Advances in Steel Structures - part 14 pptx

110 J.G. Teng et al. programme on thin walled columns with consisting of between four to forty sides. Only even numbers of sides were considered. The mode of collapse observed was elastic buckling of the fiat plate elements followed by plastic collapse of the junction between adjacent elements. Columns with more than eighteen sides were found to collapse in a manner similar to that of a circular tube. Avent and Robinson (1976) conducted an elastic stability analysis of thin-walled regular polygonal columns by expanding nodal displacements into Fourier series. They derived buckling curves for axially loaded pin-ended columns with polygonal cross-sections. The buckling curves describe local plate buckling for short columns and Euler column buckling for longer ones. An increase in the local buckling capacity for sections with odd numbered panels was noted. The local buckling stress for sections with an even number of sides was found to match that of uniaxially compressed simply supported plates. As the number of sides is increased to 16, the critical local buckling load approaches that of a cylinder. Kurt and Johnson (1978) considered imperfections in axially loaded pin-ended columns of polygonal sections. In their study they distorted the sides of a polygon by applying a midpoint lateral displacement and then utilised the same analytical solution technique as Avent and Robinson (1976). Similar buckling curves were produced to those of Advent and Robinson and it was found that as the number of sides increases, the polygon behaviour approaches that of a cylinder. In the Euler buckling range of response, initial imperfections decrease the predicted buckling strength. In the local buckling range of response, the buckling strength is increased by the imperfections. More recently Koseko et al. (1983) undertook an experimental and theoretical study of the local buckling strength of thin walled steel members of octagonal cross-section. The finite strip method was employed for the theoretical work. Aoki et al. (1991) have since conducted experiments on columns varying from square to octagonal in cross-section. Residual stresses and geometric imperfections were measured and an empirical design formula was calibrated from the experimental results. Polygonal cross-sections appeared to be better than box sections in respect to the ultimate strength considerations. The most recent significant contribution to local buckling in polygonal section columns appears to have been made by Migita et al. (1992), who considered the interaction between local and overall buckling in polygonal section steel columns. No study to date appears to have conducted a comparison between the buckling capacity of regular polygonal columns subjected to axial compression and that under bending. This in turn has prompted the current study. Five different thin-walled polygonal section forms are considered in this study. These are square, pentagonal, hexagonal, heptagonal and octagonal sections with four to eight sides respectively. These sections are shown in Figure 1 where their principal axes are indicated. It should be noted that while all sections are symmetric about the vertical or y axis, only sections with an even number of sides are symmetric about the horizontal or x axis. The width of a plate element is denoted as b while the thickness is represented by t. Thirteen different width-to-thickness ratios are considered for each section form and a summary of the dimensions of the sections investigated here are given in Table 1. The total cross-sectional area is kept constant for all five sections for each thickness. The number of sides of each section is denoted by n in Table 1. STABILITY ANALYSIS The program THIN-WALL (THIN-WALL 1996, Papangelis and Hancock 1995), developed by the University of Sydney, is used to study the local buckling of columns with polygonal sections subjected to axial compression or bending. The program is based on the well-established finite strip method of analysis (Cheung, 1976, Hancock 1978). In the finite strip method, thin walled sections are subdivided into longitudinal strips. The displacement functions, which are used to describe the displacement Local Buckling of Thin-Walled Polygonal Columns 111 variation in the longitudinal direction, are assumed to be harmonic. Polynomial functions are used to describe the displacement variation in the transverse direction. The finite strip buckling analysis can be represented in matrix format as follows: [K]{D}- A[G]{D} = O (1) where [K] and [G] are the stiffness and stability matrices of the structure being investigated. The solution produces the eigenvalue, or the critical buckling load factor represented by t, and the eigenvector given in [D]. Substitution of the eigenvector into the assumed displacement functions gives the buckled shape of the plate assembly. l y y y X X X X / / / / /J / b Square Pentagon Hexagon Heptagon Octagon Figure 1" Polygonal Shapes TABLE 1" GEOMETRIC PROPERTIES OF POLYGONAL SECTIONS t Section (mm) Square Pentagon Hexagon Heptagon Octagon n=4 n=5 n=6 n=7 n=8 b =126 mm b = 100.8 mm b = 84 mm b = 72 mm b = 63 mm b/t A b/t A b/t A b/t A b/t A (mm2) (mm 2) (mm 2) (mm 2) (mm 2) 0.5 252 252 252 252 252 0.7 353 353 353 353 353 0.9 454 453 453 453 453 1.0 504 504 504 504 504 1.5 756 756 756 756 756 2.0 1008 1008 1008 1008 1008 2.5 1260 1260 1260 1260 1260 3.0 1512 1512 1512 1512 1512 4.0 2016 2016 2016 2016 2016 5.0 2520 2520 2520 2520 2520 6.0 3024 3024 3024 3024 3024 7.0 3528 3528 3528 3528 3528 8.0 4032 4032 4032 4032 4032 252.0 201.6 168.0 144.0 126.0 180.0 144.0 120.0 102.9 90.0 140.0 112.0 93.3 80.0 70.0 126.0 100.9 84.0 72.0 63.0 84.0 67.2 56.0 48.0 42.0 63.0 50.4 42.0 36.0 31.5 50.4 40.3 33.6 28.8 25.2 42.0 33.6 28.0 24.0 21.0 31.5 25.2 21.0 18.0 15.8 25.2 20.2 16.8 14.4 12.6 51.0 16.8 14.0 12.0 10.5 18.0 14.4 12.0 10.3 9.0 15.8 12.6 10.5 9.0 7.9 112 J.G. Teng et al. RESULTS General Figure 1 and Table 1 show the configurations and dimensions of the polygonal sections considered in this study. Two different loading scenarios were investigated: axial compression and bending. All results were generated for mild steel plate assemblies with an elastic modulus of 200,000 MPa and a Poisson's ratio of 0.3. The elastic local buckling capacity is given in terms of the dimensionless buckling stress coefficient kcr which is related to the critical stress ~, through the following familiar expression (Bulson 1970, Trahair and Bradford 1998): n-:E 1 O'cr =kcr 121"-v2'xz'/tx2~l j[b ) (2) where E is the elastic modulus and v the Poisson's ratio. In the parametric study described below, the plate slenderness (width-to-thickness ratio) was varied from a minimum of about 10 to a maximum of 252. For each plate slenderness, a corresponding elastic local buckling capacity was determined. 4.4 - .3 .~ 4.2 0 m 4.1 9 ~ 4- 3.9 3.8 ~' A A A # o i _r r ooN O & , o square .'. pentagon [] hexagon e heptagon x octagon I I I I 50 100 150 200 slenderness, b/t Figure 2" Local Buckling Stress Coefficient versus Slenderness for Polygonal Sections Subjected to Axial Compression t 250 Axial Compression Figure 2 shows the elastic local buckling stress coefficient for the five polygonal sections investigated as a function of the plate slenderness or b/t ratio. Here the sections are subjected to uniform axial compression. From this figure it can be observed that the square, hexagonal and octagonal section columns have approximately the same buckling capacity. The heptagonal and pentagonal columns have a markedly increased buckling resistance. This increase can be attributed to the local buckling configuration of the individual plate elements. For a plate slenderness greater than about fifty, the buckling stress coefficient stays virtually constant. However, when the slenderness drops below about Local Buckling of Thin- Walled Polygonal Columns 113 fifty, the buckling stress coefficient starts to reduce. The square, hexagonal and octagonal sections have a buckling stress coefficient of about four and this is consistent with that of a plate simply supported on all four edges and subjected to uniaxial compression (Bulson 1970). Representative buckling modes are shown in Figure 3 for the five polygonal sections subjected to axial compression. The dotted lines in this figure represent typical buckled configurations while the continuous lines represent the original shapes. The square, hexagonal and octagonal sections all buckle in a similar manner with each plate element buckling in an opposite direction to the adjacent plate elements. These three sections have an even number of sides, that is, four, six and eight sides respectively. For sections with an odd number of sides, this alternating inward-outward bucking mode is incompatible with the number of sides. For these sections, two consecutive plate elements must buckle in the same direction, be that inward or outward or two half waves have to appear in one of the plate elements. These variations in buckling modes lead to a higher buckling capacity as seen for the pentagonal and heptagonal columns. " " "" f"-"i i t ',\ /: ' ) Figure 3" Typical Local Buckling Modes under Axial Compression 6.5 f A A A A .8. 6 5.5 ^ , 0 0 0 0 0 [] [] [] [] .~ex[l X .'< x~ 4.5 4 I I 0 50 100 150 200 slenderness, b/t Figure 4: Local Buckling Stress Coefficient versus Slenderness for Polygonal Sections Subjected to Bending -0 I I o(D o square (x,y- axis) -'- pentagon (y- axis) [] hexagon (y- axis) e heptagon (y- axis) x octagon (x,y- axis) I I 250 Bending Figure 4 shows the elastic local buckling stress coefficient for the five polygonal sections in bending as a function of the plate slenderness. For each section, a buckling analysis was carried out for bending about the x-axis in a positive and negative direction, and bending about the y-axis. The lowest of the 114 J.G. Teng et al. three buckling stresses is taken as the critical buckling stress. Figure 4 shows that the octagonal section has the lowest buckling resistance followed by the hexagonal, square, heptagonal and pentagonal section. The critical axis of bending for each section, which produces the lowest buckling coefficient is also reported in Figure 4. Figure 5 shows typical buckling modes of the five polygonal sections when subjected to bending in the weakest of the three directions. J! Y Figure 5: Typical Local Buckling Modes under Bending Comparison between Axial Compression and Bending The calculated buckling coefficient for bending is greater than the corresponding value for axial compression. This is as expected and is also reflected in the buckling modes of Figures 3 and 5. All sections subjected to bending experienced about a 25% increase in the buckling stress compared to axial compression. For low b/t ratios the buckling capacity of all sections, subjected to axial compression or bending, reduces. This is believed to be due to membrane deformations in the plates which are not accounted for in classical theories for plate buckling. CONCLUSIONS The elastic local buckling capacity of polygonal sections has been investigated in this paper. Square, pentagonal, hexagonal, heptagonal and octagonal sections have been investigated, with elastic local buckling coefficients presented for a variety of plate width-to-thickness ratios. It has been shown that the dimensionless buckling stress coefficient is influenced by two parameters: the nature of the applied loading and the number of sides of the section. The buckling stress coefficient is higher for bent sections than axially compressed ones, and this difference can be quite significant. Sections with an odd number of sides have an enhanced buckling capacity over those with an even number of sides, with pentagonal sections being the strongest under either axial compression or bending. It should be noted that for sections subject to bending, the bending moment was applied in three different directions to find the weakest axis of bending. Further work should be carried out to establish if another axis of bending exists which leads to an even lower buckling stress. ACKNOWLEDGEMENTS The authors gratefully acknowledge the contribution to this work made by Mr K. K. Wong who carried out the initial calculations for the results presented here during his final year project supervised by the first author. The second author wishes to thank The Hong Kong Polytechnic University for providing him with a Postdoctoral Fellowship. REFERENCES Aoki, T., Migita, Y and Fukumoto, Y. (1991). Local Buckling Strength of Closed Polygonal Folded Section Columns. Journal of Constructional Steel Research 20, 259-270. Local Buckling of Thin-Walled Polygonal Columns 115 Avent, R.R. and Robinson, J.H. (1976). Elastic Stability of Polygonal Folded Plate Columns. Journal of the Structural Division, ASCE 102(ST5), 1015-1029. Bulson, P.S. (1969). The Strength of Thin Walled Tubes Formed from Flat Elements. International Journal of Mechanical Sciences 11, 613-620. Bulson, P.S. (1970). The Stability of Flat Plates, Chatto & Windus, London, U.K. Cheung, Y.K. (1976). Finite Strip Method in Structural Analysis, Pergamon Press, Oxford, U.K. Hancock, G.J. (1978). Local, Distortional, and Lateral Buckling of I-Beams. Journal of the Structural Division, ASCE 104(ST11), 1787-1798. Koseko, N., Aoki, T. and Fukumoto, Y. (1983). The Local Buckling Strength of the Octagonal Section Steel Columns. Proceedings of Structural Engineering~Earthquake Engineering, JSCE 330, 27- 36. Kurt, C.E. and Johnson, R.C. (1978). Cross Sectional Imperfections and Column Stability. Proceedings of the Structural Division, ASCE 104(ST12), 1869-1883. Migita, Y., Aoki, T. and Fukumoto, Y. (1992). Local and Interaction Buckling of Polygonal Section Steel Columns. Journal of Structural Engineering, ASCE, 118(10), 2659-2676. Papangelis, J.P. and Hancock, G.J. (1995). Computer Analysis of Thin-Walled Structural Members. Computers and Structures, 56(1), 157-176. THIN-WALL (1996), Cross-Sectional Analysis and Finite Strip Buckling Analysis of Thin Walled Structures: Users Manual- Version 1.2, Centre for Advanced Structural Engineering, Department of Civil Engineering, The University of Sydney, Sydney, Australia. Trahair, N.S. and Bradford, M.A. (1998). The Behaviour and Design of Steel Structures to AS 4100, Third Edition, E&FN Spon, London, U.K. Wittrick, W.H. and Curzon, P.L.V. (1968). Local Buckling of Long Polygonal Tubes in Combined Bending and Torsion", International Journal of Mechanical Sciences 10, 849-857. This Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left BlankThis Page Intentionally Left Blank ULTIMATE LOAD CAPACITY OF COLUMNS STRENGTHENED UNDER PRELOAD H. Unterweger Department of Steel Structures, Technical University Graz Lessingstrasse 25, A- 8010 Graz, Austria ABSTRACT The ultimate load capacity of symmetric columns, which are strengthed under preload using steel plates, is presented. The analytic calculation model is an ideal column with geometric imperfections including second order effects. The decrease of the bending stiffness, due to the development of plastic zones in the cross section, is taken into account by a modification factor, based on comprehensive FE- calculations. By using modified buckling reduction factors it is possile to find out directly the extent of strengthening steel plates, depending on preload and type of cross section. KEYWORDS strengthen of steel columns, flexural buckling, strengthen steel plates, ultimate load capacity INTRODUCTION Due to increasing loads or changing of service conditions sometimes members of existing structures must be strengthen. Examples for primary compressed members are: - columns of buildings - chords of truss girders of old bridges, due to increased traffic loads The strengthening design of the cross section of the member (base - section), using steel plates (welded or bolted to the member) has to consider the preload in the member due to at least permanent actions. In actual codes in Europe (e.g. Eurocode (1993), DIN (1990)) no procedures for strengthening of mem- bers are included. 117 118 H. Unterweger For members without risk of stability failure the determination of the extent of the strengthening plates gives no problems (see also Fig. 2). But for members, whrere flexural buckling is relevant, the design procedure seems not clear at all. Publications in this field are very small and limited to the counties of Eastern Europe, summed up by Rebrov & Raboldt (1981). This paper presents the main results of a comprehensive study by the author (1996). ASSUMTIONS AND EXTENT OF THE PRESENTATION The presented results refer to columns with pinned ends on both sides (Figure 1). In the following only axial loading (constant axial force N) is taken into account. Additional limited bending moments can be also taken into account as shown in Unterweger (1996). Figure 1 : System and type of cross sections of the presentation. The base sections are universal rolled columns with H - shape. The strengthening plates are situated either on the outsides of the flanges (type 1) or on both sides of the web (type 2) and they have the same length as the columns. The result are also applicable to other section types like welded H - sections, hol- low sections or channels, if they are symmetric to the buckling axis (see figure 1). The slenderness ratio of the individual parts of the cross section is limited in such way that local buckling is not possible until the plastic cross section resistance is reached (classified as class 1 and 2 in Eurocode). CALCULATION PROCEDURE Columns without buckling failure mode For columns with small slenderness, e. g. short columns or columns which are supported by walls or bracings, the cross section resistance is relevant for design. The determination of the extent of the strengthening plates AA for a given axial force Nv+ AN, as shown in Figure 2, is very simple. If only the elastic resistance is taken into account the preload N v acts on the base section A 0 and only the additional load AN acts on the whole section A (Eqn. 1). That means that yielding of the base section limits the loading capacity. If the plastic resistance of the cross section is taken into account the whole section also acts for the preload N v (Eqn. 2). Therefore stress redistribution between base section and strengthening plates, due to plastification of the base section, is necessary (~,f, ~'m are partial safety factors). Ultimate Load Capacity of Columns Strengthened under Preload 119 Figure 2 : Stress distribution due to axial force N and verification procedure for elastic (Eqn.1) and plastic cross section resistance (Eqn. 2). With increasing preload the differencies between elastic and plastic resistance grow. At the theoretical border line case - yielding of the base section under preload - the strengthening plates are completely ineffective. Therefore the design procedure based on plastic resistance is simpler and much more eco- nomic, but leads to more or less plastifications of the base section. The aim of this study is to show that also for slender columns plastification of the base section can be taken into account to exploit the full bearing capacity of the strengthening plates. Columns under flexural buckling In general for columns flexural buckling is the relevant failure mode for design, which will be discussed in the following. The partial factors are omitted (Tf, Tm). Engineering solution Considering the design procedure for columns, based on slenderness depending buckling reduction fac- tors K: (e.g. European buckling curves a § d in Eurocode) the design procedure for determining the load capacity N R of a strengthen column follows Eqn. 3 (stress equation) Nv AN + <fy N R = N v+AN (3) K: o. A o ir A- ' . Sections Subjected to Bending -0 I I o(D o square (x,y- axis) -& apos ;- pentagon (y- axis) [] hexagon (y- axis) e heptagon (y- axis) x octagon (x,y- axis) I I 250 Bending Figure 4 shows. Members. Computers and Structures, 56(1), 15 7-1 76. THIN-WALL (1996), Cross-Sectional Analysis and Finite Strip Buckling Analysis of Thin Walled Structures: Users Manual- Version 1.2, Centre. discussed in the following. The partial factors are omitted (Tf, Tm). Engineering solution Considering the design procedure for columns, based on slenderness depending buckling reduction fac- tors