Study Of Steady-state Response Of Hysteretic System Under Stochastic Excitation

Hysteresis phenomenon exits in various of structural or mechanical engineering.A large number of devices exhibiting significant hysteresis behavior,such as non-traditional materials and smart structures,have been applied in practical engineering.Hysteretic loop is created between hysteretic restoring force and displacement in periodic motion.Hysteretic force depends on both the instantaneous deformation and the past history of deformation.Thus,the hysteretic system belongs to a type of complex nonlinear system.In the last few decades,hysteretic random vibration has been studied extensively by various approximate techniques,such as the equivalent linearization technique and the stochastic averaging method.However,no exactly closed-form solution is available so far.This paper introduced the iterative method of weighted residuals firstly.This method is applied to study several classical single-degree-of-freedom nonlinear systems and obtained the closed form solutions of steady-state response of these systems.Numerical results of the examples show that the proposed procedure could lead to the exact solution when the exact solution exists or yields to a highly accurate solution as compared with the Monte-Carlo simulation data if the exact solution is not available.As applications,the iterative method of weighted residuals is further extended to study the Gaussian white noise excited classical Bouc-Wen hysteretic system and generalized Bouc-Wen system in this paper.In the study of classical Bouc-Wen hysteretic system,a Gaussian PDF is obtained with equivalent linearization technique firstly,which is used to construct a weighting function.Then,the method of weighted residuals is utilized to determine the non-Gaussian PDF of exponential polynomial type.Finally,an iterative procedure is introduced to improve the accuracy of the solutions obtained from the method of weighted residuals.As an illustrative example,the steady-state stochastic response of the steel fiber reinforced ceramsite concrete column under horizontal random seismic loading is studied,in which the hysteretic parameters associated with Bouc-Wen hysteretic model is identified from the pseudo-static test.Compared to the Monte Carlo results,the accuracy of results obtained from equivalent linearization method is poor.The results obtained from weight residue method can show the nonlinearity of Bouc-Wen systems,but its accuracy is still unsatisfactory.The iterative method of weight residuals can lead to results which have higher accuracy.The proposed closed-form solution of steady-state PDF of Bouc-Wen hysteretic system can be benchmark to examine the accuracy of solutions obtained by other methods.In the study of generalized Bouc-Wen hysteretic system,a full scale ceramsite concrete frame structure under random excitation is studied.First,the hysteretic dissipation test curve is obtained with application of the quasi-static method.The undetermined parameters in the generalized Bouc-Wen system are identified with application of gradient descent method.A Gaussian PDF of the corresponding classical Bouc-Wen system is obtained firstly with equivalent linearization technique.Then,the 2-order Gaussian steady-state PDF of generalized Bouc-Wen system is obtained by using the iterative method of weighted residuals.Finally,the 2-order Gaussian steady-state PDF is applied to form the new weighting function,and the 4-order non-Gaussian steady-state PDF is obtained.Compared to the Monte Carlo simulation results,the 4-order non-Gaussian steady-state PDF has higher accuracy,especially in the tails of the results.Therefore,the iterative method of weight residuals can be an effective approach to estimate the reliability of single-layer ceramsite concrete frame structures under random earthquake excitation...