On the Stability and Strength of Steel Columns Affected by Distortional Buckling

Transcription

1 PAPER NUMBER: On the Stability and Strength of Steel Columns Affected by Distortional Buckling E. Batista 1, D. Camotim, L. C. Prola and E. Vaz 1 1 COPPE, Federal University of Rio de Janeiro, Caixa Postal 5, Rio de Janeiro, RJ, BRASIL Department of Civil Engineering, Technical University of Lisbon, Av. Rovisco Pais, 19 Lisboa Codex, PORTUGAL ABSTRACT The results of an investigation concerning the effect of distortional buckling on the behaviour of thin-walled cold-formed steel columns displaying rack sections are presented. A parametric study is first performed to study the variation of the critical stress and characteristics of the associated buckling mode with (i) the column length and (ii) the thickness and relative widths of the plates forming the cross-section. In particular, cross-sectional proportions leading to the occurrence of a critical distortional buckling mode are identified. The linear stability results were used to define an experimental test program, which is presently under way and is reported next. The experimental set-up and test procedure are briefly described and the test results are displayed and discussed. These results deal with (i) the observation and characterization of the distortional buckling mode, (ii) the estimation of the postbuckling reserve associated to this mode and (iii) the determination of column ultimate strength values. KEYWORDS Cold-formed columns, thin-walled sections, rack sections, stability analysis, local buckling, local plate buckling, distortional buckling, experimental analysis. INTRODUCTION In order to evaluate the structural efficiency of thin-walled cold-formed steel structural elements, namely axially compressed columns, it is rather important to investigate their local and global buckling behaviour. Although it is clearly established that the local buckling behaviour of most cold-formed columns may be conditioned by the occurrence of either local plate or distortional buckling modes, an important difference still exists on the amount of information available about each of these buckling modes. While the local plate mode is rather well understood, there is a comparatively smaller number of results, both analytical and experimental, dealing with the behaviour associated to the distortional mode. Moreover, only very few of the available results can be used for design purposes (Lau and Hancock, 197, and Hancock et al., 199). The objective of this paper is to present analytical and experimental results concerning the behaviour of axially compressed columns displaying a cross-section designated as "rack" section (traditionally used in storage racks). This section is represented in figure 1 and may be pictured as a C-section with flanges

2 stiffened by L-shaped stiffeners (stiffeners with two lips). In particular, it is intended to study, characterize and present experimental evidence of the behaviour associated to the occurrence of the distortional mode. w b t b3 w3 w1 b1 w b Figure 1: "Rack" section geometry Initially, a parametric study is performed to investigate the variation of the critical stress and characteristics of the associated buckling mode with (i) the column length and (ii) the thickness and relative widths of the plates forming the cross-section. Both local (local plate and distortional) and global (flexural-torsional) buckling modes are dealt with and the stability analysis is performed by means of a computer program based on the finite strip method (Prola and Camotim, 1995). First, assuming buckling modes with only one halfwavelength, the relevant buckling mode configurations are identified and the length values associated to the transition between local and global critical modes are estimated. Then, for lengths corresponding to local critical modes, the cross-sectional proportions (particularly the width of the horizontal stiffener lips, b ) for which the distortional mode is associated to the critical local stress are identified and characterized. Next, the experimental test program presently under way at COPPE- Federal University of Rio de Janeiro is reported. The rack columns were pinned at both ends and three different cross-sections geometries were considered, all designed, on the basis of the the parametric study mentioned before, to display a critical distortional buckling mode. The experimental set-up, test procedure and data processing are briefly described and the test results already obtained are displayed and discussed. These results deal mainly with (i) observing and characterizing the distortional buckling mode, (ii) estimating the postbuckling reserve associated to this buckling mode and (iii) determining the columns ultimate strength. STABILITY ANALYSIS The results of a parametric study dealing with the buckling behaviour of simply supported axially compressed columns with "rack" sections are presented. The main objective of this study is to investigate the combined influence of (i) the scaled column length (a ), (ii) the plate slenderness (b 1 /t) and (iii) the scaled flanges and stiffeners widths (b, b 3 and b ) on the nature of the local critical buckling mode and value of the corresponding bifurcation stress and, in particular, to identify the values of the stiffener lip widths b corresponding to the transition from a critical local plate buckling mode to a distortional one. Bifurcation Stress The two curves shown in figure illustrate the different types of variation of the buckling coefficient k b, related the lower bifurcation stress σ b by σ b = (k b π E)/[1 (1 - ν ) (b 1 /t) ], (1) with the logarithm of the scaled column length a (E and ν are the material Young's modulus and Poisson's ratio), for two different plate slenderness b 1 /t values and assuming a buckling mode with only one halfwavelength. Each curve displays two local minima, the first one corresponding to a local plate mode (LPM -

3 see figure 3(a)) and the second one to a distortional mode involving both the flanges and stiffeners (DM - see figure 3(b)). Depending on the value of b 1 /t, the critical local mode may either be the LPM or the DM. The corresponding stress value is designated by σ crl (buckling coefficient k crl ). For slender columns (large lengths), the critical buckling mode is a global flexural-torsional one (FTM - see figure 3(c)). In order to estimate the length values associated to the transition between local and global critical modes, it is necessary to consider the influence of the number of half-wavelengths present in the buckling mode. These length values, designated by a LG, are indicated in figure. For column lengths lower than a LG, one has σ cr = σ crl, the critical buckling mode is local and may display more than one half-wavelength. On the other hand, for column lengths higher than a LG, σ cr < σ crl, the critical buckling mode is global and displays one half-wavelength. Column lengths in the vicinity of a LP lead to interaction between local and global modes. b 1 /t=5 b 1 /t= Kb Kb K crl K crl, a a LG (a), a a LG (b) Figure : Variation of k b with a and b 1 /t (b =.5; b 3 =.; b =.3): (a) b 1 /t=5 (b) b 1 /t= (a) (b) (c) Figure 3: Column buckling modes: (a) LPM (b) DM (c) FTM Local Critical Bifurcation Stress It is now intended to study the combined influence of b 1 /t, b, b 3 and b on the critical value of the local buckling coefficient k crl and related buckling mode shape. The different parameter values considered take into account the cold-formed sections commercially produced in Brazil and are b 1 /t= 5; 5;, b =.5;.;.7, b 3 =.15;.;.5 and a continuous variation of b ( b.). Figure shows the combined influence of b 1 /t, b 3 and b on the values of k b and nature of the local critical mode. The solid and dashed lines stand for the values of k b associated to the LP and D modes, respectively. It is shown that, for a sufficiently high b value, the critical buckling mode always changes from LP to D. This is due to the fact that, in a sense, one may say that both the flanges and the horizontal stiffener lips are stiffened by the vertical stiffener lips and, therefore, a large enough b value always triggers the occurrence of the DM. The curves displayed in figures (a) - (c) enable the estimation of the maximum allowable stiffener lips width b in order to prevent the occurrence of a critical distortional mode (designated as (b ) tr from here on). It is clearly observed that the (b ) tr value increases with both

4 the plate slenderness b 1 /t and the vertical stiffener lips width b 3. It should be noticed that, for b 1 /t= and b 3 =.5, the critical mode is always LP within the range of b/b1 values considered (see figure (c)) b 1 /t=5 b 1 /t= Kb Kb Kb b 1 /t= b1/t = 5 b1/t= 5 b1/t = K b K b K b b 3 = b 3 = b 3 = b b (a) (b) (c) Figure : Variation of k crl with b 1 /t, b 3 and b (b =.) (a) b 1 /t=5 (b) b 1 /t=5 (c) b 1 /t= The aim of figure 5 is to illustrate the combined influence of b, b 3 and b on the values of k b and nature of the local critical mode. The solid and dashed lines stand again for LP and D modes. As before, it is observed that the critical mode always changes from LP to D for a sufficiently high b value. The value of (b ) tr increases with the vertical stiffener lips width b 3 and decreases with the flanges width b (a wide flange obviously makes the cross-section more prone to distortional buckling). It should be noticed that, for b =.7 and b 3 =.15, the critical mode is always D within the range of b/b1 values considered (see figure 5(c)). The corresponding curve displays a local maximum at about b =.5. b b/ b 1= =..5 5 b/ b 1=. b =. b/ b 1= /b 1 =..7 7 Kb Kb Kb K cr l K cr l K cr l b 3 = b 3 =. 15 b. 3 = b (a) (b) (c) Figure 5: Variation of k crl with b, b 3 and b (b 1 /t=5) (a) b =.5 (b) b =. (c) b =.7 Finally, it is worth mentioning that another type of distortional buckling mode, which involves only the stiffeners, may occur for rack sections with other geometries (Camotim and Prola, 199). Such a mode may become critical for high b 3 and low b values (the horizontal stiffener lips are unable to prevent the displacement of the stiffener junctions). b b Transition between Critical Local Plate and Distortional Modes Figures and 5 show that (i) the values of the plate slenderness b 1 /t and the horizontal stiffener lips width b must be high enough to guarantee that the LPM is the local critical buckling mode and that, for given

5 b 1 /t and b 3 values, (ii) the use of horizontal stiffener lips with b values higher than (b ) tr has a clearly detrimental effect on the value of the local critical stress σ crl. This fact makes the determination of such b/b1 values of great importance. Table 1 shows the different (b ) tr values obtained for the range of parameter values considered in this study and it is possible to observe the aforementioned trends. Finally, it should be pointed out that it would be of great practical interest to develop a simple analytical expression which (i) provides reasonable conservative estimates of the combined influence of b1/t, b/b1 and b3/b1 on the value of (b/b1)tr and (ii) covers a range of those parameter values as large as possible. TABLE 1 (b ) tr - TRANSITION BETWEEN CRITICAL LOCAL PLATE AND DISTORTIONAL MODES b =.5 b =. b =.7 b 3 = b 1 /t = b 1 /t = b 1 /t =.3 >. >..3.3 > EXPERIMENTAL ANALYSIS It is well known that, in order to gain a good insight on the behaviour of cold-formed steel columns, it is absolutely essential to perform experimental investigations. Therefore, the results of the stability analysis just presented were used in the planning of a test program which is being carried out at the Structures Laboratory of COPPE - Federal University of Rio de Janeiro. The main objectives of this test program are: (i) to obtain experimental evidence of the distortional buckling behaviour identified, (ii) to estimate the level of postbuckling reserve associated to this mode and (iii) to start a process of gathering reliable data concerned with the ultimate strength of columns affected by distortional buckling. These data will enable an assessment of the adequacy of the available ultimate strength prediction methods, almost all of which are based on linear stability results, to estimate the maximum load-carrying capacity of rack columns. Moreover, if the available methods are proven to be inadequate, the measured ultimate strength values may most certainly be used to establish and calibrate appropriate design tools. Experimental Set-up and Test Procedure The experimental set-up consisted essentially of an Amsler test machine with a 9 kn load capacity. All the tests were performed under load control, with about. MPa/sec as the compressive rate. In order to adequately simulate pinned end conditions, spherical hinges were used on both the lower and the upper templates. It is important to say a few words about (i) the column centralization procedure and (ii) the acquisition and processing of the experimental data. In order to guarantee the concentricity of the load application, a rather careful centralization procedure was adopted, which involves the performance of the following operations: (i) machine finishing of both end cross-sections of the specimen to be tested, (ii) careful representation of the exact geometry of each end cross-section, which is achieved by placing a sheet of paper on it and drawing on top, (iii) calculation of the exact centroidal position of each end cross-section, (iv) fixation of the sheets of paper to the templates in such a way that the cross-section centroidal positions coincide with the test machine axis and, finally, (v) careful positioning of the column over its drawn end cross-sections. This is a simple, efficient and low cost centralization procedure, the main advantage of which resides in the fact that it does not require the use of strain gages. This procedure has already been used in several experimental analyses concerning the stability behaviour of thin-walled cold-formed members (Batista and Rodrigues, 199, and Batista et al., 199).

6 Two sets of experimental data were considered. The first one, intended to follow closely and measure accurately the development of the distortional buckling mode, consisted of (i) the value of the applied load and (ii) the values of six axial strains (strain gages) and four lateral displacements parallel to the web (displacement transducers), all at mid-height of the column. In the second one, intended solely to estimate the columns ultimate strength, only the applied load value was measured. The strain gages were located, on both wall sides, near the free edges of the horizontal stiffener lips and at mid-width of the web (plates b and b 1, in figure 1). The displacement transducers were placed at the flange-stiffener junctions and at the horizontal stiffener lips free edges. The readings of the test machine, strain gages and displacement transducers were fed into a data acquisition system (microcomputer, A/D converter borders and software), which made possible to follow the test progress on line and to transfer all the test data to a computer file. Figure shows a schematic representation of the experimental set-up and test procedure, including the location of the strain gages and displacement transducers. P Compressive load SG5 SG Mid height cross-section D3 D1 SG1 SG SG SG3 D D D D Strain gages D3 D1 Displacement transducers Data Acquisition System Analysis of the experimental results Sheet of paper with the cross-section centroidal axis Template of the test machine Figure : Schematic representation of the experimental set-up and test procedure Test Program Three sets of five cold-formed columns (specimens) were tested, all manufactured by means of a press braking process and displaying the mean geometrical and material characteristics indicated in tables and 3. TABLE GEOMETRICAL CHARACTERISTICS OF THE SPECIMENS Specimen set w 1 w w 3 w t Column length Concerning the geometrical characteristics, it should be pointed out that (i) the only difference between the cross-sectional geometries of the two first sets of columns is the value of the plate thickness and that (ii) the

7 geometrical parameter values are b.59, b 3.15, b.33 (all three sets) and b 1 /t 5.7 (set 1);. (set ); 55. (set 3) (the nominal dimensions shown in table were taken on the outside of the cross-section see figure 1). Finally, the material properties were obtained from standard tensile tests, with four tests being performed for each cross-section geometry. The average values obtained are presented in table 3. TABLE 3 MATERIAL CHARACTERISTICS OF THE SPECIMENS Specimen set σ y (MPa) σ u (MPa) ε r (%) σ y / σ u Since the choice of the lengths of the columns to be tested was made with the objective of obtaining experimental evidence concerning the distortional buckling behaviour, the variation of σ b with a was first investigated. The curves shown in figure 7 correspond to the stability behaviour of columns displaying the three sets of geometrical characteristics just described. The coordinates of the six local minima associated to the local plate and distortional modes (lengths a LP and a D, and bifurcation stresses σ LP and σ D ) are indicated in table. The column lengths selected (see table ) correspond to the three distortional minima. One may designate these specimens as distortional stub columns, in order to underline the fact that their slendernesses lie between those of the traditional stub columns (stability conditioned by the local plate mode) and of the long columns (stability conditioned by the global mode). One last word to mention that, due to the different cross-sectional areas, the relations between the values of the axial compressive load P (directly comparable with the experimental results) are slightly different from the ones between the corresponding stress values σ. The values of P LP and P D are also included in table and take into account the actual (measured) area of the cross-section A b(mpa) b(mpa) b(mpa), , , a a a (a) (b) (c) Figure 7: Variation of σ b with a for the three specimen sets tested: (a) set 1 (b) set (c) set 3 Specimen set TABLE ANALYTICAL LOCAL BUCKLING BEHAVIOUR OF THE TESTED SPECIMENS A (mm ) a LP σ LP (MPa) P LP (kn) a D σ D (MPa) P D (kn)

8 As mentioned before, each specimen set comprised five distortional stub columns. Only one of them was tested fully instrumented (six strain gages and four displacement transducers). The remaining four were tested without any instrumentation and, therefore, only provided data concerning the ultimate strength. Results The experimental results reported in this paper deal (i) with the shape of the distortional buckling mode, (ii) with the equilibrium path characteristics (up to collapse) and (iii) with the values of the columns ultimate strength. Concerning the buckling mode shapes of the columns tested, it was clearly observed that all the specimens buckled in the distortional mode (lateral displacements took place both at the flange-stiffener and stiffener-stiffener junctions), as antecipated by the stability analysis performed. A typical configuration of the distortional buckling mode observed during the tests is shown and schematically represented in figure. P P Observed distortional buckling mode Plate element b1 ~ L/ Plate colapse mechanism on plate element b1 ~ L/ Spherical hinge (a) P Frontal view (b) Lateral view P Figure : (a) Photograph of the distortional buckling mode observed during tests (b) Schematic representation of the observed buckling mode and collapse mechanism Figure 9(a) shows the characteristics of the equilibrium path (load versus lateral displacements of the stiffener lips free edges) experimentally observed and recorded during the test of the fully instrumented specimen from set 1. The distortional stub column was quasi-statically loaded up to collapse and it is worth mentioning that no special meaning should be attributed to the descending branches of the curves, as the test was performed under load control. In fact, the descending branches drawn are a consequence of the speed of the data acquisition system, which enabled the recording of the measured quantities during the initial stages of collapse. It was observed that the collapse took place some time after the onset of the distortional buckling mode and that it was characterised by the formation of a well defined plastic hinge at mid-height of the column, as illustrated in figure (b) (a collapse mechanism very similar to the one observed for stub columns buckling in the local plate mode). On the other hand, the load-stress results presented in figure 9(b) show that the material remained in the elastic range practically throughout the whole test. The two facts just stated lead to believe that the collapse is due to a progressive loss of the column axial stiffness, which takes place in the distortional postbuckling range and, finally, precipitates the formation of the plastic hinge. As a rather good

9 correlation was observed between the experimental and theoretical values of the distortional critical stress (see figure 9), one may say that the initial geometrical imperfections were rather small. This means that the ratio between the mean axial stress values at collapse (σ uexp ) and at the onset of distortional buckling (σ crth ) provides a reasonable estimate of the cross-section postbuckling reserve associated to the distortional mode. The averages of such ratio values (σ uexp /σ crth ) avg, for all the specimens in each set, are presented in the sixth column of table 5 and show the order of magnitude of the postbuckling reserve experimentally observed. In the particular case depicted in figure 9, the distortional stub column tested displayed a postbuckling reserve of about 5%. Displacements P crth = 1 Stress (MPa) P crth = 1 Load P (kn) - -1 Load P (kn) - -1 (a) (b) Figue 9: Experimental results for the fully instrumented tested specimen of set 1 (a) Load-displacements D1 and D (b) Load-stresses from strain gages SG3 and SG Finally, the results concerning the distortional stub columns ultimate strength values are presented in table 5, together with the corresponding average strength and stress values. It is observed that the ultimate strength values P uexp do not vary significantly within the columns of a given specimen set. All the tests not involving a collapse mechanism with a plastic hinge at mid-height of the column were not considered (five tests discarded, three from set 3 and one from sets 1 and ). In the discarded tests, the plastic collapse mechanism was localised near the specimens end, close to the test machine template. The ultimate strength of a column P u is usually written, in terms of the column squash load P y =Aσ y, as P u = Q A σ y, () where Q is a reduction coefficient which takes into account the erosion of the column compressive strength due to its local buckling behaviour. Although the number of columns tested is clearly unsufficient to serve as basis to a meaningful statistical analysis, table 5 also shows the average values of the experimentally obtained coefficients Q exp =(σ uexp /σ y ), as well as the values of the relative slenderness of the columns with respect to distortional buckling, defined as λ d =(σ y /σ crth ).5. The values of Q and λ d are certainly related and further studies, both experimental and analytical, are needed in order to shed more ligth on this matter. TABLE 5 EXPERIMENTAL RESULTS Specimen P uexp σ y (P uexp )avg (σ uexp )avg set (kn) (MPa) (kn) (MPa) (σ uexp /σ crth )avg (Q exp )avg λ d

10 CONCLUDING REMARKS The results of a linear stability analysis of thin-walled rack -section columns were presented. A parametric study was performed to investigate the variation of the critical stress and characteristics of the associated buckling mode with (i) the column length and (ii) the thickness and relative widths of the plates forming the cross-section. It was shown that the behaviour of the columns may be conditioned by local plate, distortional or flexural-torsional buckling modes. The local stability of the cross-section was investigated, in order to identify the cross-section proportions for which the distortional mode is associated to the critical local stress. In particular, the value of the horizontal stiffener lips width, corresponding to a transition between critical local plate and distortional buckling modes, (b/b1)tr, was determined for a number of cases. The linear stability results were used to define an experimental test program presently under way at COPPE - Federal University of Rio de Janeiro. The experimental set-up, test procedure and data processing were briefly described and the test results already obtained were displayed and discussed. The distortional buckling mode was experimentally observed and characterized. Moreover, it was possible to obtain experimental evidence concerning the existence of an important postbuckling reserve associated to its occurrence and to provide an estimate for it. The ultimate strength values of the 1 distortional stub columns successfully tested were also presented. Further studies, both experimental and analytical, are required in order to enable the establishment of column design rules which either prevent the occurrence or take into account the influence of distortional buckling. REFERENCES Batista, E. de M. and Rodrigues, F.C., Buckling curve for cold-formed compressed members. Journal of Constructional Steel Research, 199, (), Batista, R.C., Batista, E.M. and Pfeil, M.S., Effective cross-section properties for thin-walled cold-formed members. Proceedings of SSRC Annual Technical Session and Meeting, Pittsburgh, 199, Camotim, D. and Prola, L.C., On the stability of cold-formed steel elements with rack sections. Proceedings of 5 th International Colloquium on Structural Stability - SSRC, Rio de Janeiro, 199, 1-3 Hancock, G.J., Kwon, Y.B. and Bernard, E.S., Strength design curves for thin-walled sections undergoing distortional buckling. Journal of Constructional Steel Research, 199, 31(,3), 19-1 Lau, S.C. and Hancock, G.J., Distortional buckling formulas for channel columns. Journal of Structural Engineering - ASCE, 197, 113(5), Prola, L.C. and Camotim, D., Local and global stability of cold-formed steel structural elements with stiffened S-sections. Proceedings of IV ENMC (in portuguese), Lisboa, 1995, 95-1