Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 47, 2, pp. 435-456, Warsaw 2009 STABILITY AND LOAD CARRYING CAPACITY OF MULTI-CELL THIN-WALLED COLUMNS OF RECTANGULAR CROSS-SECTIONS Marian Królak Katarzyna Kowal-Michalska Radosław J. Mania Jacek Świniarski Technical University of Lodz, Department of Strength of Materials and Structures, Łódź, Poland e-mail: Marian.Krolak@p.lodz.pl The paper concerns theoretical, numerical and experimental analysis of the stability and ultimate load of multi-cell thin-walled columns of rec- tangular and square cross-sections subjected to axial compression (uni- form shortening of the column). The theoretical analysis deals with the local and global stability ofmulti-cell orthotropic columns of a rectangu- lar profile with rectangular cells. It has been shown that for a multi-cell column made of the same material and having the same cross-section area, the value of local buckling stress of the columnwalls grows rapidly with an increase of the cell number. The experiment conducted for iso- tropic columns has also proved a significant growth of the ultimate load with the increase of the cell number. The paper gives some conclusions which can be useful in design of thin-walled box columns. Key words: stability, ultimate load, thin-walled, multi-cell column 1. Introduction Thin-walled columns and beams of flat walls are built of long rectangular plates connected on longitudinal edges. Compressed walls of such structures with a small thickness-to-width ratio (t/b < 1/200) undergo local buckling with small critical stress, comparing to the yield limit. To increase the local buckling load of these structures, longitudinal ribs are addedwhose stiffness is properly chosen (Maquoi and Massonet, 1971; Massonet and Maquoi, 1973). Parallel to the acting load, the ribs divide the columnwall into a fewplate-like 436 M. Królak et al. stripes.Their width is smaller than the plate itself. This causes a several times or even greater growth of the local buckling load when the cross section area increases not much. In the present paper it has been suggested to increase the local buckling strength of a thin-walled single cell compressed column or a bent beam (box, girder)by substituting themwithmulti-cell structures. In thepaper, themulti- cell structures of equal cross-section area, equal dimensions and made of the same material as a single cell structure will be considered to easily prove the increase of local buckling resistance for the first ones. In the world literatu- re, there are surprisingly few works (Chen and Wierzbicki, 2001; Kim, 2002) dealing with the problems of stability, post-buckling behaviour and ultimate stress of prismatic beam-columns of multi-cell cross-sections. 2. Local buckling analysis of thin-walled orthotropic columns with a rectangular cross-section under edgewise compression Let us consider a long prismatic multi-cell column subjected to compression with a cross-section shown in Fig.1b, and a single cell column (box column) presented in Fig.1a. Fig. 1. Considered cross-sections of analysed columns The following notation will be introduced: n – number of cells adjacent to one wall b1v,b1h – height/width of column cross-section t1v, t1h – thickness of vertical/horizontal walls of single cell column Stability and load carrying capacity... 437 bnv,bnh – height/width of cell tnv, tnh – thickness of vertical/horizontal walls of multi-cell column L – length of column. In further considerations we assume that: all walls aremade of an orthotropicmaterial of elastic constants E1,E2, G, ν12, ν21 E1 – elastic modulus in the longitudinal direction (direction of compression) E2 – elastic modulus in the direction perpendicular to the co- lumn axis G – Kirchhoff’s modulus ν12,ν21 – Poisson’s ratios. In order to compare the values of local buckling stress for single-cell and multi-cell columns it is assumed that theoverall dimensionsof the cross-section areas of the analysed columns are the same. In this work, the geometric and material parameters of thin-walled single and multi-cell columns are chosen in such a way that the local buckling of all walls occurs at the same value of compressive loading. In such a case, all walls can be treated as simply supported along all edges. In further considerations it is assumed that the column overall dimen- sions provide the global stability of the structure. It is assumed too, that the local-global buckling interaction does not occur. In paper Królak and Kowal- Michalska (2004b) detailed theoretical analysis of global and local buckling of thin-walled multi-cell rectangular profiles (particularly square) made of an orthotropicmaterial is described. For columnswhich fulfill the above assump- tions given in Królak and Kowal-Michalska (2004b), Volmir (1968), the follo- wing approximate relations were obtained: • For single-cell column buckling stress σloccr1 = k π2E1 12γ (t1h b1h )2 = k π2E1 12γ (t1v b1v )2 (2.1) where k is the buckling load factor, which for long plates simply sup- ported along all edges is taken as k=4, γ – coefficient which depends on elastic constants; its inverse has the form 1 γ = 1 2 √ E1 E2 + 1 2 E2ν12 E1 + G E1 ( 1− E2ν 2 12 E1 ) 1− E2ν 2 12 E1 (2.2) 438 M. Królak et al. where the elasticity constants fulfill Betty’s relation ν12E1 = ν21E2. Formulas (2.1) and (2.2) were obtained after some transformations from the local bucklingcritical stress relation of the longorthotropicuniformly compressed plate, presented in detail in Volmir (1968). For the special case of isotropic plates, there is γ=1−ν2. • Formulti-cell orthotropic columns (with n cells), analogously to formu- las (2.1), we obtain the buckling stress from σloccrnh = k π2E1 12γ (tnh bnh )2 =n2k π2E1 12γ (tnh b1h )2 (2.3) σloccrnv = k π2E1 12γ (tnv bnv )2 =n2k π2E1 12γ (tnv b1v )2 From the set a priori assumption on the same cross-section area of the single- cell andmulti-cell column there result simple relations tnv = n 3(n−1) t1v tnh = n 3(n−1) t1h (2.4) Comparing formulas (2.1) and (2.3) and remembering relations (2.4), for the considered multi-cell columns, the critical local buckling stress can be calcu- lated from the approximate formula σloccrn =αnσ loc cr1 (2.5) where obviously αn = n4 9(n−1)2 n=2,3,4, . . . (2.6) Relation (2.6) is valid for columns of rectangular and square cross-sectionsma- de of iso- and orthotropicmaterials.We note that increase of the parameter n (number of cells adjacent to one wall) causes an increase in the ratio tn/bn. According to formulas (2.1), (2.3) and (2.5), it is connected with a greater value of local buckling stress, so the coefficient αn shows howmany times the critical local buckling stress of the multi-cell column with n cells is greater than the critical buckling stress of the single cell column made of the same material and with the same cross-section area. The relation αn as a function of the parameter n for rectangular cross-section columns is presented on a graph in Fig.2. For n=1, the coefficient αn =1. In Królak et al. (2007), formulas and graphs for different profiles for thin- walledmulti-cell columns are given. Some of them –not onlywith rectangular cells – are presented in Fig.3. Stability and load carrying capacity... 439 Fig. 2. Values of the coefficient αn as a function of the number of cells n Fig. 3. Cross-sections of analysed columns In Fig.4, for different cross-sections of multi-cell columns graphs of the coefficient αn as a function of the parameter n are presented. Fig. 4. Coefficient αn as a function of the number of cells 440 M. Królak et al. 3. Global buckling of multi-cell columns subjected to axial compression Theglobalbuckling stress fora compressedmulti-cell columnsimply supported at both ends is defined by Euler’s formula σglcrn = π2(Dn)min AL2 (3.1) where (Dn)min is the minimal flexural stiffness of the column with the n-cell parameter, A – cross-section area, and L – column length. The cross-section areas of single-cell and multi-cell columns are equal, and so A = 2(b1ht1h + b1vt1v). For themulti-cell column, the formula of global buckling stress can be written as σglcrn =βnσ gl cr1 (3.2) where βn =(Dn)min/(D1)min. The variation of coefficient βn as a function of the parameter n (number of cells) is plotted in Fig.5. Fig. 5. Variation of the coefficient βn as a function of the number of cells It turns out fromFigure 5 that themaximal decrease of the global buckling critical stress occurs for n=3(it is circa 33%of the critical stress of the single- cell column). For n> 3, the stress σglcrn rises gradually with the growth of the parameter n and reaches nearly 0.8σ gl cr1 for n=9. 4. Experimental test on analysed columns 4.1. Description of tested thin-walled columns The experimental tests of segments of columns were carried out on six models with rectangular and square cross-sections, that is: Stability and load carrying capacity... 441 • one model of a thin-walled column with single-cell square cross-section, • threemodels of thin-walledmulti-cell columnswith square cross-sections and square cells, • two models of thin-walled multi-cell columns with rectangular cross- sections. The model of the single-cell column (marked as model 1) was made of a ste- el sheet of t1 = 1.24mm thickness. The cross-section of this model and its dimensions are presented in Fig.6. Fig. 6. Cross-section of model 1 and its dimensions The presented model was made of two parts of correctly bent sheets con- nected in two corners by spot welding. The sticking out of the model outline welded elements were cut perpendicular to the model height not to carry the compressive load. The total height of the tested model (the length of the co- lumn segment between nodal lines of local buckling of its walls) was equal to the wall width (L = b1). The material constants for the steel sheet were obtained from a tensile test and were as follows: E =2.0 ·105MPa, ν =0.3, R0.2 = 179MPa, Rm = 313MPa. The cross-section area of the model was A = 4b1t1 = 4 · 312.5 · 1.24 = 1550mm 2. The ratio of the wall width-to- thickness was b1/t1 =252. Three models of square multi-cell columns (models number 2, 3, 4) were made of a steel sheet of t1 =0.5mmthickness. Five square cells were adjacent (n=5) to the external wall. The cross-section of such amodel is presented in Fig.7. The elements of these models were joined by spot welding. The internal walls of cells (between external and internal walls of the column) were made as a ’I-shape’ profiles. They were cut of a steel sheet by laser technology and their ’flanges’ were bent in the way shown in Fig.8. 442 M. Królak et al. Fig. 7. Square cross-section of the multi-cell model Fig. 8. Internal wall of the multi-cell model Thedimensions of themanufacturedmodels slightly differ from each other because of what occurred during sheet bending and spot welding processes. Table 1.Model dimensions Model b b5 t5 h A number [mm] [mm] [mm] [mm] [mm2] 2 318 63.6 0.5 318 1526 3 322 64.4 0.5 322 1546 4 316 63.2 0.5 316 1517 The cross-section areas of all models – single- and multi-cell of square shapes were nearly equal to 1500mm2. The material properties of the ste- el sheet of thickness 0.5mm, which was used for manufacturing of multi- cell models were as follows: E = 1.97 · 105MPa, ν = 0.3, σH = 140MPa, Stability and load carrying capacity... 443 R0.2 = 202MPa, Rm = 315MPa. In Fig.9, there is a photography of one of the squaremulti-cell models. There were significant imperfections in tested models, especially in one of themmadeas a trial one.Thesewere imperfections of model walls and differences of cell dimensions. Fig. 9. Model of a square multi-cell column Two models of thin-walled multi-cell columns with a rectangular shape andwith rectangular cells (model 5 and 6) weremade of brass sheets of t5h = 0.3mmand t5v =0.6mm thicknesses. Number of cells (parameter n=5)was equal for bothmodels.Thewalls were joined by the break-head rivetswith the diameter 2.2mm. The rivets were used because it was impossible to connect the brass sheets by spot welding. The cross-sections of brassmodels and their dimensions are shown in Fig.10. Fig. 10. Cross-section of the brass multi-cell column 444 M. Królak et al. The models differ between themselves with the rivets spacing and their layout. Figure 11 presents a photograph of the manufactured brass model. Fig. 11. Multi-cell model made of brass sheets Thematerial properties of brass sheets were as follows: E =1.0·105MPa, ν =0.3, σH =100MPa, R0.2 =180MPa, Rm =256MPa. 4.2. Test stand description A special test stand presented in Fig.12 was designed and manufactured to perform experimental investigation of the multi-cell models. Fig. 12. Experimental stand The upper movable plate of the stand can slide along four vertical bars which are fixed to the lower plate. Both plates are made of steel of thick- ness 45mm. The high thickness of the plates provides small strains during Stability and load carrying capacity... 445 compression of the column, and the guide bars provide the mutually parallel position of the plates. As a result, a uniform shortening of all walls is achie- ved (symmetry of column). Between all loaded edges of themodel and both – upper and lower steel plates, soft aluminum and foamed PCV plates are ad- ditionally placed to provide uniformly distributed compressive stresses in the pre-buckling state. Thickness of these plates is 5 and 3mm for aluminum and PCV, respectively.Moreover, thePCVplates approximate the simply support conditions on loaded edges and remove pointwise contact between the edges and loading plates as well as possible stress concentration. In Fig.13, dimples pressed in the PCV plate duringmodel compression are shown. Fig. 13. Dimples in a foamed PCV plate 4.3. Conditions of experimental tests The experimental tests were carried out on the multipurpose material te- sting machine INSTRON controlled by a PC unit. The applied equipment and software ensured automatic measurement of the loading force and the di- splacement of the upper plate of the test stand fixed in the testing machine. Thewalls of the testedmodel were bondedwith strain-gages connected to the multi-channel bridge SPIDER (HBM), also controlled by PC. The deflections of selected points of external walls were measured by dial gauges with a low axial force of the plunger. The deflections were evaluated in points where the bucklingwave summitswere expected.The column in the testingmachine and the testing system are presented in Fig.14. 4.4. The aim of the experiments Thepurpose of the experimental testwas the analysis of behaviour of thin- walled multi-cell columns in following ranges: pre-buckling, buckling, post- 446 M. Królak et al. Fig. 14. The test stand andmeasuring equipment buckling elastic, elastic-plastic and failure. The critical load of global buckling and ultimate loadwere themain values whichwere to bemeasured.The other parameterswhich influence thebehaviour of columns, especially imperfections, were also observed. The experimental tests were to prove: • much higher resistance of multi-cell columns to local buckling than single-cell ones (with equal cross-section area), • higher load carrying capacity of multi-cell columns than single-cell ones (better exploitation of material properties in multi-cell columns), • correctness of derived formulas for local buckling stress of multi-cell co- lumns. The above problems are referenced in the following sections. 4.5. Results of experimental tests 4.5.1. Square cell models Thesquare shapecross-sections of columnmodels (or segments of columns) were presented in Figs. 6 and 7, while their dimensions are drawn up in Ta- ble 1. The load shortening curves (L-S curves) of these models obtained from the INSTRON testingmachine are compiled in Fig.15. These are plots of the compressive force (without taking into account the upper pressing plate we- ight, which was nearly 1kN) as a function of the upper plate displacement. Thisdisplacementdependson shorteningofmodelsheight anddimplespressed Stability and load carrying capacity... 447 in the aluminum and foamed PCV plates. The aluminum plates were placed between edges and upper and lower steel pressing plates for models 1, 2, 3, 4, when for the last two models the PCV plates were added. Fig. 15. L-S curves for squaremulti-cell columns The results of tests are compared in Table 2. Table 2.Results of experiments Model A σcomcr P com cr P exp ult σ exp ult number [mm2] [MPa] [kN] [kN] [MPa] 1 1550 11.00 12.6 51.6 33.3 2 1526 44.69 68.2 101.6 66.6 3 1546 43.58 67.4 99.43 64.3 4 1517 45.26 68.6 94.64 62.4 The values of ultimate loads P exp ult (maximum value of the force in the L- S plot without weight of the upper pressing plate considered) were obtained directly from the experiment. The average ultimate stresses were determined from division of the ultimate force by model cross-section area. The average ultimate stress is ca 5.2 times smaller than the yield limit of the material used formodel construction. Itmeans that for a single-cell thin-walled column the strength of the material is exploited only in small amount. The obtained in the experiments the ultimate loads for three models of multi-cell columns (models 2, 3, 4) differ from themselves by 7.1% at most, while the average ultimate stresses σ exp ult by circa 6.5%. These stresses are 3.13 times less than the yield limit of the material used for model manufacturing, which means that thematerial ofmulti-cell column is exploitedbetter.Theusageofmaterial properties in the single-cell columnreached 19.25,while in themulti-cellmodel 448 M. Królak et al. ca 31.9%. The hundred percent usage of the material properties corresponds to full plasticity of cross-section area. Thecritical force for local bucklingof the single-cell columnwasdetermined both from the LSC of the column for loads in the range 0-50kN (Fig.16), and from the deflection plot of onewall center point as a function of load (Fig.17). Fig. 16. Method of buckling load evaluation based on L-S curve Fig. 17. Method of buckling load evaluation based on deflection plot As shown in Figs. 16 and 17, the critical values of buckling load obtained in experiments are in the range of Pexpcr =12.3-12.6kN. The value of local buckling stress determined in laboratory tests is in the range of σexpcr =7.9-8.1MPa. In the tested models of multi-cell columns, classical local buckling pheno- menawere not observed. On the obtained LS curves, there is no characteristic inflection point which refers to the critical load (to change the column com- pressive stiffness).Theoretically, the classical buckling shouldoccur in columns Stability and load carrying capacity... 449 without imperfectionwith so chosen geometry of thewalls and boundary con- ditions as to all walls would immediately lose stability. It appeared from the carried out experiments that the multi-cell column walls with different im- perfections (geometric and in the load input) buckle gradually under uniform compression.Mainly due tonon-uniformlydistributed load (causedby the lack of ideal contact between the wall edges and the compressive plates) the more loadedwalls buckled earlier. At themomentwhen the lastwalls buckled, those buckled earlier worked in the post-buckling range.When the critical stresswas comparatively high, it could be even an elastic-plastic state. From themoment when almost all or even all walls buckled, the increments of column shortening stay higher for the same increments of compressive load (what is caused by greater and greater deflections of walls). For this reason, the authors suggest to determine the critical load of local buckling (stress or force) of uniformly compressedmulti-cell columns as the maximum load value, over which for an equal increment of compression the wall deflections grow (or grow the shor- tening increments). The Load Shortening Curve of multi-cell columns can be divided into a few phases: • the initial phase, the structure ’settles’ in bearings, • thepre-bucklingphase,whenthenumberofbuckledwalls rises gradually, • the critical state – themaximal load value over the tangent to the linear (or almost linear) part of the L-S plot, • the post-buckling elastic range, • the post-buckling elastic-plastic range, up to load carrying capacity (ma- ximal load), • the collapse phase. For such a definition of the critical load, the critical local buckling forces amount ca Pexpcr = 80kN for the tested three models of square multi-cell columns. After a detailed analysis of all L-S plots of models 2, 3 and 4, the local buckling forces and stresses were determined as follows P exp cr2 =79kN σcr2 =51.8MPa P exp cr3 =81kN σcr3 =52.0MPa P exp cr4 =80kN σcr4 =52.7MPa 4.5.2. Models of rectangular cross-section columns The cross-section of two models of multi-cell columns with rectangular shapes are shown in Fig.18, and their dimensions are given in Table 3. 450 M. Królak et al. Fig. 18. Load Shortening Curve Table 3.Rectangular cells dimensions Model bv bh bv5 tv5 bh5 th5 L A number [mm] [mm] [mm] [mm] [mm] [mm] [mm] [mm2] 5, 6 250 125 50 0.6 25 0.3 250 900 These two models were made of brass sheets. The thickness and width of cell walls were selected to have the same thickness-to-width ratios t5h b5h = t5v b5v =1.2 ·10−2 The models differed from each other with the rivet spacing and layout. The L-S plots of these models are presented in Fig.19. The results of experiments are compiled in Table 4. Table 4.Results of experiments Model σcomcr P com cr P exp ult number [MPa] [kN] [kN] 5 52.06 46.85 50.70 The average ultimate stresses are approximately 3.32 times lower than the yield limit of used brass sheets material. Therefore, thematerial properties of brass were exploited in 30.2%. The critical loads for both rectangular multi- cellmodelsweredetermined according to thedefinition given in 3.5.1 (Fig.18). The critical stresses of bothmodels aremutually close (theydifferby 2%only). Stability and load carrying capacity... 451 Fig. 19. L-S curve of rectangular multi-cell models 5. Results of numerical computations For the compressed multi-cell column of a square or rhomboidal shape with equal width of all walls simply supported on loaded edges, the critical stress of local buckling can be approximated (calculated) from (2.3) or (2.4). The buckling load factor k can be determined numerically from the ana- lysis of one cell of the column (it is valid for cells of arbitrary cross-section shapes and wall thickness). Themost accurate value of local buckling critical stress can be determined with FEM software considering the whole column. Fig. 20. Influence of initial imperfections Assuming that a single wall of the square or rhomboidalmulti-cell column is a long compressed plate with all edges simple supported, the load carrying capacity can be calculated as an average ultimate stress for thewall of equal to 452 M. Królak et al. the other walls thickness. In Fig.20, it is shown that imperfection amplitudes have significant influence on the average ultimate stress. The plots presented in Fig.20 were obtained by an analytical-numerical methodwith the assumption that thematerial is linearly elastic and perfectly plastic andhad followingparameters: E =2·105MPa, ν =0.3,Re =203MPa. These are properties of a steel sheet of thickness equal to 0.5mm, which was used to manufacture models of the square multi-cell columns. The results of computations of ultimate loads performed with FEM pac- kages are reliable only when the real tensile test data are implemented in the numerical calculations. A case of numerical computations carried outwith the FEM package ANSYS is submitted in the next two figures. Fig. 21. L-S curve for the squaremulti-cell model obtained in FEA Fig. 22. L-S curve for the rectangularmulti-cell model obtained in FEA Stability and load carrying capacity... 453 6. Comparison of experimental and numerical results The results of local buckling stresses and ultimate loads obtained in experi- mental tests and in numerical computations are presented in Table 5. 7. Conclusions From the carried out experimental tests, the following conclusions can be drawn: • For compressed multi-cell columns of equal dimensions, equal cross- section areas, made of the same materials and exactly in the same way supported: – local buckling stress ofwalls increaseswith the increase of the para- meter n– thenumberof cells adjacent to the single columnexternal wall (which is not equivalent to the increase of the total number of column cells), – for some value of the parameter n, the whole column cross-section can plasticize before local buckling of it occurs, – for a multiple number of adjacent cells, the width of the single wall becomes small and local buckling of the whole wall can occur instead of the single-cell wall, – when the parameter n increases, the ultimate load of the column increases too, but this progress ismuch slower than that of the local buckling critical stress, – the classical local buckling phenomena did not occur for the tested models ofmulti-cell columns because of different types of imperfec- tions, – with the progress of compression, the number of buckled cell walls rises, – a change of stiffness of themulti-cell column takes place in practice after all walls have buckled (some of them are in the elastic post buckling range or even elastic-plastic state), – the suggestedmethod to determine the local buckling critical stress based on the experimental L-S curves seems to be justified for such structures, 4 5 4 M . K r ó l a k e t a l . Table 5 Mo- del No. Cross- -section shape Cell Wall thick- ness [mm] Cross- Ultimate Critical Average Critical Ma- ter- ial wall -section load load ultimate stress stress width area Exp. Comp. Exp. Comp. Exp. Comp. Exp. Comp. Introd. [mm] [mm2] [kN] [kN] [kN] [kN] [MPa] [MPa] [MPa] [MPa] form. 1 Squ. n=1 312.5 1.24 1550 51.6 66.0 12.6 17.1 33.3 42.6 8.1 11.0 11.4 steel 2 Squ. n=5 63.6 0.5 1526 101.6 156.2 67.0 63.0 66.6 102.3 43.9 41.3 44.7 steel 3 Squ. n=5 64.4 0.5 1546 99.4 157.0 68.0 62.2 64.3 101.6 44.0 40.2 43.6 steel 4 Squ. n=5 63.2 0.5 1517 94.6 155.2 62.0 63.4 62.4 102.3 40.9 41.1 45.3 steel 5 Rectangle bv =50 tv =0.6 900 50.7 79.0 39.5 44.1 56.3 87.8 43.9 49.0 52.0 brass n=5 bh =25 th =0.3 6 Rectangle bv =50 tv =0.6 900 47.2 79.0 40.2 44.1 52.4 87.8 44.7 49.0 52.0 brass n=5 bh =25 th =0.3 Stability and load carrying capacity... 455 – great diversity of geometrical imperfections of multiple column walls causes that their influence on decreasing the ultimate load is less significant than for the single-cell column, – the global buckling stress of the multi-cell column decreases in the case when the parameter n equals 2 or 3 and increases with n, approaching gradually the critical stress of the single-cell column, • For a greater number of cells, the interaction of different modes of buc- klingmay take place, which is difficult to predict in theoretical conside- rations and which was proved in the numerical analysis. The experimental tests gave quite good agreement of the results for compati- ble models and between the experimental and numerical data, which is often difficult to achieve in stability investigations. Acknowledgemment The current work has been done in the frame of the research project KBN 4T07A02829. References 1. Chen W., Wierzbicki T., 2001, Relative ments of single cell, multi-cell and foam-filled thin-walled structures in energyabsorption,Thin-Walled Structures, 39, 287-306 2. Grądzki R., Kowal-MichalskaK., 1985,Elastic and elasto-platic buckling of thin-walled columns subjected to uniform compression,Thin-Walled Struc- tures Journal, 3, 93-108 3. KimH.S., 2002,New extrudedmulti-cell aluminumprofile formaximumcrash energy and weight efficiency,Thin-Walled Structures, 40, 311-328 4. Kołakowski Z., Kowal-MichalskaK., 1999, Selected Problems of Instabi- lity in Composite Structures, Technical University of Lodz, Łódź, Poland 5. Królak M., Kowal-Michalska K., 2004a, Stability and load-carrying ca- pacity of multi-cell columns subjected to compression, Proc. of IV Coupled Instabilities in Metal Structures, CIMS 2004, Pignatario M. (Edit.), Rome, 213-222 6. Królak M., Kowal-Michalska K., 2004b, Stability and ultimate load of multi-cell orthotropic columns subjected to compression,Proc. of the 8-th SSTA Conference, Jurata, Poland, 235-239 456 M. Królak et al. 7. KrólakM.,Kowal-MichalskaK.,ManiaR., Świniarski J., 2006,Bada- nia doświadczalne stateczności i nośnościmodeli cienkościennych słupówwielo- komorowychpoddanych równomiernemuściskaniu,MateriałyXXII Sympozjum Mechaniki Eksperymentalnej Ciała Stałego, Jachranka, 297-302 8. KrólakM., Kowal-MichalskaK., Świniarski J., 2007, Stateczność i no- śnośc cienkościennych słupówwielokomorowych równomiernie ściskanych,Ma- teriały XX Konferencji Problemy Rozwoju Maszyn Roboczych, Zakopane 9. MaquoiR.,MassonetCh., 1971,Nonlinear theoryof post-buckling resistan- ce of large stiffened box girders, IABSE Publications, 31, 11, 91-140 10. Massonet Ch., Maquoi R., 1973, New theory and tests on the ultimate strength of stiffened box girders,Proc. Int. Confer. on Steel Box Girders Brid- ges, The Inst. of Civil Engrs, Loudres 11. Volmir S.A., 1968, Stability of Deforming Systems, Science Moscow [in Rus- sian] Stateczność i nośność wielokomorowych cienkościennych słupów o prostokątnych przekrojach poprzecznych Streszczenie Pracapoświęcona jest teoretycznej, numerycznej i doświadczalnej analizie statecz- ności i nośności granicznejwielokomorowychcienkościennych słupówo prostokątnych i kwadratowych przekrojach poprzecznych poddanych osiowemu ściskaniu (odpowia- dającemu równomiernemu skróceniu słupa). Rozważania teoretyczne dotyczą lokal- nej i globalnej utraty stateczności wielokomorowych ortotropowych słupów o obrysie prostokątnym, z prostokątnymi komorami.Wykazano, że dla wielokomorowego słupa wykonanego z tego samego materiału i o takim samym polu przekroju poprzeczne- go wartości lokalnego naprężenia krytycznego ścian słupawzrastają gwałtownie wraz ze wzrostem liczby komór. Doświadczenia przeprowadzone dla izotropowych słupów potwierdziły wzrost zarówno naprężeń krytycznych, jak i nośności badanych modeli wraz zewzrostemkomór.Wpracy podano pewnewnioski, któremogą być przydatne przy projektowaniu cienkościennych słupów o przekrojach skrzynkowych. Manuscript received July 23, 2008; accepted for print November 18, 2008