Dynamic method for measuring critical buckling load of sway frames

Transcription

1 Proceedings of the 9th International Conference on Strctral Dynamics, EURODYN 14 Porto, Portgal, 3 Jne - Jly 14 A. Cnha, E. Caetano, P. Ribeiro, G. Müller (eds.) ISSN: 311-9; ISBN: Dynamic method for measring critical bckling load of sway frames Boris Blostotsky 1, Elia Efraim 1 1 Department of Civil Eng., Faclty of Engineering, Ariel University, 47 Ariel, Israel bx@ariel.ac.il, efraime@ariel.ac.il ABSTRACT: The critical sway bckling load is the pper limit of the allowed vertical frame loading. This load is defined by the frame configration, cross-sections of the colmns and horizontal beams as well as configration and stiffness of the connections between them. In known non-destrctive methods of determining the side-sway frames critical load an implementation of vertical loading is technically difficlt that is especially essential at fll-size tests. The vale of the applied load is limited also by significant deformation of the frame approaches the critical load vale. The paper deals with a new non-destrctive method for experimental determining the critical sway bckling load of frames. For ensring mobility of the frame in the sway mode the vertical loading is carried ot by the flexible traction element monted between the frame beam and fixed base throgh the loading device. The critical vertical load of the "frame traction element" combined system is higher than of single frame. This fact allows to load the system with loading that corresponds to critical load of a frame withot loss of system stability. Procedre of the experiment incldes: determining of free vibration natral freqency of the combined system "frame traction element" for different tensions of traction element, determining the lateral stiffness of the system, calclation the corresponding stiffness of the frame and its critical load. The accracy of the method is verified by comparison the experimental and theoretical reslts. EY WORDS: Dynamic test; Critical load; Sway mode; Bckling. 1 INTRODUCTION Testing of portal frames in sway mode is performed in order to determine the maximal allowable load limited by plastic deformations of frames elements or their connections [1,] and a critical load limited by frames stability [3,4]. Motivation to determine accrate vale of critical force is cased by its sing as follow : - critical load is maximal allowable load for given configration of frame and can be sed as criterion of perfection or efficiency of the strctre; - it is sed in calclation prescribed by standards for design of strctral elements nder combined action of compression and bending [5]; - it is sed for verification of theoretical analysis of stability at varios end conditions of frames colmns [6]. In the dynamic method of non-destrctive testing of frames for critical load [7,8,9] the relation of instability to strctral stiffness is sed. Many theoretical and experimental investigations confirm decreasing of free vibration freqency with increasing of compression forces [1]. The critical load determined based on freqency measrement reslts for varios loads by extrapolation p to nll freqency according to linear dependence of freqency sqared on compression load. At frames testing in sway mode the vertical loading is performed by sing of additional weights [9,11], directly via a lever system [1] or sing special mechanism (gravity load simlator) [13]. In the present work a new non-destrctive method for determination of frames critical bckling load in side-sway mode is proposed. The method allows performing measrements dring the tests nder loads that exceeds the frame critical loads withot losing its stability. Technological advantage of this method is the ability to create loading correspond to critical loads of real fll-size frames. The aim of the present work is developing of more exact non-destrctive experimental method for determination of critical load of frame in sway mode. Increasing in accracy is achieved by condcting experiments in diapason of vertical loads incldes the critical load as well as by sing of loading system withot friction. TE IDEA AND DESCRIPTION OF TE METOD Figre 1 presents the setp scheme for testing of sway frames with colmns (A) flly restrained at the base and with hinged or fixed attachment to girder (B). The traction element (C) (cable or rod) has a pinned connections to the base and the frame's beam and incldes the device (D) for prodcing of tension force. Since the lateral vibrations of the frame case vertical displacements of the girder, maintaining a stable tension is achieved by sing hydralic or pnematic loading device or by inclding additional spring element in case of mechanical loading device (D). The experiment consists in determining the natral freqency of the system vibrations at different vales of tension force in traction element and accordingly compression forces in the frame colmns. Adopting the lateral movement of the frame girder as degree of freedom, the generalized mass M' of the system can be presented in the following form: M ' = M + M + M + M (1) where M f - the mass of the frame girder; f c e a 3851

2 Proceedings of the 9th International Conference on Strctral Dynamics, EURODYN 14 B E stiffness of the frame as the part of the loaded system is determined by lateral displacement of the girder de to nit lateral force with constant vertical direction of traction force. A C Q/ Q/ F D l T Figre 1. Setp scheme for testing of sway frames with colmns flly restrained at the base and fixed or pinned attachment to the girder. M' c - the generalized colmn mass; M' e - the generalized mass of traction element inclding tension device and flexible rod or cable; M' a - the additional mass attached for increasing the generalization accracy. The main inaccracy of mass generalization consists in determination of generalized colmn mass that depends on shape of vibration mode [14]. For colmns with constant cross-section M c ' =.4M c for rigid pinned end conditions, and M c ' =.365M c for rigid connection at both ends. These vales are obtained based on form of static deformation nder niformly distribted load. The inflence of the error in calclated generalized colmn's mass can be redced by increasing translational moving mass by adding of mass M a to a girder (Figre 1). The stiffness of the system "frame traction element" is calclated based on measred natral freqency as follows: ( f ) M s = n π () The stiffness of the frame only is calclated based on correlation with systems stiffness. The frames critical load determined based on frame's stiffness vales. 3 DEPENDENCE OF TE FRAME STIFFNESS AND SYSTEM STIFFNESS ON VERTICA OAD In figre Figre the calclation scheme for stiffness and critical loads of the system "frame traction element" is presented for case of frame with rigid girder and rigid connection of colmn with the base and pinned or rigid connection with the girder. The general load of the system incldes a weight load Q of system elements that cases initial axial compression in the colmns, and a tensile force T in the traction element. The axial load on each colmn is P = P / = T / + / (3) 1 Q where P is general load on the frame. The lateral stiffness of system "frame traction element" is determined by lateral displacement de to nit lateral force taking into accont changing of direction of traction force T cased by lateral displacement of frame's girder. The lateral Figre. Calclation scheme for stiffness and critical loads of the system "frame traction element" The problem of lateral displacement determination is solved by integration of elastic deformation eqation of colmn in sway mode [15], bt taking into accont the varying the traction force direction. For convenience of the analysis a dimensionless parameters of frame and system stiffness as well as non-dimensional load parameters are introdced as follow: 3 s = s ; EI 3 = (4) EI.5 P = (5) EI.5 Q = (6) EI where - colmn height, E - the elasticity modls, I - moment of inertia of colmns cross-section. For rigid-rigid bondary conditions at colmns ends the soltion of elastic deformation eqation is + = 1 (7) tan s The lateral stiffness of frame with rigid connection at both colmns ends can be obtained from soltion (7), when the condition of constant vertical direction of traction force assigned by : = tan( / ) Figre 3 presents graphs of dependencies non-dimensional stiffness characteristics s ' of the system "frame traction element" and ' of the frame alone (when /=) on nondimensional parameter and / ratio for rigid-rigid end conditions. 3 (8) 385

3 Proceedings of the 9th International Conference on Strctral Dynamics, EURODYN 14 ' s /= /=1.5 /=1 /=.5 /= rigid-rigid end conditions, q= vertical loading Figre 3. Graphs of dependencies non-dimensional stiffness characteristics s ' of the system on non-dimensional parameter for different ratios /. The vales of s '= ' = 4 at = are correspond to lateral stiffness of frames withot axial forces in colmns (case of Q = and T = ). The vales of s ' and ' at = are correspond to lateral stiffness of frame nder strctral weight load only. The intersections of the graphs with axis determine the vales s,cr and cr that correspond to the critical loads of system and frame respectively. The critical force P s,cr in the colmns and the tension force T s,cr in traction element that correspond to the critical state of the system in sway mode are calclated from (5) and (3) when = s,cr. The critical force P cr of frame only is determined by (5) when = cr. An eqation for determination s,cr for rigid-rigid end conditions followed from (7) satisfying conditions s = [16] and has the following form: 1, tan s s, cr cr s, cr + = Soltion of (8), that corresponds to nll stiffness of frame, gives the critical vale cr = π for frame with rigid-rigid connections (for frame with rigid-pinned connections cr =π/). The roots of eqation (9) depend on two parameters: - nondimensional parameter of initial weight load Q, and / ratio - non-dimensional parameter of the system "frame traction element". Taking the initial load on colmn as a part of its critical load as P cr we have for rigid-rigid end conditions (9) Q= q (1) = q π (11) The vales of s,cr for q=,.5 are presented in table Table 1. According to determined vales of parameters s,cr and cr and given vale of weight load non-dimensional parameter, the domain of dependency (8) for ' is and domain for ' s from (7) is s,cr >. The case of /= corresponds to permanent vertical direction of the tensile force T, so there is no dependence of the roots vales π and π/ on initial weight load q and they eqal to the critical load parameters of the frame alone. Table 1. The vales of non-dimensional parameter s,cr of system "frame traction element" for different vales of / ratio and two states of initial weight load. Colmn ends / conditions rigid-rigid q= π π q=.5 π rigid-pinned q= π/ π q=.5 π/ When cr << s,cr the load acting on the system larger then frame's critical load. Dring the vibrations the forces acting on girder from colmns coincide with direction of girder's displacement [16], the frame stiffness is negative [15]. The system's stability is provided by lateral component of inclined force in traction element that larger than colmns' reaction. At = s,cr the colmns' reactions are eqal and opposite to the lateral component of the tensile force and this state is limit of system stability. For example, for rigid-pinned or rigid-rigid colmns connections according to roots vales presented in table Table 1 for /=1 and q=.5 P s,cr /P cr =.78 and for /= and q=.5 P s,cr /P cr =7.9. Stability of the system at loads which exceed frame's critical vale allows testing the frames nder load that closer or exceeds its critical bckling load. imitations of this possibility are absence of plastic deformations in frame elements or in connections between them and bckling of colmns in non-sway mode. 4 DETERMINATION OF CRITICA OAD BASED ON TESTING RESUTS As testing reslts the vales of lateral stiffness of the system "frame traction element" are determined for different vales of tensile force in traction element and as a conseqence vertical load on the colmns. Based on these vales determining of the frame critical load is reqired. If the testing reslts inclde measrements at loads exceeding the frame's critical load, the last one is determined by interpolation of the reslts. Otherwise by extrapolation. In both cases this vale is defined as load vale corresponding to nll stiffness [15] [16]. For developing the mathematical model the expression for ' s from (7) taking into accont (3) with (8) can be presented as follow: T s = + (1) The physical meaning of (1) is that lateral stiffness of the system is the sm of stiffness of the frame and the flexible traction element. According to (1) for each experiment based on measred vales of s and P the frame lateral stiffness is obtained as T = s (13) Figre 4 presents the dependence graph of lateral stiffness ratio / ='/' on non-dimension load parameter P/P cr =(/ cr ). 3853

4 Proceedings of the 9th International Conference on Strctral Dynamics, EURODYN 14 /o P/Pcr Figre 4. Dependence of lateral stiffness ratio / = '/' on non-dimension load parameter P/P cr =(/ cr ) traction element force stabilization spring girder colmn velocity sensor The graph illstrates the possibility of approximation by straight line that cross over the point (,1) on stiffness axis and point (1,) on load axis that is described by following eqation: + P = 1 P cr (14) The relative deviation of linearly approximated vales / from the exact theoretical vales for all vales / in loads range (,1)P cr not exceed 1.4%, and for loads range (, 1.)P cr (for case of testing nder loads that exceed in % the critical) the deviations are lower than 1.8%. 5 TESTING OF TE "FRAME TRACTION EEMENT" SYSTEM The objectives of this investigation: - verification of frames lateral stiffness dependency on load with constant vertical direction <P<P cr ; - verification of lateral stiffness dependency of frame and system nder loading by flexible traction element; - precision assessment of determining frame critical load; The experimental setp (Figre 5) consist of two colmn made of alminim sheet with thickness mm, thickness mm and height 55 mm, rigidly connected at the base. At the pper ends the colmns are rigidly connected to the girder with mass 15 gr, the bending stiffness of which significantly greater than colmns stiffness. The flexible traction element pin-jointed to the girder and consists of dynamometer and screw tension device. The traction device has elastic element for stabilization of tensile force dring frame vibrations. The system's mass, inclding masses of all movable system elements, generalized to the degree of freedom of the girder's lateral movement is 35 gr. The generalized mass of the frame alone is 195 gr. Investigation of stiffness dependence on variable load with constant vertical direction is condcted by loading the girder by variable weight [9]. inear extrapolation based on reslts of measred stiffness sing least sqares method p to vale of = give 51.9N. The deviation of reslts from regression line not exceeds 3%. screw tension device dynamometer Figre 5. The experimental setp for testing of "frametraction element" system The investigation of system stiffness dependence on load prodced by traction device is condcted with variation of traction element length (/=1.96 and.75). The first measrement of freqency and calclation of the system and the frame stiffness is performed nder the girder's weight load withot tensile load in traction device. The following measrements are condcted nder inflence of tensile force in traction element p to load P max =1.5P cr. The measrements and calclations reslts are presented in figre Figre 6. In case of /=.75 (=94 mm) with increasing of loading the system natral freqency and stiffness are decrease. The system is stable nder maximal load that exceed the critical load of frame alone. The dependence (P) coincide linear with aforementioned accracy of linear approximation inclding the range of negative stiffness. In case of /=1.96 (=8 mm) with increasing of loading the system's natral freqency and stiffness are increase. Assessment of accracy and reliability of critical load determination is condcted by comparison of experimental vales obtained as reslts of three types of loadings and calclated theoretical vales. The elasticity modle that determined by measring of stiffness of console rod is 65 GPa. The comparison of reslts is presented in table Table. From the comparison of experimental vales follows that estimation of critical load vale P cr = 5.5 N is reliable and sfficiently accrate. Satisfactory vale of critical load at constantly vertical loading has been achieved throgh carefl alignment of colmns before loading, that allowed loading the frame with load p to.8p cr. Considerable deviation of theoretical vales towards increase is explained by nonconformity of colmns end conditions to theoretical perfectly 3854

5 Proceedings of the 9th International Conference on Strctral Dynamics, EURODYN 14 rigid-rigid de to elasticity of colmn's connections with the base and the girder. (a) Stiffness [N/m] (b) Stiffness [N/m] P [N] P [N] s _lin s _lin Figre 6. The reslts of the calclations of frame stiffness based on system stiffness s : (a) loading by force directed towards the pole prodced by traction element with ratio /=1.96; (b) loading by force directed towards the pole prodced by traction element with ratio /=.75; Table. Comparison of experimental and theoretical vales of frame's critical force Critical *D, % load Pcr, N Vertical loading oading by force /= directed towards /= the pole Average vale Theoretical vale * D- Deviation from average experimental vale 6 CONCUSIONS The method for loading frames in sway mode and measring their critical load vale is proposed. According to the method the first natral freqency of the frame loaded by varying vertical force sing tensile traction element is measred. The lateral stiffness of the system "frame traction element" is determined and based on it the lateral stiffness of frame only is calclated. The critical load of the frame is determined by linear interpolation of the stiffness-vertical load dependency for nll vale of frame stiffness. The constant vertical load prodced by the weight of the frame elements and variable load prodced by traction element are considered. Analysis of physical states of the system "frame traction element" depend on the frame loading and ratio of colmns heights to traction element length is performed. It is established that depend on this ratio the lateral stiffness of the system can increase or decrease, bt it always higher that the frames lateral stiffness. The system is stable nder the loads that exceed the critical one for the frame only. The testing load is limited by bckling of the system, bckling the colmns in non-sway mode, plastic deformations of frame elements and their connections. Assessment of accracy of the linear approximation model of frame stiffness dependence on vertical load has been performed by comparison with its exact soltion. Inaccracy of the linear model of frame stiffness dependence on vertical load not exceed 1.4% in the load range (-1.)P cr and 1.8% in the range (-1.)P cr. The proposed method allows improving reliability and increase the determining accracy of sway frame critical load. Frame loading method sing flexible traction element allows prodce loads corresponds to critical loads of real fll-size frames withot special devices for decreasing of friction inflence. REFERENCES [1] V. evi, G.C.Jr. Driscoll,.W., Analysis of restrained colmns permitted to sway, Jornal of the Strctral Division, Proc. ASCE, Vol. 93, ST1, 1967, Reprint Nc. 31 (67-1)". Fritz aboratory Reports. Paper 9. [] B. P. Parikh, The Elastic-plastic analysis and design of nbraced mltistory steel frames, Ph.D. Dissertation, ehigh University, 1966 [3].W., Stability of elastic and partially plastic frames, PhD Dissertation, ehigh University, 196. Fritz aboratory Reports. Paper 1763, 196 [4] P.G. okkas, A search on the instability of frame strctres tested to bckling nder side sway, 4th GRACM Congress on Comptational Mechanics GRACM, Patra, 7-9 Jne, [5] Eropean Committee for Standardization (CEN), Erocode 3: Design of steel strctres Part 1.1: General rles and rles for bildings (Eropean standard EN :5:E), Brssels; 5 [6].Tomski, J. Przybylski, M. Golebiowska-Rozanow, J. Szmidla, Stability and vibration of a two-member frame nder generalized load, Proceedings of Eropean Session of International Colloqim "Stability of Steel Strctres", Bdapest, I/49-I/416, 1995 [7] M.J. Jacobson, M.. Wenner, Predicting bckling loads from vibration dat, Experimental Mechanics, 8(1): 35N-38N, 1968, doi: 1.17/BF37414 [8] J. Singer, J. Arbocz, T. Weller, Elements of a Simple Bckling Test A Colmn nder Axial Compression, in Bckling Experiments: Experimental Methods in Bckling of Thin-Walled Strctres: Basic Concepts, Colmns, Beams and Plates, Volme 1, John Wiley & Sons, Inc., New York, 7 [9] B. Blostotsky, E. Efraim, Y. Dachkovsky, Dynamic method test of bckling load of one-storey sideway permitted frame. Proceeding of 5th International Mechanical Engineering Form, Jne, 1, Prage, Czech Repblic, pp. 4-1, 1. [1].N. Virgin and R.. Plat. Effect of axial load on forced vibrations of beams, Jornal of Sond and Vibration, 168:395 45, [11]. Vallytham, V.J., rian, S.P., Narayanan, M.S iew, N. Ab Bakar. Prediction of Bckling load of steel racking frame sing nondestrctive method. Steel & Composite Strctres Proceedings of the 4th International Conference, 1, doi:1.385/ _SS-Th4 3855

6 Proceedings of the 9th International Conference on Strctral Dynamics, EURODYN 14 [1] Y.C. Yen,.W., G. C. Jr. Driscoll, Tests on the stability of welded steel frames, Welding Research Blletin, No. 81,p1, September 196, Reprint No. 6 (6-1)" (1961). Fritz aboratory Reports. Paper [13] E. Yarimci, J.A. Pra,.W., Techniqes for testing strctres permitted to sway, Experimental Mechanics, Vol. 7 (8), pp , 1967 [14] R.W. Clogh, J. Penzien, Dynamics of strctres, Mc Craw-ill, NewYork,1991 [15] i Y. and X.. Storey-Based Stability Analysis of Mlti-Storey Unbraced Frames, Strctral Engineering and Mechanics An International Jornal, 19(6), pp , 5 [16]. X,. Wang, Stability of Mlti-storey Unbraced Steel Frames sbjected to Variable oading, Jornal of Constrction Steel Research, 63(11), pp ,