INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume 2, No 4, 2012

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1 INTERNATIONAL JOURNAL OF CIVIL AND STRUCTURAL ENGINEERING Volume, No 4, 01 Copyright 010 All rights reserved Integrated Publishing servies Researh artile ISSN Strutural Modelling of Stability of Plane Sway Frames Department of Civil Engineering, Rivers State University of Siene and Tehnology, Port-Harourt doi: /ijser ABSTRACT This paper presents the results of testing of experimental models with varying beam to olumn stiffness ratios with the aim of establishing their relationship on the stability of seleted rigid frames. Test frames with beam to olumn stiffness ratios of 0.5, 0.67, 1.0, 1.33 and.0 were fabriated, and subjeted to ompressive loading in the laboratory. The results obtained from the tests after appliation of appropriate sale fators are ompared to the prototype ritial loads obtained from the sidesway stability funtion approah by Horne and Merhant. Results from the analysis show auray as high as 98.5% when ompared to the results obtained from numerial methods. The results of the tests further emphasise the ritiality of sale fators when using modelling approah for solution of strutural stability problems. Keywords: Frames, Stability, Stiffness, Modelling, Critial Load. 1. Introdution Frames an be lassified as one of the most important naturally ourring and manmade support systems in existene and their appliations have ontinued impat tremendously on the eonomis of physial infrastruture development worldwide. The effiieny of a frame no mater its onfiguration, is haraterized by its load arrying apaity of its olumns, and this apaity is defined by the ritial load that will ause the frame to beome unstable. Researhers have ontinued to investigate means of aurately determining these ritial loads in order to optimise the use of frames. Numerous approahes exist for the evaluation of apaities of frames, popular amongst whih are analytial methods. However with inreasing size, geometri omplexity and aompanying variation in boundary onditions, the onept of strutural modelling as a means of solution to frame stability problems an be used as an effetive alternative approah. Strutural modelling is defined as the assembly of strutural element(s) built to a redued sale whih is to be tested, and for whih the laws of similitude must be employed to interpret test results. This definition highlights the model-prototype similitude relationship whih is atered for by the sale fator employed during modelling, and subsequently in interpretation. The appliation of strutural models in Strutural dates bak several enturies. An extensive review of their development is a subjet of suh publiations as disussed by Sabins et al. Reeived on Marh, 01 Published on May

2 Modelling is applied mainly as a solution to investigation of omplex strutural systems, employing onepts as similitude dimensional analysis.. Strutural Modelling of Plane Sway Frame Destrutive testing remains one of the most aurate means of verifying load arrying apaity of strutures. Logially, the loser a model is to its atual size, the more aurate the test data obtained from it, if the loading on the struture is aurately simulated. Life size modelling is very hallenging in terms of ost of fabriation, loading equipment, instrumentation and testing. In most ases, prototype testing is almost impossible from the eonomi standpoint. The relevane of strutural models in the study, design and analysis of strutural systems is justified by the great savings resulting from signifiant redution in loading and size of models in omparison with those required in the orresponding prototype tests. Furthermore, the appliation of strutural models enables a wider partiipation in strutural researh in those areas where sophistiated laboratory failities and loading systems are not available. The issues of size effets have been raised in various works. However, the authors are in the agreement with the position that size effets are an intrinsi harateristi of models sine the atomi and moleular dimensions an never be modelled to a given dimensional sale fator. The reliability of strutural model is further substantiated by the fat that arefully rafted test models have been employed to reprodue, with reasonable high auray, both qualitatively and quantitatively the behaviour of strutural systems. Strutural modelling has also reeived serious attention in the study of Ephraim et al, who investigated the load arrying apaity of sandrete masonry wall on a series of 1:10 strutural model. Meanwhile, Uzoewhulu has reently mobilized the modelling tehnique in a suessful study of the lateral resistane of reinfored onrete frames, in-filled with ementstabilized laterite bloks under raky loading. It is interesting to observe that the results of these investigations show a lose agreement of prototype and model behaviour up to the ultimate states of failure. While analytial approahes to stability of plane sway frames have reeived signifiant attention in the researhes of Merhant, Horne et al, Hoenderkamp [6] and Orumu. Tehnial literature also reveals a onsiderable number of experimental works with models of sway frames. A great number of these researhes are assoiated with investigation of stability of sway frames in reinfored and pre-stressed onrete materials, as shown in the modelling approah by Fisher et al ; they however do not apply the priniples of similitude relationship to their work, or appliation of similitude priniples for extrapolation of laboratory data for appliation to life strutures through the use of sale fators. This researh is an effort by the authors to extend the strutural modelling proess to the stability studies of frames using the operational and preditive requirements evolved from similitude mehanis. 3. Methodology The aim of this researh is to produe diret models to represent prototype frames with dimensional properties in three ategories. The models are to be tested to failure, and the 1099

3 results interpreted using the priniples of similitude to verify the saled up result with results obtained for the prototype by mathematial means. P P P P B I C θ Β M B M C θ I 1 L h A D A D A B C ACTION M AB M BA M BC M CB M CD M DC D Rotate B sk 1 sk 1 B 4k k 0 0 -s (1+) B Rotate 0 0 k 4k k 1 B s k 1 s k 1 -s (1+) k b C Sway -s (1+) k 1 h -s (1+) k 1 h 0 0 -s (1+) k 1 h -s (1+) k 1 h 1 4s 1( + k ) m h NOTES 1. k 1 = k b. k = k The study is thus presented in two parts, namely: dimensional analysis and laboratory experimental tests. The details of these are overed in the relevant subsetions, whih follow. 3.1 Dimensional analysis The governing equation of the ritial load for the one storey plane frame is assumed to be of the form F(P, E, L, H, A, I)= 0 equation 1 Where the assumed independent variables have the following definitions: P- Critial Load L- Span of Frame 1100

4 H- Height of Frame Strutural Modelling of Stability of Plane Sway Frames E- Modulus of Elastiity of frame material A- Cross setional Area of olumn I- Moment of Inertia of olumn. Assuming the modulus of elastiity E and the representative length L as dimensionally independent variables, equation 1 an be reast into an expliit form with dimensionless ratios: PL kb G, = 0 equation E k El P P = k = φ k 1 El b k φ k b equation 3 equation 4 Sine the onstant P must be the same for model and prototype, the preditive ondition is obtained by foring the dimensionless ratios: P P p m = E E l m m pl p k φ p k φm b b k k equation 5 Thus, the preditive sale fator for the ritial load S p = S L S E P m Assuming the same material in model and prototype, i.e. S E =1, the derived similitude onditions for loads, material properties and geometry are given in Table 1. Table 1: Prototype and Model Properties for sample P1 S/no Property Dimension Sale Fator Prototype Model Loading Critial Load F S L P CR(P1) P CR(model) Stress FL kN/mm 460kN/mm Material Property Modulus of FL - 05 kn/mm 05 kn/mm 1 Elastiity Poisons Ratio Density FL kN/mm kN/mm

5 3.0 Geometry Length Column L S L L 1 S L L S L 50 5 Beams L 3 S L L 4 S L L 5 S L Experimental materials and proedure The experimental aspets of the researh were onduted in the Strutures laboratory of the Rivers State University of Siene and Tehnology, Port-Harourt, Nigeria Materials The materials employed in the fabriation of the test modes is high yield arbon steel reinforement bars, and its properties are summarised in the Table 1 above. 3.. Model properties The plane frames were fabriated as two storey plane frames. The frames are lassified aording to their beam to olumn stiffness ratios in the value: 0.5, 0.67, 1.0, 1.33 and.0. The olumns are fabriated as ontinuous members, with the beams framing into the olumns at the first and seond floors, onneted via full fillet welds at the beam olumn interfae Experimental proedure A total of five plane frame samples were fabriated to the speifiations outlined in setions 3..1, and 3... The objetive of the test is to load the frame to failure in a bid to determine experimentally the ritial load of the frame. Axial load appliation from the loading frame to the test frame is ontrolled in suh a way that eah of the frame olumns reeives approximately the same amount of load. This loading was ahieved via the use of the fabriated load transfer frame illustrated in Figure. Figure 1: Load transfer frame Figure : Load transfer frame 110

6 3..4 Experimental set-up and instrumentation The experimental set-up is as illustrated in Figure 3, whih shows a shemati of the set-up. Speimens were plaed on the loading frame platform. Spirit level and shim plates were used to ensure plumbness of the frames as they were being plaed on the frame. When plumbness was ahieved, the test frames were firmly seured to the loading frame base by bolting through the slots provided in the test frame supports. Figure 3: Shemati of Test Set-up The load transfer mehanism was then plaed on the frame ensuring that eah of its four legs plaed diretly on the top of eah of the frames olumn. The loading frame was also heked for plumbness, using a spirit level and shim plates. After verifiation of plumbness of the test frame and load transfer mehanism, the test frame was loaded with inrements of 5kN until failure. The instrumentation employed for the tests was mainly the load gauge of ELE make, appropriately alibrated prior to use. Plate 1 shows the test set-up and the testing proess of a typial frame. Plate 1: Test Set-up 1103

7 4. Disussion of results The results obtained from the prototype frames and ompared to the predited results are presented in this setion. Table below shows a summary of the theoretial results obtained for the prototype (P1, P, P3..) under onsideration, the equivalent test results obtained from the diret model and the predited results obtained using the sale fators for various beam to olumn stiffness ratio for both prototype and model k b /k ; The model ritial load P1 were obtained from the laboratory exerise; the prototype ritial load were alulate using the Horne and Merhant method; and the predited ritial load expressed as the model ritial load multiplied by the sale fator obtained from the prototype-model similitude harateristis. S/no. SAMPLE k b /k Table : Result summary CRITICAL LOAD P1 P P3 l b =3.0m l b =0.3m l b =4.5m l b =6.0m Prototype Model Predited Prototype Predited Prototype Predited l =1.5m l =0.15m l =.5m l =3.0m Results obtained are typial of diret modelling tehniques show that as the beam to olumn stiffness ratio inreases there is a orresponding inrease in the ritial loads for both the model and the prototype, and the predited ritial loads. Figure 4: Graph of P r vs. k b /k highlighting the variane in results between predited ritial loads and the prototype ritial loads for frames under investigation. The graphs presented in Figure 4 show plots of the predited ritial loads and prototype results obtained by numerial means. The graph shows the high orrelation (98.5% to 93.5%) 1104

8 of when ompared, exhibiting variane of only minimum of 1.5% to maximum of 6.5%. The results highlight how sale fators, whih satisfy similitude requirements, affet auray of predited results. The sale fator for the elasti modulus is maintained as unity, representing the similarity in material types between the model and the prototype (this is typial of diret models), while the size fator is modified aordingly aording to the prototype to model sale ratio. The high degree of auray of results obtained from modelling, ompared to the results obtained by analytial means learly gives redene to the importane of satisfying similitude requirements when using modelling as a basis for any researh for strutural purposes. Many laboratory strutural investigations used for investigative, verifiation and validation purposes reah their onlusions without heking to satisfy similitude requirements. The subjet of modelling is subjet worthy of detailed attention in design odes, espeially when design engineer have to takle design of omplex strutures. While setion of ACI 318 permits the use of modelling in shelled strutures, the extent of possible range of appliation of modelling tehniques even to simpler problems is however undersored. Importane of modelling appliation to strutural design and researh an be of great eonomi benefit and advantage to developing ountries where redued sale models an be used at a redued ost of fabriation, load requirement and instrumentation plus boosting partiipation in researh and development. 5. Conlusion On the basis of the analysis and disussion of results obtained in the above experimental work, the following onlusions were arrived at. 1. Diret strutural model is highly appliable in the study of stability of two storey plane frames.. The model yields a near linear relationship between the ritial load on frame and its beam to olumn stiffness ratio. This is an agreement with established trend. 3. There is a lose agreement, about 98%, between loads predited by the model study and those obtained analytially, both for the experimental frame as well as the theoretially simulated frames onsidered in the study. 4. The importane of modelling tehniques an be diretly employed in design espeially in ases where there are hanges in geometri harateristis of the elements under onsideration. 6. Referenes 1. Sabins G.M, Harris G. H, White R. N and Mirza Saheed., (1983), Strutural Modelling and Experimental Tehniques. Prentie Hall International In.. Ephraim M. E and Wenapere D.A., (005), Strutural Modelling of Compressive Strenght of Sandrete Masonry Strutures. Proeedings of the Nigerian Institution of Agriultural Engineers, 7, pp

9 3. Uzoewulu N. L., (006), Investigation of Strength of Sway Frame with Cement Stabilized Laterite Blok Infill. Masters degree thesis submitted to the Rivers State University of Siene and Tehnology Port-Harourt. 4. Horne M.R. and Merhant W., (1965), The Stability of Frames, Pergamon Press, New York, N.Y. pp Merhant W., (1955), The Critial Load of Tall Building Frames. The Strutural Engineer, 33(3) 6. Orumu S.T., (1997), The stability of sway frames by method of superposition of frames: Masters degree thesis submitted to the Rivers State University of Siene and Tehnology Port-Harourt. 7. Lightfoot, E., (1956), The analysis of Wind Loading of Rigid Jointed Multi-storey Building Frames, Civil Engineering, 51, 757, Gregory Fisher and Vitor C. Li., (003), Intrinsi Response Control of Moment- Resisting Frames Utilizing Advaned Composite Materials and Strutural Elements. ACI Strutural Journal 100-S18 PP Aristizabal-Ohoa, J.D., (1997), Storey stability of Braed, partially Braed, and Unbraed frames: Classial Approah. Journal of Civil Engineering, ASCE, 13(6), pp Zalka, K.A., (1999), Full-Height Bukling of Frameworks with Cross-braing. Proeedings of the Institute of Civil Engineers 134, pp Ephraim, M.E and Chinwah J.G, Orlu I.D., (1990), Mehanism Approah to Composite Frame and Infill. Proeedings of the seond International Conferene on Strutural Analysis and Modelling (SEAM ), University of Siene and Tehnology, Kumasi, Ghana, 1, pp Alami, Z.X and Ferguson, P.M., (1963), Auray of Models Used in Researh on Reinfored Conrete, Proeedings of the Amerian Conrete Institiute, 60(11), pp Amerian Soiety for Testing and Materials, (1978), ASTM A36-77a, Standard Speifiation for Strutural Steel, Annual Book of Standards, Part 4, Philadelphia, pp