Response Analysis Of Nonlinear Structures Research Articles

The dimension-reduction strategy via mapping for probability density evolution analysis of nonlinear stochastic systems

Dynamic response analysis of nonlinear structures involving random parameters has for a long time been an important and challenging problem. In recent years, the probability density evolution method, which is capable of capturing the instantaneous probability density function (PDF) of the dynamic response and its evolution, has been proposed and developed for nonlinear stochastic dynamical systems. In the probability density evolution method, the strategy of selecting representative points is of critical importance to the efficiency especially when the number of random parameters is large. Enlightened by Cantor’s set theory, a strategy of dimension-reduction via mapping is proposed in the present paper. In the strategy, a two-dimensional domain is firstly considered and discretized such that the grid points are assigned with probabilities associated to the joint PDF. These points are then sorted and set on a virtual line according to a certain principle. Partitioning the sorted points on the virtual line into a certain number of intervals and selecting one single point in each interval, the two random variables can be transformed to a single comprehensive random variable. The associated probability of each point is simultaneously transformed accordingly. In the case of multiple random parameters, the above dimension-reduction procedure from two to one could be used recursively such that the random vector is finally transformed to one single comprehensive random variable. Numerical examples are investigated, showing that the proposed method is of high efficiency and fair accuracy.

A MODAL SUPERPOSITION METHOD FOR NON-LINEAR STRUCTURES

The dynamic response of multi-degree of freedom (MDOF) non-linear structures is usually determined by the numerical integration of equations of motion. This is computationally very costly for steady state response analysis. In this study, a powerful and economical method is developed for the harmonic response analysis of non-linear structures. In this method, the equations of motion are first converted into a set of non-linear algebraic equations, and then the number of equations to be solved is reduced by employing linear modes of the system. The resulting coupled algebraic equations are solved iteratively. The concept of the pseudo-receptance matrix is introduced in calculating the steady state response. It is shown that the reduction in computational time can be considerable as very accurate results can be obtained by using only a limited number of modes. Several example problems are solved in order to illustrate the performance of the method. Effects of higher modes and higher harmonic components of the response are discussed in detail. In order to illustrate the accuracy and speed of the method suggested, the results obtained by applying the so called Iterative Modal Method are compared with those obtained by the Newmark method which is used as a classical time domain analysis technique.

Dynamic response analysis of nonlinear structures using step-by-step integration techniques

The paper describes step-by-step integration techniques to predict the dynamic response of nonlinear tension structures. The analytical procedure and the mathematical formulation of selected potential algorithms used for the solution of the equation of motion are presented. Methods of introducing the wind and earthquake loading into the step-by-step method are described. The response analysis of vibrations of a flat pretensioned cable net excited by simple harmonic loading and by a falling load are also presented, followed by a numerical example of an experimental portal frame solved by the linear acceleration method.

Forced harmonic response analysis of nonlinear structures using describing functions

A semianalytical quasilinear method based on the describing function formulation is proposed for the harmonic response analysis of structures with symmetrical nonlinearities. The equations of motion are converted to a set of nonlinear algebraic equations. The linear and nonlinear parts of the structure are dealt with separately, the former being represented by the constant linear receptance matrix, and the latter by the generalized quasilinear matrix which is updated at each iteration. Several examples dealing with cubic stiffness, piecewise linear stiffness, and coulomb friction type of nonlinearities are presented

Seismic response analysis of nonlinear structures

A solution method for the response of a class of nonlinear viscoelastic shear building structures subjected to stochastic excitation has been developed by means of a stochastic equivalent linearlization technique. The nonlinear viscoelastic properties of the structures are modelled in terms of an equation of motion which linearly involves the auxiliary variables as part of the restoring force and in terms of the auxiliary equation which describes a nonlinear relationship among the story displacements and auxiliary variables and their time derivatives. This auxiliary equation is linearized using a stochastic linearization technique. The integration of the equation of motion together with the linearized auxiliary equation is carried out numerically using the state-vector formulation. In doing so, an iterative upgrading of the values of the linearization coefficients is performed simultaneously for all the stories in the first time interval until a convergence criterion is satisfied. The iterative process is then repeated in the time intervals that follow, until the entire time interval for which the dynamic analysis is performed is covered. The analysis is based on the modal method which, however, requires the use of complex eigenvalues and eigenvectors. The proposed analysis produces covariance functions of the story displacements and velocities among other response quantities. These covariance functions play an important role in estimating structural reliability. The covariance function thus developed agrees very well with that obtained by the Monte Carlo technique.