Advances in Steel Structures - part 36 docx

330 E. Mele et al. 7. SUMMARY AND CONCLUSIVE REMARKS In this paper an overview of a test program carried out on three series of welded connections has been provided, the global behaviour and the failure modes of the connections have been described, and the main differences in the performance of the three series of specimens have been emphasised. Some major aspects deriving from this preliminary analysis of the experimental data concern the effect of the loading history, of the column section and panel zone design on the cyclic behaviour, maximum rotation, energy dissipation and failure mode of the connections. The quite high values of the maximum global rotations of the connections, especially if compared to the rotation capacities exhibited by US-type welded connections in past testing programs, can be related to the following aspects. 9 According to the major findings reported by (Roeder & Foutch, 1996), the connection rotation capacity and ductility strongly decreases as the beam depth increases. Thus higher rotations are expected for beam-to-column connections usually adopted in Europe, where the depth of the beam section (db= 300 -450 ram) is significantly less than the ones utilised in the US practice (db= 500- 1000 mm), due to the current adoption of perimeter frames configuration. 9 Fully welded connections, as the ones adopted in the tested specimens, have already shown, in past experimental tests, higher rotation capacity than the BWWF connections (Tsai & Popov, 1995; Usami et AI., 1997). 9 A significant contribution of panel zone deformation has been observed throughout the tests, suggesting the possibility of utilising the joint panel for providing energy dissipation and stable behaviour of the connections even at large number of cycles. The design implications of this last aspect are currently being evaluated by the authors through the comparison with similar experimental data available in the inherent b~liography, through the evaluation of the provisions supplied by the seismic codes and through theoretical analyses. REFERENCES Bertero V.V., Anderson J.C., Krawinkler H. (1994). Performance of steel building structures during the Northridge earthquake. Ethq. Eng. Res. Center, Rep. UCB/EERC-94/09, University of California, Berkeley. Calado L., Mele E., De Luca A. (1999). Cyclic behaviour of steel semirigid beam-to-column connections, submitted for publication on: s Struct. Eng. ASCE. Mahin S.A., Hamburger R.O., Malley J.O. (1996). An integrated progrmn to improve the performance of welded steel frame buildings. Proc. 11 th WCEE, World Conf. Earthq. Eng., Elsevier Science Ltd., Paper No.ll14. MaUey J.O. (1998). SAC Steel Project: summary of Phase-I testing investigation results. Eng. Structs, 20:4-6, 300-309. Mele E., Calado L. Pucinotti R. (1997). Indagine sperimentale sul comportamento ciclico di alcuni collegamenti in acciaio. Proc. 8 ~h National Conf. Ethq. Engrg. ANIDIS, Taormina, Italy, 1031-1040. Nakashima M., Suita K., Morisako K., Maruoka Y. (1998). Tests on welded beam-column subassemblies. I: global behaviour. II detailed behaviour. J. Struct. Eng. ASCE, 124:11, 1236-1252. Plumier A. et AI. (1998). Resistance of steel connections to low-cycle fatigue. Proc.ll th ECEE, Balkema. Roeder C.W., Foutch D.A. (1996). Experimental results for seismic resistant steel moment frame connections. J. Struct. Eng. ASCE, 122:6, 581-588. Tanaka A., et AI. (1997). Seismic damage of steel beam-to-column connections - evaluation from statical aspects. Proc. STESSA '97, 2 "a Int. Conf. on Steel Structures in Seismic Areas, Kyoto, Japan, 856-865. Taucer F., Negro P., Colombo A. (1998). Cyclic testing of the steel frame. JRC ELSA Spec. Publ. No.L98.160, Dec. 1998 Tsai K.C., Popov E.P.(1995). Seismic steel beam-column moment connections. In: Metallurgy, Fracture Mechanics, Welding, Moment Connections and Frame Systems Behavior. Rep.SAC/BD-95/09, SAC Joint Venture, Sacramento, Cal. Usami T. et A1. (1997). Real scale model tests on flange fracture behaviour of beam adjacent to beam-to-column joint and the seismic resistance after repairing and strengthening. Proc. STESSA '97, 2 ~ Int. Conf. on Steel Structures in Seismic Areas, Kyoto, Japan, 955-962. ADVANCED METHOD FOR MODELLING HYSTERETIC BEHAVIOUR OF SEMI-RIGID JOINTS Y. Q. Ni, J. Y. Wang and J. M. Ko Department of Civil and Structural Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong ABSTRACT Modelling of the hysteretic behaviour at beam-to-column connections is an important issue for static and dynamic analysis of steel structures with semi-rigid joints. Some empirical models, such as the Ramberg-Osgood model, the Richard-Abbott model, and the Lui-Chen model, have been proposed for this purpose. To complete the interior and branch hysteresis curves, a set of empirical rules are indispensable to these models. However, some of the empirical hysteresis rules may conflict with the experimental observations. In this study, a mathematical hysteresis model-the Preisach model-is introduced to describe the hysteresis behaviour of steel semi-rigid connections. This phenomenological model can completely specify hysteresis curves without need of any empirical rules or additional conditions, and is really capable of representing hysteresis with non-local memory. A time-domain incremental method is presented to evaluate the transient dynamic response of flexibly jointed flames in terms of the Preisach model. A comparison of the analytical results with those by the Ramberg- Osgood model demonstrates the suitability of the Preisach model for this application. KEYWORDS Hysteresis, semi-rigid connection, Preisach model, phenomenological modelling, nonlinear dynamic response. INTRODUCTION The strength, stability, ductility and energy dissipation capacity of steel frames can be significantly influenced by the behaviour of beam-to-column connections. The semi-rigid characteristics of typical connections in steel frames have been widely recognized. To assess actual structural behaviour, it is necessary to incorporate the effect of connection flexibility in the analysis of these structures. This is particularly meaningful for the dynamic analysis because hysteretic damping at flexible connections may contribute significant energy dissipation. Some empirical models, such as the Lui-Chen exponential model (Lui and Chen 1986, Chui 1998), the Ramberg-Osgood model (Sveinsson and McNiven 1980, Chui and Chan 1996), and the Richard-Abbott model (Richard and Abbott 1975, 331 332 Y.Q. Ni et al. Deierlein et al. 1990), have been widely used to represent the nonlinear moment-rotation behaviour of semi-rigid connections. These models are sufficient for describing the hysteresis loops under cyclic loading. However, for cases of transient loading or loading between variable limits, these models are incomplete and empirical rules have to be introduced to stipulate the paths of interior and branch hysteresis curves. It is intractable to encode these rules in computer program design and sometimes the hysteresis curves followed by the empirical rules conflict with the experimental observations. The Bouc-Wen differential model has also been used to describe the hysteretic behaviour of semi-rigid steel connections subjected to dynamic loading (Mak 1995). This analytical model is mathematically tractable due to its complete capability in tracing transient hysteretic response. Given an arbitrary time history of the displacement, the hysteretic force can be completely specified by the Bouc-Wen model without need of empirical rules or additional conditions. However, it has been demonstrated that the differential-type models, including the Bouc-Wen model, can only represent the hysteresis with local memory (Ni et al. 1999). As a result, these models do not allow the crossing of minor loops which can arise in the measured hysteresis curves. In the past decade, hysteresis phenomenon has been studied by mathematicians as a new branch of mathematics research (Macki et al. 1993). They explored the hysteretic nonlinearity in a purely mathematical form by introducing the concept of hysteresis operators. One of the mathematical hysteresis operators is the Preisach operator, or called Preisach model (Visintin 1994). The Preisach model formulates hysteretic constitutive relations in a conceptually simple and computationally elegant way. This model has several appealing features, including its ability to capture nonlocal memory, which make it capable of accurately modelling various hysteretic characteristics following a phenomenological approach. The Preisach model has been applied to describe the magnetomechanical hysteresis in ferromagnetic materials (Mayergoyz 1991), and the inelastic constitutive laws of ductile materials (Lubarda et al. 1993, Sumarac and Stosic 1996), piezoceramics and shape memory alloy (Hughes and Wen 1997, Song et al. 1999), and nonlinear vibration isolators (Wang et al. 1999). The present study introduces the Preisach model to describe the moment-rotation hysteresis curves of semi-rigid joints and subsequently analyzes the transient dynamic response of flexibly connected steel frames. The Ramberg-Osgood model in conjunction with the empirical rules is first used to produce a set of hysteresis loops. By taking these numerical hysteresis loops as 'experimental' curves of semi- rigid joints, an identification technique is implemented to establish the corresponding representation in terms of the Preisach model. A time-domain incremental method is adopted to evaluate the transient dynamic response of a portal frame under seismic excitation and other dynamic loads. The hysteresis loop and dynamic response characteristics using the Preisach model are compared with those using the Ramberg-Osgood model to validate the suitability of thePreisach model. HYSTERESIS MODELS Experiments by many researchers have confirmed the hysteretic behaviour of typical beam-to.column connections under cyclic loading or dynamic excitation. The hysteretic behaviour provides energy absorption capacity that is beneficial for resistance against the earthquake, wind and other dynamic loads. The most direct indication of hysteretic behaviour is the hysteresis loops. The hereditary nature of hysteretic systems shows the multi-valuedness of the hysteretic force (joint moment) corresponding to one value of displacement (joint rotation) due to different past histories of deformation. Therefore, the hysteretic force depends not only on the instantaneous deformation but also the past history of deformation. The majority of existing models can only capture local history of the hysteresis. For most Modell&g Hysteretic Behaviour of Semi-Rigid Jo&ts 333 practical hysteretic system, all the dominant extrema of the entire history leave their marks upon future states of the hysteresis. The models, which can capture nonlocal-memory hysteresis, usually give a finer fit for the experimental hysteresis curves. In this section, two hysteresis models, which will be used to represent the hysteretic behaviour of semi-rigid joints, are discussed. Ramberg-Osgood Model One of the hysteresis models commonly used to represent semi-rigid joints is the Ramberg-Osgood model (Jennings 1964). This model describes the hysteretic force-deflection skeleton (virgin) curve by a three-parameter polynomial as F F 1,-1 u(F) = ~-(1 + A I- ~- ) (1) where u and F represent the deflection and hysteretic force respectively; K, A and n are parameters controlling the curve shape. Eqn. 1 allows a smooth transition from the elastic to the plastic region and some freedom in the shape of the hysteresis. The ascending and descending branches of the hysteresis loops are described by the same basic equation as the skeleton curve but scaled by a factor of two, namely, F-Fr F-Fr in-1 u(F) = u r 3r (1 + A I ) (2) K 2K where Ur and Fr are the deflection and force at the reference point of the curve. Eqns. 1 and 2 are sufficient for describing the hysteresis loops under cyclic loading or repeated loading between fixed limits. However, for cases of transient loading or loading between variable limits, Eqns. 1 and 2 are incomplete because they give no indication of how the skeleton and branch curves can be linked together to give the response to other than cyclic loading. To complete the model, empirical hysteresis rules have to be introduced. Sveinsson and McNiven (1980) have given a detailed description of the empirical rules in thirteen phases by defining two types of skeletal curves and eleven types of branch curves (interior curves and bounding curves). As shown in Eqn. 2, since the current state (u, F) is only related to a specific previous state (Ur, Fr) at the reference point (reversal point) of the hysteresis curve, the Ramberg-Osgood model represents hysteresis with local memory. Preisach Model The Preisach model is constructed as a superposition of a continuous family of elementary rectangular loops, called relay hysteresis operators as shown in Figure 1. That is r(t) = ~ /~(a, fl)G~[u](t)dc~lfl (3) S where/.t(a,/5') is a weight function, called Preisaeh function, with support on a limiting triangle S of the (a, fl)-plane with line a- fl being the hypotenuse and point (a o,/30 = - ao) being the vertex as G~o[u](t) r u(0 ., _ .,, Figure 1. Relay Hysteresis Operator 334 Y.Q. Ni et al. Figure 2. Input Sequence and Preisach Plane (Limiting Triangle) with Interface L(t) shown in Figure 2. The triangle S in the half-plane a > fl is named Preisach plane. /.t(a, ,B) is equal to zero outside S. G~/~ is the relay hysteresis operator with thresholds a >/3. It is a two-position relay with only two values +1 or -1 corresponding to 'up' and 'down' positions respectively, i.e., +1 on [at, oo) G~p[u](t) = 1 on (-0% fl) (4) The Preisach model can be interpreted as a spectral decomposition of a complicated hysteretic constitutive law that has nonlocal memory, into the simplest hysteresis operators G~ with local memory. Corresponding to an arbitrary input sequence u(t) shown in Figure 2, the triangle S can be subdivided in two sets at any time instant t: S+(t) consisting of points (a, r) for which the corresponding Gaz-operators are in the 'up' position; and S-(t) consisting of points (a, r) for which the corresponding G,~z-operators are in the 'down' position. The interface L(t) between S+(t) and S-(t) is a staircase line whose vertices have a and fl coordinates coinciding respectively with local maxima Mk (k = 1, 2, ) and minima mk (k = 1, 2, ) of the input sequence at previous instants of time. The nonlocal selective-memory is stored in this way. Thus, the output r(t) at any instant t can be equivalently expressed as r ( t ) = ~ ~ /.t ( et , fl ) d crd fl - ~ ~ /z ( et , fl ) d crd fl (5) s+(t) S-(t) The Preisach function/.t(a, ,8) is usually determined by identification from experimental hysteresis loops. One of the advantages of the Preisach model is that it can be expressed in a real-time numerical simulation form. With the numerical formulation, the Preisach model is stated as n(t)-I - + Z (rM rM )+(rM.,u rM ) for fi(t)<O r(t) = k=~ (6) n(t)-I + ~-](rM~ m~-rMk.m~_,)+(ru-ru.,,_,) for fi(t)>_O k=l +r(t) rat F ir:~ or:er Figure 3. Determination of ra~ on First-Order Curve Modelling Hysteretic Behaviour of Semi-Rigid Joints 335 where r- is the 'negative saturation' value of the output; ruk.m k is the value on the first-order curve corresponding to the extrema {M k , m k } as shown in Figure 3. The numerical implementation of the Preisach model is performed in the following steps. Firstly, the Preisach plane (limiting triangle) is discretized into a squared mesh. A series of first-order transient curves are entered that form the discrete sets of (a,/3, rap). The alternating series of dominant extrema {Mk, mk} are then determined according to the time history of the input and updated at each new time instant. Using {Mk, mk} and mesh value raft, all the terms in the parentheses in Eqn. 6 are computed by numerical interpolation. The current output is then evaluated from Eqn. 6 with respect to monotonic increasing and monotonic decreasing cases respectively. TRANSIENT DYNAMIC RESPONSE OF A PORTAL FRAME A one-storey portal frame with semi-rigid joints, as shown in Figure 4, is used to demonstrate the dynamic response analysis results in terms of the Preisach model, and to compare the results with those obtained by the Ramberg-Osgood model. The moment-rotation characteristics of the flexible connections are incorporated in the analysis as a spring point element. The equation of motion of the structure can be expressed as Figure 4. Schematic and Modelling of a Portal Frame mii(t) + cft(t) + ku(t) = f (t) (7) in which m, c and k are the mass, viscous damping coefficient and stiffness respectively; f(t) is an external excitation. Due to the nonlinearity of the semi-rigid connections, the stiffness k varies with response and needs to be updated at each time step. On the assumption of linearly elastic columns and beams, the transient value of k can be expressed as k = 12EIc 3EIc 1.0 Hc 3 (1.0 ) H c 4EI~ K 4- c H~ 1.0 + KcLb /6EI b where Kc is the tangent stiffness of the connections obtained from the hysteresis loops. (8) In this study, the Ramberg-Osgood model with the parameters K = 1.26x 10 7 N-m/rad, A = 9.316x 10 9 and n = 5.5 is used to produce the 'experimental' hysteresis curves. The Preisach model is then established by use of the 'experimental' first-order curve data. A time-domain incremental method is presented to evaluate the transient dynamic response of the flexibly connected structure. 336 Y.Q. Ni et al. Figures 5 and 6 illustrate the lateral displacement dynamic response of the frame and the moment- rotation hysteresis curves under the horizontal pulse excitationflt) = 50 kN for 0 < t < tcr "- 0.1S. Due to the hysteresis damping at connections, the dynamic response amplitude gradually attenuates with time. It is seen that both the predicted displacement response and the hysteresis loops in terms of the Preisach model agree well with those in terms of the Ramberg-Osgood model. Usually, the local-memory hysteresis models cannot give a satisfactory representation of the minor hysteresis loops subjected to nonzero-mean excitation. Figures 7 and 8 show the displacement dynamic response and the corresponding hysteresis curves under a nonzero-mean cyclic loading. It is observed that, although the Ramberg-Osgood model and the Preisach model produce consistent displacement response, the steady-state minor hysteresis loops produced by the Ramberg-Osgood model have nearly zero enclosed area (the loading stiffness is almost identical to the unloading stiffness), showing obvious disagreement with actually observed minor loops. Contrarily, the Preisach model produces more reasonable minor hysteresis loops with a certain energy dissipation capability. In order to verify the ability of the Preisach model to predict transient dynamic response, the dynamic response of the frame subjected to seismic excitation is analyzed. The ground acceleration excitation is the E1 Centro Earthquake with the peak acceleration value of 1.25g. Figure 9 shows the transient dynamic response of the structural lateral displacement in terms of the Ramberg-Osgood model and the Preisach model respectively. Figure 10 shows the corresponding moment-rotation hysteresis curves. The displacement dynamic response history obtained by the Preisach model coincides favourably with that obtained by the Ramberg-Osgood model. In particular, both the positive and negative response amplitudes predicted by the Preisach model are almost identical with the corresponding values predicted by the Ramberg-Osgood model. Also, the hysteresis loops arising from the two models match well with each other. Figure 5. Lateral Displacement Response under Pulse Excitation (tcr "- 0.1 S) Figure 6. Trajectories of Hysteresis Loops under Pulse Excitation (tcr = 0.1 S) Modelling Hysteretic Behaviour of Semi-Rigid Joints 337 Figure 7. Nonzero-Mean Cyclic Loading and Corresponding Displacement Response Figure 8. Trajectories of Hysteresis Loops under Nonzero-Mean Cyclic Loading Figure 9. Lateral Displacement Response under Ground Seismic Excitation Figure 10. Trajectories of Hysteresis Loops under Ground Seismic Excitation 338 Y.Q. Ni et al. CONCLUDING REMARKS This paper reports on the nonlinear dynamics of flexibly connected steel frames by using the Preisach model to represent the hysteretic behaviour of semi-rigid joints. The Preisach model possesses two salient attributes: (1) it is a phenomenological model and can describe transient hysteresis curves without needing any empirical rules; (2) it is a nonlocal-memory hysteresis model and is therefore capable of accurately depicting the minor loops and interior curves. The nonlinear dynamic response of a flexibly connected frame by using the Preisach model was analysed under various dynamic loads, and a good agreement with the results by the Ramberg-Osgood model was observed. This validated the suitability of the Preisach model for representing the hysteretic behaviour of semi-rigid joints. ACKNOWLEDGEMENT The funding support by The Hong Kong Polytechnic University to this research is gratefully acknowledged. References Chui P.P.T. (1998). Geometric and Material Nonlinear Static and Dynamic Analysis of Steel Structures with Semi-Rigid Joints. Ph.D. Thesis, The Hong Kong Polytechnic University, Hong Kong. Chui P.P.T. and Chan S.L. (1996). Transient Response of Moment-Resistant Steel Frames with Flexible and Hysteretic Joints. Journal of Constructional Steel Research 39, 221-243. Deierlein G.G., Hsieh S.H. and Shen Y.J. (1990). Computer Aided Design of Steel Structures with Flexible Connections. Proc. 1990 National Steel Construction Conference, AISC, Chicago, USA. Hughes D. and Wen J.T. (1997). Preisach Modeling of Piezoceramic and Shape Memory Alloy Hysteresis. Smart Materials and Structures 6, 287-300. Jennings P.C. (1964). Periodic Response of a General Yielding Structure. ASCE Journal of the Engineering Mechanics Division 90, 131-163. Lubarda V.A., Sumarac D. and Stosic S. (1993). Preisach Model and Hysteretic Behaviour of Ductile Materials. European Journal of Mechanics, A/Solids 12,445-470. Lui E.M. and Chen W.F. (1986). Analysis and Behaviour of Flexibly-Jointed Frames. Engineering Structures 8, 107-118. Macki, J.W., Nistri, P. and Zecca, P. (1993). Mathematical Models for Hysteresis. SlAM Reviews 35, 94-123. Mak W.H. (1995). System and Parameter Identification of Semi-Rigid Connections in Steel Structures.Ph.D. Thesis, The Hong Kong Polytechnic University, Hong Kong. Mayergoyz I.D. (1991). Mathematical Models of Hysteresis, Springer-Verlag, New York, USA. Ni Y.Q., Ko J.M. and Wong C.W. (1999). Nonparametric Identification of Nonlinear Hysteretic Systems.ASCE Journal of Engineering Mechanics 125, 206-215. Richard R.M. and Abbott B.J. (1975). Versatile Elastic-Plastic Stress-Strain Formula. ASCE Journal of the Engineering Mechanics Division 101, 511-515. Song C.L., Brandon J.A. and Featherston C.A. (1999). Estimation of Local Hysteretic Properties for Pseudo- Elastic Materials. Identification in Engineering Systems: Proceedings of the 2nd International Conference, Swansea, UK, 210-219. Sumarac D. and Stosic S. (1996). The Preisach Model for the Cyclic Bending of Elasto-Plastic Beams. European Journal of Mechanics, A/Solids 15, 155-172. Sveinsson B.I. and McNivwn H.D. (1980). General Applicability of a Nonlinear Model of a One Story Steel Frame. Report No. UCB/EERC-80/IO, University of California, Berkeley, California, USA. Visintin A. (1994). Differential Models of Hysteresis, Springer-Verlag, Berlin, Germany. Wang J.Y., Ni Y.Q. and Ko J.M. (1999). Transient Dynamic Response of Preisach Hysteretic Systems. Proc. International Workshop on Seismic Isolation, Energy Dissipation and Control of Structures, Guangzhou, China. Cold-Formed Steel . to beam-to-column joint and the seismic resistance after repairing and strengthening. Proc. STESSA '97, 2 ~ Int. Conf. on Steel Structures in Seismic Areas, Kyoto, Japan, 95 5-9 62. ADVANCED. Preisaeh function, with support on a limiting triangle S of the (a, fl)-plane with line a- fl being the hypotenuse and point (a o,/30 = - ao) being the vertex as G~o[u](t) r u(0 ., _ .,,. arbitrary input sequence u(t) shown in Figure 2, the triangle S can be subdivided in two sets at any time instant t: S+(t) consisting of points (a, r) for which the corresponding Gaz-operators