Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 51, 2, pp. 497-504, Warsaw 2013 EXPERIMENTAL AND NUMERICAL (FSM) INVESTIGATIONS OF THIN-WALLED BEAMS WITH DOUBLE-BOX FLANGES Piotr Paczos Poznan University of Technology, Institute of Applied Mechanics, Poznań, Poland e-mail: piotr.paczos@put.poznan.pl In the paper, experimental and numerical investigations of thin-walled beams with double- box flangeswere presented.Theywere a continuation of researches conducted at theUnit of Strength ofMaterials and Structures at PoznanUniversity of Technology.Numerical results obtained with the Finite Strip Method (FSM) were compared with experimental ones and used for validation of analytical solutions. Key words: thin-walled beams, FSM, experimental investigations 1. Introduction In the paper, results of numerical analyses and experimental investigations of cold-formed, thin- walled beams with double-box flanges were presented. The beams were loaded with a ben- dingmoment. Software CUSFMver. 3.12 of Prof. B. Schafer (http://www.ce.jhu.edu/bschafer/ cufsm/; Schafer and Ádány, 2006), based on the Finite Strip Method, was used for numerical simulations. The obtained results were compared with the findings of experimental studies. The buckling of thin-walled beams, with clear distinction between buckling of web and flan- ges, was described in this paper. Thin-walled beams tend to buckle under bending. Therefore, it is justified to look for new shapes of cross-sections thatwould increase the strength and stability of beams. Researches on thin-walled structureshave beenconducted formanyyears. Someof themhave focused on experimental investigations, others on analytical models. Contemporary structural materials make it possible to reduce the weight of structures with a simultaneous increase in their strength and stability. Thin-walled beams tend to buckle locally due to the high wall thickness-to-depth ratio. Similar studies were done by Pastor and Roure (2008), Camotim and Dinis (2009), who investigated thin-walled beams loaded with bending moments, concentrated forcers or distributed loads. They presented formulas for critical loads and equations describing the interaction between local and global buckling. The load capacity of cold-formed thin-walled beams is usually restricted by their stability and post-buckling behaviour. Their strength was considered by Cheng and Schafer (2007) and Trahair (2009). Experimental investigations, stress anddisplacement distribution of cold-formed beamswere shownbyPaczos andMagnucki (2009).Other examples of papers directly connected with the subject of this work are Biegus and Czepiżak (2008), Magnucki and Paczos (2009), Magnucki et al. (2010), Paczos andWasilewicz (2009) orMahendran and Jeyaragan (2008). The paper concerns thin-walled beams with double-box flanges made of a cold-rolled steel sheet (Fig. 1). This is a part of broader research on thin-walled beams and search for new shapes of cross-sections that would increase the strength and stability of beams. They have been conducted for a few years at the Unit of Strength of Materials and Structures at Poznan University of Technology. 498 P. Paczos Fig. 1. The cross-section of the investigated beams 2. Description of the investigated cross-sections All the investigated beams were cold formed and made of steel sheets of known material pro- perties. Their cross-section was the same but their wall thickness varied. The beams were ma- nufactured by Polish company “Pruszyński Sp. z o.o.”, Sokolow, Poland. 2.1. Geometrical properties The dimensions of the cross-section of the investigated beams were following (Fig. 1): beam depth H = 160mm, flange width b = 0mm, wall thickness t = 1.00mm and t = 1.25mm, c=23mm, d=18mm, e=14mm, f =17mm, g=15mm. The presented double-box flanges improved the strength and stability of beams and decreased their weight. Fig. 2. Longitudinal dimensions of beams and a loading diagram The longitudinal dimensions of beams and a loading diagram are presented in Fig. 2. The marked dimensions were following: • total length Lt =2000mm, • distance between concentrated forces L=500mm, • distance between the concentrated force and the support L0 =730mm. The tensile force of the testing machine was divided into two equal concentrated forces. The beams were supported (fixed) at two places. This system of forces and supports resulted in a constant bendingmoment at the centre segment of beams. Experimental and numerical (FSM) investigations of thin-walled beams... 499 2.2. Material properties The investigated beams weremade of steel sheets. Their material properties were following: Young’s modulus E = 181GPa, Poisson’s ratio ν = 0.3 and yield strength σeH = 329MPa. They were determined by testing 5 samples prepared according to Eurocode 3. Fig. 3. The stress-strain curve of specimens – steel DX51 The results of tensile tests were presented inFig. 3. The specimensweremade of steel DX51, hot-dip galvanized, zinc coating 200g/m2. These findingsmade it possible to determinematerial properties that were used in numerical analyses and analytical models. 3. Numerical analysis – FSM In order to better understand the buckling of the investigated beams, numerical analyses were done.Moreover, the obtained resultswere comparedwith the experimental ones. The beamswe- re analysed using FSM (Finite StripMethod). In recent years, there have been a few significant works devoted to this method. For example, Adany and Schafer (2006a) presented a method for calculating critical forces of thin-walled beams based on FSM. They used GBT for decom- position of different buckling modes. Their new method was called cFSM (constrained FSM). Its application and some examples were presented by Adany and Schafer (2006b). Adany et al. (2010) demonstrated the computer programCUFSM for determination of bucklingmodes based on cFSM. Djafour et al. (2010) proposed a procedure for calculation of the constrain matrix in cFSM for global and distorsional buckling modes. It can be used for analysis of closed-section beams. The beams were modelled with 36 elements (37 nodes). They were loaded by a bending moment constant along thewhole beam, in such away that the obtained load factors were equal to actual critical stresses divided by 100. The discretization error and the convergence of results were checked by doubling and quadrupling the number of elements. Simply supported beams were analysed since this kind of support was considered during experimental investigations. The buckling modes and load factors are presented in Figs. 4 and 5. They were equal to: • 1.41 for the beam of wall-thickness t=1.00mm, • 2.20 for the beam of wall-thickness t=1.25mm. The critical stresses for the determined buckling load factors were equal to σcr =141MPa and σcr =220MPa, respectively. The bucklingmodes presented inFig. 6 referred to local buckling in an elastic range (stresses were below the yield strength). In both cases, there was an interaction between the compressed flange and web of the beam. The buckling modes were similar, but in the beam of the wall 500 P. Paczos thickness 1.00mm, the flange buckled outwards and the web inwards. In the beam of the wall thickness 1.25mm, the flange and webmoved in the opposite direction. Fig. 4. FSM – critical forces vs. half-wave length, a beamwith double-box flanges, wall thickness t=1mm Fig. 5. FSM – critical forces vs. half-wave length, a beamwith double box-flanges, wall thickness t=1.25mm Fig. 6. FSM – bucklingmodes of the beam, wall thickness is equal to (b) t=1mm and (c) t=1.25mm 4. Experimental investigations The experimental investigations were done at the Laboratory of Strength of Materials and Structures, Institute of Applied Mechanics at Poznan University of Technology, Poland. Ten beams of the wall-thickness 1.00mm and 1.25mm and length L = 400, 500, 600mm were tested. Differences between the results obtained for the beams of the same dimensions were not big. Therefore, the results were averaged and presented in tables and drawings. The principal moments of inertia (Fig. 1) were equal to: Iy = 3030108mm 4 (beams of the wall thickness t=1.00mm) and Iy =3787635mm 4 (beams of the wall thickness t=1.25mm). The following equations were used for expressing the relationship between the tensile force, bendingmoment andmaximum stresses F = 2Mbcr L0 σdos = Mbcr Wy (4.1) Experimental and numerical (FSM) investigations of thin-walled beams... 501 where: Mbcr – critical bendingmoment, L0 – distance between the concentrated force and the support (Fig. 2), F – critical force, Wy – section modulus. All the tests were conducted on a tensile testing machine ZWICK Z100 with mechanical drive and a custom test stand for channel beams (Fig. 7). At the web and compressed flange, therewere placed four foil strain gauges. The deflection of the beamswasmeasuredwith 3 beam deflection sensorsWI10 (Fig.8). Fig. 7. Test stand: diagram and picture Fig. 8. Position of strain gauges and beam deflection sensors In Fig 9, there is a graph showing the relationship between the tensile force and deflection of beamsmade of steel (DX51) sheets with thickness 1.00mm and 1.25mm. The characteristic points on the graph, were this relationship stopped to be linear, meant that at that load, the beams buckled locally. The first of points was considered as the critical load because then the cross-section of the beam got deformed. However, sometimes this deformation was not visible clearly. This assumption was confirmed and validated by FSM. Fig. 9. Tensile force vs. deflection During experiments, an interaction between the buckling of the web and compressed flange was observed.The graphs presented inFig. 9 referred to thewhole experiment until themoment 502 P. Paczos when the investigated beams collapsed.A slight increase of thewall thickness by 0.25mmalmost doubled the value of critical forces. Fig. 10. The beam before and after tests Thin-walled members usually collapse as a result of the plastic mechanism of damage (Fig. 10). If a beam is loaded with a monotonically increasing load, its behaviour can be di- vided into four stages: pre-buckling, non-linear behaviour, post-buckling (elastic-plastic range) and damage. Local buckling usually appears simultaneously with global one, and is a cause of low limit load which depends on a few factors: overall dimensions of the structural member, boundary conditions, kind of load and shape of the cross-section of the beam. The investigated beams buckled also quite quickly, and inmost cases this was a local bucklingmode (Figs. 6, 10). 5. Conclusions Theobtained results prove that it is useful to donumerical analyses, e.g. based onFSMorFEM. They help to validate and properly interpret results of experimental investigations because, in some cases, it may be difficult to determine the critical load. The initial deformation of the cross-section of a beam (buckling) may be hard to notice in the right moment because it is small. In Table 1, the results of experimental investigation and numerical analyses are gathered. The presented critical stresses referred to the determined critical loads 15kN and 30kN for beams of the wall thickness, respectively, 1.00mm and 1.25mm. Table 1.Comparison between the results of experimental investigation and numerical analyses (FSM) Kind of research Wall thickness t=1.00mm t=1.25mm Experimental investigations – stresses [MPa] 146 225 Numerical analysis (FSM) – stresses [MPa] 141 220 Difference [%] 3.5 3.2 Numerical simulations make it possible to analyse the buckling of flanges, web and the interaction between different bucklingmodes. They help to build proper analytical models, e.g. by giving indications of some buckling modes. The obtained results led to the following conclusions: • the results of numerical analyses were in agreement with the outcome of experimental investigations, the differences between stresses were below 4%, buckling modes were the same as well, Experimental and numerical (FSM) investigations of thin-walled beams... 503 • further increase of load, after buckling, caused not only bigger deformation of the cross- sections but also resulted in other bucklingmodes, • the results justify searching for new shapes of cross-section of cold-formed thin-walled beams that have not been included in standards. The comparison of experimental investigations with numerical analyses based on the Finite Strip Method and CuFSM software by prof. Shafer (http://www.ce.jhu.edu/bschafer/cufsm/) was presented in this paper.Numerical analyses based on theFinite ElementMethodwere done as well, and their results were shown at Nordic Steel Construction Conference in 2012 (Paczos et al., 2012). The weight of the beam of the wall thickness t = 1.00mm was equal to 9.52kg, whereas the beam of the wall thickness t = 1.25mm weighed 11.78kg. The critical forces of those beamswere equal to 14.4kN and 27.9kN, respectively. This means that with a relatively small increase of weight (23.7%), their strength rose by 93.7%. A similar phenomenon was observed during experimental investigations of classic cold-formed channels with a reinforced web. The critical forces of those channels were equal to 1.4kN and 2.8kN for beams of the wall-thickness t=1.00mm(7.92kg) and t=1.25mm(9.92kg), respectively, i.e. the 25% increase in theweight of the beam doubled their strength. However, in the case of channels with boxed flanges 20%, heavier beams had 10 times bigger strength (increase from 1.4kN to 14.4kN). This work presents preliminary investigations done in order to better understand a problem and search for new, untypical shapes of the cross-sections of cold-formed thin-walled beams that could increase their strength and stability. Acknowledgement This paper was supported by Polish Ministry of Science and Higher Education, grant no. 2073/B/T02/2010/39. References 1. AdanyS., JooA.L., SchaferB.W., 2010,Bucklingmode identification of thin-walledmembers by using cFSM base functions,Thin-Walled Structures, 48, 806-817 2. Adany S., Schafer B.W., 2006a, Buckling mode decomposition of single-branched open cross- sectionmembers via finite strip method: Derivation,Thin-Walled Structures, 44, 563-584 3. Adany S., Schafer B.W., 2006b, Buckling mode decomposition of single-branched open cross- section members via finite strip method: Application and examples, Thin-Walled Structures, 44, 585-600 4. Biegus A., Czepiżak D., 2008, Experimental investigations on combined resistance of corru- gated sheets with strengthened cross-sections under bending and concentrated load, Thin-Walled Structures, 46, 303-309 5. Camotim D., Dinis P.B., 2009, Coupled instabilities with distortional buckling in cold-formed steel lipped channel columns, Stability of Structures XIIth Symposium, Zakopane 7-11 September 2009, Technical University of Lodz, 13-32 6. ChengYu, SchaferB.W., 2007, Simulation of cold-formed steel beams in local and distortional bucklingwith applications to the direct strengthmethod, Journal of Constructional Steel Research, 63, 581-590 7. Djafour M., Djafour N., Megnounif A., Kerdal D.E.A., 2010, Constrained finite strip method for open and closed cross-sectionmembers,Thin-Walled Structures, 48, 955-965 8. http://www.ce.jhu.edu/bschafer/cufsm/ 504 P. Paczos 9. MagnuckiK., PaczosP., 2009,Theoretical shape optimization of cold-formed thin-walled chan- nel beamswithdropflanges inpurebending,Journal of Construction Steel Research,65, 1731-1737 10. MagnuckiK., PaczosP.,Kasprzak J., 2010,Elastic buckling of cold-formed thin-walled chan- nel beams with drop flanges,ASCE: Journal of Structural Engineering, 136, 7, 886-896 11. Mahendran M., Jeyaragan M., 2008, Experimental investigation of the new built-up liteste- el beams, Proc. 5th International Conference on Thin-Walled Structures, Vol. 1, M. Mahendran (Edit.), QueenslandUniversity of Technology, Brisbane, Australia, ICTWS, 18-20 June 2008, 433- 441 12. Paczos P., Kasprzak J., Magnucki K., 2012, Local elastic buckling of cold formed thin- walled channel beams with non-standard flanges, Nordic Steel Construction Conference 2012, 5- 7 September 2012, 811-820 13. Paczos P., Magnacki K., 2009, Experimental investigations of cold-formed thin walled c-beams withdropflange,Sixth InternationalConference onAdvances in Steel Structures,S.L.Chan (Edit.), ICASS 2009, Vol. I, Hong Kong, 16-18December 2009, 395-400 14. Paczos P., Wasilewicz P., 2009, Experimental investigations of buckling of lipped, cold-formed thin-walled beams with I-section,Thin-Walled Structures, 47, 1354-1362 15. Pastor M.M., Roure F., 2008, Open cross-section beams under pure bending I Experimental investigations,Thin-Walled Structures, 46, 476-483 16. Schafer B.W., Ádány S., 2006, Buckling analysis of cold-formed steel members using CUFSM: conventional and constrained finite strip methods, 18-th International Specialty Conference on Cold-Formed Steel Structures, October 26-27, 2006, Orlando, Florida 17. Schafer B.W., 2006, Designing cold-formed steel using the direct strength method, 18-th In- ternational Specialty Conference on Cold-Formed Steel Structures, October 26-27, 2006, Orlando, Florida 18. Trahair N.S., 2009, Buckling analysis design of steel frames, Journal of Constructional Steel Research, 65, 1459-1463 Badania doświadczalne i numeryczne (FSM) belek cienkościennych z półkami skrzynkowymi Streszczenie Wpracyprzedstawionobadaniadoświadczalne i numerycznebelek cienkościennychz półkami skrzyn- kowymi. Zagadnenie to jest kontynuacją badań przeprowadzonychw ZakładzieWytrzymałosci Materia- łów i Konstrukcji Politechniki Poznańskiej. Wyniki numeryczne otrzymane z wykorzystaniem metody pasm skończonych porównano z wynikami uzuskanymi z eksperymentu, a nastepnie użyto do weryfikacji rozwiązań analitycznych. Manuscript received January 27, 2012; accepted for print December 20, 2012