Single-Degree-of-Freedom Analytical Predictive Models for Lead-Core Bearing Devices

Transcription

1 Single-Degree-of-Freedom Analytical Predictive Models for Lead-Core Bearing Devices T.A. Zhelyazov Technical University of Sofia, Sofia, Bulgaria. ABSTRACT: The contribution discusses constitutive models aimed at predicting the mechanical response of lead-core bearing devices for passive seismic isolation of structures. The scope is limited to single-degree-of-freedom models which supply a relation between the shear displacement (shear strain) and the shear force (shear stress). Two constitutive relations are considered. In the first model a relation between shear displacement and shear force is postulated in which a hysteresis parameter is introduced. The evolution of the hysteresis parameter is defined separately. In the alternative constitutive model effects due to different interacting mechanisms are superimposed and a damage variable is introduced. A procedure for mechanical parameters identification is also presented. Keywords: Lead core bearing device; Constitutive model; Analytical models; Material degradation 1. INTRODUCTION Single-Degree-of Freedom (SDOF) predictive models for lead-core bearing (LCB) devices typically formulate a relation between the shear displacement (or shear strain) and the shear force (or the shear stress). Shear stress is usually defined as the ratio between the shear displacement and the total height of the rubber layers in the bearing devices.lcb devices typically consist of components possessing different mechanical properties: lead-core, rubber and steel elements. In this context SDOF models should provide a macro-characteristic representative for the mechanical response of this multiplecomponent system on the macro scale. Constitutive models for LCB devices have evolved from the bilinear model in which pre-yield stiffness is simply replaced by the post-yield stiffness after the yielding of the lead, to more sophisticated models involving differential equations. These are the Bouc-Wen model and a differential equation model in which a damage variable is introduced. Two constitutive models are discussed in this contribution - the Bouc-Wen model (i) and relation involving a damage variable (ii).the Bouc-Wen model consists of an equation that defines a skeleton curve (a relation between shear displacement and shear force) and another equation which specifies the evolution of the hysteresis parameter. In the constitutive model accounting for damage, the total shear stress acting in LCB device is split into several terms that quantify the contributions of the different mechanisms. The above models are implemented in numerical procedures created by using Python software. Upon appropriate model parameter calibration these numerical tools can be used in designing structures which contain seismic isolators such as LCB devices. Only analytical and numerical aspects of the discussed constitutive models are considered here. This contribution appears thanks to a continuous collaboration, started in 013 between the author and the Earthquake Engineering Research Centre, a part of the University of Iceland.. BOUC-WEN MODEL The relation between the shear force Q in the LCB device and the shear displacement u is obtained by equation (1); the hysteresis parameter evolution is defined by equations () [Bouc-1967, Bouc-1971, Wen-1976 Constantinou, Tadbakhsh-1985, Song, Der Kiureghian-006] as follows:

2 Qy Q u 1 Q yz (1) d y dz d y du Z Z 1 du Z du A () In equation (1) u stands for the current displacement and Z for a dimensionless hysteresis component which should satisfy equation (). The other material parameters in (1) are defined as follows: α is the post-yielding to pre-yielding stiffness ratio, d y is the displacement for which the yielding in the lead core takes place, and Q y is the shear force, corresponding to the yielding. In equation () β, γ and A are dimensionless model parameters and η controls the transition phase at yielding of the lead core. Further, the modified mid-point method (see for example Press et al-007) is used to integrate numerically the ordinary first-order differential equation (): dz u f Z, u (3) z0 Z u i (4), z z hf Z ui u (5) 1 0, zk zk hf Z ui kh u 1 1 (k=1...n-1) (6) Z 1 u z z hf Zu H, n n i u 1 (7) This is an algorithm to calculate the value of the function Z at point u, provided the value at point u i is known. In equation (7) H = u - u i ; n is the number of substeps into which the interval H is divided. Overdot denotes a derivative with respect to time. To solve equations (1) and () a numerical procedure is created in Python software. The algorithm contains a subroutine (equation (3) - equation (7)) for numerical integration by using the modified mid-point method. A trial shear force shear displacement relationship is shown in Figure 1.

3 Figure 1. Mechanical response of a lead-core bearing in terms of shear displacement- shear force curve obtained by using numerical procedure based on the Bouc-Wen model. Model parameters used to obtain this constitutive relation are summarized in Table 1. Table 1. Bouc-Wen model: mechanical constants A , STRESS-STRAIN CONSTITUTIVE RELATION ACCOUNTING FOR MATERIAL DEGRADATION An alternative approach, compared to the model discussed in the previous section can be found in [Dall Astra and Ragni-006], [Govindee and Simo-199] and [Haupt and Sedlan-001]. This type of constitutive relations can be situated in line with models that are compatible with thermodynamics fundamentals [Fabrizio and Morro-199]. An internal variable is introduced to assess mechanical degradation. Total shear stress is split into three terms: a term which represents an elastic contribution ( e), and two terms describing overstresses relaxing in time ( 1 and ): The elastic contribution is assessed as: e 1 (8) F( ) e (9) with F() being a polynomial depending on the shear strain. Overstresses are defined as functions of the shear strain, model parameters (E 1, E ) and internal variables ( 1, ): F E,, ) (10) 1 ( 1 1 F E,, ) (11) (

4 Evolutions of internal variables v,1 and v, are defined as follows (Dall Astra and Ragni, 006):,1 1,1 1,, (1) q e H,, 1, and in the above equations are model parameters. Generally they are identified through curve fitting on the basis of acquired experimental data. In equation (13) H stands for the Heaviside function: (13) 1 if q 0 H q (14) 0 if q 0 As it can be seen in equation (1), it is presumed that the quantity 1 is a function of the shear strain, shear strain rate and a damage parameter- q e. Damage parameter supplies information about material degradation. Upon appropriate calibration on the basis of the current value of the damage parameter degrading phenomena in material can be rationally estimated. In the present study the evolution of the damage parameter is defined by using a governing equation proposed by Dall Astra and Ragni (006): 0.5 qe if qe 0.5 q e (15) 0 if 0.5 qe 1 As already stated damage parameter should enable the evaluation of the current state of mechanical damage after a given period of exploitation. The loading history or the loading path which has led to the current state of mechanical degradation is taken into account through the variable - the shear strain and its time rate of change - see equation (15). In the present study only numerical simulations of typical identification test e.g. repetitive loading paths as the one shown in (Figure ). Figure. Loading history: strain evolution The evolution of the damage parameter obtained by integrating equation (15) is depicted Figure 3. As it can be seen only the first quarter of one loading cycle is considered - variation of the shear strain from zero to 1.5 at a damage strain rate 0.033(3 ). For the damage parameter to be more informative, the model should be calibrated against experimental data and the degrading phenomena - assessed through direct observation if appropriate.

5 Figure 3. Degradation of the material in function of the shear strain 4. IDENTIFICATION OF THE MODEL PARAMETERS The numerical procedure employed for the identification of model constants is outlined in this section. The identification procedure is based on genetic algorithms [Goldberg ]. Curve fitting typically consists in finding a set of model parameters for which the results obtained by implementing the model fit best a trial set of data points. Generally the trial set of data points is an experimentally obtained relation that characterizes the modelled behaviour. Chart-flow of the procedure is shown in Figure 4. Figure 4. Procedure for identification of m\the model parameters The identification procedure is aimed to obtain a target set of model parameters - the set for which results obtained by model implementation match best the trial set of data points. Initially the model constants are grouped in m sets. Each model constant is varied through those sets. Each set forms an entry for the model and on the basis of the data contained in each set the evolution of the modelled parameter can be obtained. Thus the output of the model defined by equation (1) and equation () is the evolution of the shear force in a lead-core bearing device in function of the shear displacement. The results obtained by numerical integration are compared to the experimental data. Naturally, some sets will yield a response which is more convergent to the trial data and other - which are less convergent. Therefore the adequacy of the results obtained by each set of model constants is checked on the basis of an error-estimation-like criterion. The distances between data points from the model

6 results and data points from trial set are estimated; a smaller average distance for a given set compared to another is estimated as a better fit to the experimental data. Furthermore, the initial sets are manipulated to improve convergence with the trial data set; a tuple of sets S is compiled through iterations from the initial sets of model parameters S. For every (t) iteration the chosen material model is implemented, e.g. equations (1) and () are numerically solved by using the modified mid-point method (equations from (3) to (7)) - see Analysis box in Figure 4. The Model results are further compared to the trial set of data points ( Comparison box in Figure 4). In the Modification module, in the framework of the employed genetic algorithm, the current sets of model constants are refined through mutations and crossing-over to optimize the fit with the trial data set in the next iteration. From the final sets (t) S the one yielding an optimal fit with trial data set can be chosen. The implementation of the identification procedure is illustrated by choosing a trial set of data point. Specifically the reference function is the Shear force- Shear displacement relationship plotted in Figure 1. This constitutive relationship is obtained by employing the model parameters summarized in Table 1.Several steps of the identification procedure are shown in Figure 5 a)- f). (i) a) b) c) d) e) f) Figure 5. Successive approximations (plotted in grey) to the trial curve (in black)

7 The trial curve is plotted in black whereas the set of model parameters showing the best convergence within a given iteration- in gray. It can be seen that the best guess for a given iteration is getting closer to the trial data set while 5. CONCLUDING REMARKS Analytical models aimed at reproducing the mechanical response of a lead-core bearing devices for passive seismic isolation have been considered in this paper. Two numerical procedures have been developed by using Python software on the basis of models proposed in literature: the Bouc-Wen model and a constitutive relation which takes into consideration the mechanical degradation in the lead-core bearing device. A curve fitting algorithm has been also presented. The purpose of the fitting procedure based on genetic algorithms is the identification of the parameters in the constitutive law chosen for the bearing device. Generally this identification is done through comparison with experimental data. The proposed numerical procedures can be subsequently employed in the design of damped structures - structures containing devices for passive seismic isolation such as lead-core bearing devices. After adding a procedure for identification of the model parameters the model in which a damage variable is introduced can be used for assessment of degrading phenomena taking place in the leadcore bearing device. REFERENCES Bouc R (1967) Forced vibration of mechanical systems with hysteresis. In: Proceedings of the Fourth Conference on Nonlinear Oscillation, Prague, Czechoslovakia, pp 315 Bouc R (1971) Modèle mathématique d'hystérésis: application aux systèmes à un degré de liberté. Acustica (in French) 4:16 5 Wen YK (1976) Method of Random Vibration of hysteretic system. J Eng Mechanics Division, ASCE, 10():49-63 Constantinou MC, Tadbakhsh M (1985) Hysteretic Dampers in Base Isolation: Random Approach. Journal of structural Eng 111(4): Song J, Der Kiureghian A (006) Generalized Bouc Wen model for highly asymmetric hysteresis. J Eng Mechanics ASCE 13(6): Press WH, Teukolsky SA, Vetterling WT, Flannery BP (007) Numerical Recipies, The Art of Scientific Computing, 3rd edn. Cambridge University Press Dall Astra A, Ragni L (006) Experimental tests and Analytical Model of high Damping Rubber Dissipating Devices. Engineering Structures 8(13): Govindee S, Simo JC (199) Mullins effect and the strain amplitude dependence of the storage modulus. Int J Solids Struct 9: Haupt P, Sedlan H (001) Viscoplasticity of elastomeric materials: experimental facts and constitutive modelling. Arch Appl Mech 71: Fabrizio M, Morro A (199) Mathematical problems in linear viscoelasticity. Philadelphia: SIAM Studies In Applied Mathematics Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Reading (MA): Addison- Wesley