the nonlinear stress-strain diagrams the effect of residual stresses is always positive, whereas for the bilinear material diagram always negative. The influence of membrane residual stresses was always negligible. Figure 16. Influence of residual stresses on load-carrying capacity of stub column with λ p =1,8 according to Ramberg- Osgood factor n. 6 CONCLUSIONS An experimental and numerical investigation of residual stress distribution and its influence on column behaviour of structural stainless steel hollow sections is described. Sectioning method was used for the residual stress patterns measurement and generalized predictive formulas of distribution of all residual stress components (membrane and bending) along the web of SHS were determined with high correlation of the formulas with tests. Extensive numerical parametric studies based on received results and using geometrically and materially non-linear FE analysis were performed. The investigation concerned long and stub columns to determine the influence of residual stresses on global and local buckling. Paradoxically, it was found that inclusion of residual stresses may lead to an increase in loadcarrying capacity. This was attributed principally to the influence of the bending residual stresses on the material stress-strain curve. It was found that despite the secant modulus being consistently reduced in the presence of residual stresses, the tangent modulus was increased in some regions of the stress strain curve. For cases where column failure strains coincided with these increased tangent modulus regions (which was the whole slenderness range for local buckling and the slenderness in vicinity of λ = 1.0 for global buckling), higher buckling loads resulted. For the real cross-sections, where measured material characteristics were employed, the influence of residual stresses were between +10 % to -16 % for global buckling and up to +9 % for local buckling (plus means higher load-carrying capacity). Parametric study with varying Ramberg-Osgood strain hardening factor n showed that inclusion of residual stress pattern may lead to an increase of the load-carrying capacity of compressed members having non-linear stress-strain diagram, but always to

decrease of capacity for materials with bilinear stress-strain diagram (e.g. for common carbon steel). Although the behaviour of stainless steel columns with and without bending residual stresses has been investigated in this study, it should be noted that these stresses will inherently be present in the stress strain behaviour of material extracted from structural sections and, therefore, need not generally be explicitly re-introduced into numerical models. The influence of membrane residual stresses in these sections is usually very low and may be neglected in most of analyses. ACKNOWLEDGEMENTS The financial support of the Czech Ministry of Education (Grant MSM 6840770003) is gratefully acknowledged. REFERENCES Cruise, R.B. 2007. The influence of production route on the response of structural stainless steel members. Ph.D. thesis. Department of Civil and Environmental Engineering, Imperial College London. Cruise, R.B. & Gardner, L. 2006. Measurement and prediction of geometric imperfections in structural stainless steel members. Structural Engineering and Mechanics 24(1): 63 89. Cruise, R.B. & Gardner, L. 2008. Residual stress analysis of structural stainless steel sections. Journal of Constructional Steel Research 64(3): 352-366. Gardner, L. 2002. A new approach to structural stainless steel design. Ph.D. thesis. Department of Civil and Environmental Engineering, Imperial College London. Gardner, L. 2005. The use of stainless steel in structures. Progress in Structural Engineering and Materials 7(2): 45 55. Gardner, L. & Ashraf, M. 2006. Structural design for nonlinear metallic materials. Engineering Structures 28(6): 926 934. Gardner, L. & Nethercot, D.A. 2004a. Experiments on stainless steel hollow sections - Part 1: Material and crosssectional behaviour. Journal of Constructional Steel Research 60(9): 1291 1318. Gardner, L. & Nethercot, D.A. 2004b. Experiments on stainless steel hollow sections - Part 2: Member behaviour of columns and beams. Journal of Constructional Steel Research 60(9): 1319 1332. Jandera, M., Gardner, L. & Machacek, J. 2008, Residual stresses in cold-rolled stainless steel hollow sections. Journal of Constructional Steel Research 64(11): 1255-1263. Jandera, M. 2009, Residual stresses in stainless steel box sections. Ph.D. thesis, Faculty of Civil Engineering, Czech Technical University in Prague (in Czech). Key, W. & Hancock, G.J. 1993. A theoretical investigation of the column behaviour of cold-formed square hollow sections, Thin-walled Structures 16: 31 64. Li, S.H., Zeng, G., Mac, Y.F., Guo, Y.J. & Lai X.M. 2009. Residual stresses in roll-formed square hollow sections. Thin- Walled Structures 47(5): 505-513. Quach, W.M., Teng, J.G. & Chung, K.F. 2009a. Residual stresses in press-braked stainless steel sections, I: Coiling and uncoiling of sheets. Journal of Constructional Steel Research 68(8-9): 1803-1815. Quach, W.M., Teng, J.G. & Chung, K.F. 2009b. Residual stresses in press-braked stainless steel sections, II: Pressbraking operations. Journal of Constructional Steel Research 68(8-9): 1816-1826. Rasmussen, K.J.R. & Hancock, G.J. 1988. Deformations and residual stresses induced in channel section columns by presetting and welding, Journal of Constructional Steel Research 11(3): 175 204. Rasmussen, K.J.R. & Hancock, G.J. 1993a. Design of coldformed stainless steel tubular members, I: Columns. Journal of Structural Engineering 119(8): 2349 2367. Rasmussen, K.J.R. & Hancock, G.J. 1993a. Design of coldformed stainless steel tubular members, II: Beams. Journal of Structural Engineering 119(8): 2368 2386. Rossi, B., Degée, H. & Pascon, F. 2009. Enhanced mechanical properties after cold process of fabrication of non-linear metallic profiles. Thin-Walled Structures 47(12): 1575-1589. Theofanous, M. & Gardner, L. 2009. Testing and numerical modelling of lean duplex stainless steel hollow section columns. Engineering Structures 31(12): 3047-3058. Young, B. & Lui, Y. 2005. Behaviour of cold-formed high strength stainless steel sections. Journal of Structural Engineering 131(11): 1738-1745.