The National Science Foundation's (NSF) Tokyo Office periodically receives and disseminates reports on research developments in Japan that are related to the Foundation's mission. NSF-sponsored researchers currently working in Japan prepare many of these reports. These reports provide information for use by the global science and engineering community.
Mr. Greg Baker, a Ph.D. candidate in the Department of Civil Engineering and Geological Sciences, University of Notre Dame, Notre Dame, Indiana, prepared the following report. Mr. Baker was a participant in the 1998 Monbusho Summer Program sponsored by NSF and the Ministry of Education, Science, Sports and Culture (Monbusho). Professor Yozo Fujino of the Department of Civil Engineering at the University of Tokyo, Tokyo, Japan hosted Mr. Baker. Mr. Baker can be reached via email at: gbaker1@vyasa.helios.nd.edu
Introduction
There are many important structures in which we have come to take for granted as just "being there." Bridges are examples of such. They are important lifelines of transportation to practically everyone in the world. In the United States there are an estimated 375,000 bridges and 130,000 in Japan that are fifteen meters or longer. Many of these bridges are in regions of seismic hazard. What kind of protective measures can be taken to reduce the effects of seismic motion?
There have been many construction designs and technologies developed to reduce the effects of earthquakes on structures in recent years. One particular ideology that has proven effective is the area of base isolation. The goal of seismic isolation is to "isolate" or "remove" the structure from ground motion energy and therefore reduce the probable damage level of the structure. Several key factors play important roles for successful base isolation. First of all, the base isolation system must provide flexibility in shear, that is to say that it increases the fundamental period of the overall structure and therefore removes the structure`s dominant natural frequency from the dominant spectral density acceleration content of the earthquake. Secondly, the isolation system must provide damping or energy dissipation to balance the effects of increased flexibility. Next the system must provide an adequate restoring force to the structure to return it to the original position. Another important quality is the ability to exhibit a high hysteretic effect without sacrificing stiffness at low strains. Therefore, wind loads do not initiate movement of the isolation system. In general, the selection of the appropriate base isolation system for a particular structure depends on the factors that are most important to the designer. These can include, but are not limited to, base shear; base displacement; higher frequency acceleration attenuation (above 2 Hz); and structural importance (design for excessive ground motion). At the same time costs, maintenance and life expectancy of the system must be addressed. The main performance criteria are generally base shear and base displacement. Unfortunately, when making the selection of a base isolation system, one factor must usually be sacrificed in order to achieve a high level of another factor.
Bearing Modeling
In order to implement base isolation devices into structures a model of their behavior is needed. Current design for construction of base isolated structures uses simplified models with linear and bi-linear force-displacement hysteretic loops of the base isolated devices so that global deformation characteristics can be approximated. This coupled with design constraint approximations for the maximum shear strain and displacement has been accurate enough for implementation. However, for a complete picture of issues such as load carrying capacity, design life and rate and history dependence effects, better models are needed. To design models with more accuracy, more sophistication is needed.
Wen`s Models
Over time a series of models by different authors has been developed for more accurate hysteretic modeling using varying degrees of complexity involving random vibration. This type of modeling produces a set of nonlinear stochastic differential equations, of which in order to be put in a more usable form must be linearized. A major problem found in producing an equivalent linearized model was finding a good closed form solution. One method that has been used to overcome this problem was to apply the Krylov-Bogoliubov approximation. Unfortunately, this method assumes that the response is contained in a narrow band. Wen`s particular contribution to this area of modeling was to develop a model that could represent a wide variety of hardening or softening with smoothly varying or nearly bilinear hysteresis for a considerable range, particularly those systems under random vibration with high damping (wide-band system) (Wen 1976). He was also successful in making the equivalent linearization of his model without the use of the Krylov-Bogoliubov approximation (Wen 1980). This model was then extended to include degradation of the stiffness or strength for multidegree of freedom shear beam structures (Baber, et al. 1981). A similar hysteretic model was later developed to describe hysteretic effects in two dimensions. This model was applied to reinforced concrete columns (Park, et al. 1985).
Constitutive Modeling
These models, as good as they are, are still limited. To arrive at a model that will give an accurate description of the global and local effects, such as global or local ensuing failure and local stresses, a material model must be developed. The material properties can then be included into a non-linear finite element analysis to give an accurate model of the devices. This introduces the problem of modeling, or developing a constitutive law, for the rubber materials themselves. Constitutive modeling is a set of equations that constitute the material response at any point which are generally later combined with finite-element modeling for analysis.
Two types of rubber material are examined, low damping (natural) and high damping rubber. For natural or elastomeric rubber the generally accepted constitutive law which includes very high strain rate is the extended Rivlin-Saunders, Hart-Smith theory model given by Alexander (1968).
Unified Constitutive Modeling
Unlike natural rubbers, high damping rubber possesses properties that make its behavior more difficult to predict. These include the effects of strain memory, the breakdown of weak interparticle bonds between rubber and carbon black, the disentaglement and orientation of rubber molecule chains and the change and restoration of damping during and after applied loading sequences. To model highly nonlinear effects such as these, a unified constitutive model gives more generality. A unified constitutive law is one that unifies separate constitutive laws into one package because of the assumption that the primary mechanisms of creep and plasticity result from the same thing, the motion of dislocations in the microstructure. Unifed constitutive models generally are considered to provide:
Unfortunately, developing a model of this kind does pose a few problems. The complexity of the model can create mathematical difficulties for finite-element programs as the number of internal variables increases. Likewise, modeling of the internal variables themselves can be challenging because certain phenomena or interaction of phenomena are difficult to understand fully. These problems pose challenging questions to the modeler that must be addressed (Miller 1987).
A balance must be created between defining a simple model, yet one that accurately describes the phenomena in the material. In this respect, unified constitutive modeling is more of an art than a science. The current project at the University of Tokyo is looking at several alternatives to modeling this material which include six unified models given in Unified Constitutive Equations for Creep and Plasticity by A. K. Miller (1987), Chaboche`s model and others.
Acknowledgments
This research was supported by The Monbusho Summer Program, and the National Science Foundation. The assistance of Professor Yozo Fujino, Yoshida-san, and all of those in the Bridge and Structures Laboratory at the University of Tokyo is gratefully appreciated.
References
Baber, T.T., and Wen, Y.K. (1981). "Random Vibration of Hysteretic Degrading Systems." J. Engrg. Mech., ASCE, Vol. 107, No. 6, pp.1069-1087.
Miller, A.K. (1987). Unified Constitutive Equations for Creep and Plasticity., Elsevier Applied Science, New York.
Park, Y.J., Wen, Y.K. and Ang, A. H-S. (1986). "Random Vibration of Hysteretic Systems under Bi-directional Ground Motions." Earthquake Engineering and Structural Dynamics., Wiley, New York, Vol. 14, pp. 543-557.
Skinner, R.I., Robinson, W.H., and McVerry, G.K. (1993). An Introduction to Seimic Isolation., Wiley, New York.
Wen, T.K., (1976). "Method for Random Vibration of Hysteretic Systems." J. Engrg. Mech., ASCE, Vol. 102, No. EM2, Proc. Paper 12073, pp. 249-263.
Wen, T.K., (1980). "Equivalent Linearization for Hysteretic Systems under Random Excitation." Journal of Applied Mechanics., Vol. 47, pp. 150-154.