Nestor Distefano

System identification in nonlinear structural seismic dynamics

Abstract This paper deals with the development of efficient identification methods for the determination of parameters associated with the nonlinear response of structural systems subject to seismic conditions. The equation of motion of the system is given by M u + h(/.u, u, a) = p(t) , where the external force vector p and the mass matrix M are assumed to be known. Also given is an observation vector w , consisting of some or all of the components of u , the vector of the nodal displacements. The vector function h , denoting the restoring forces of the mechanical system, is parametrically given in terms of a constant vector a . Three different methods are presented for the determination of the parameters governing the vector function h . The first one is a direct approach requiring the unknown coefficients to appear linearly in the model equation. The remaining two are based on methods of control and optimization theory, and are exempt from the limitations of the direct method. The relative merits of each approach are discussed and extensive numerical experimentation is presented. Only numerical results for one degree of freedom systems are reported in this paper.

Sequential identification of hysteretic and viscous models in structural seismic dynamics

Abstract In this paper the sequential identification approach outlined in [1]is employed to estimate the parameters of a bilinear hysteric model of the kinematic type. Emphasis is placed on the development of an estimation routine geared to bypass the difficulty originating from the indetermination of the identification problem in terms of the three parameters associated with the piecewise linear model employed. Numerical experimentation using simulated data is presented to show the feasibility and accuracy of the method. An additional example involving the modeling of a real steel frame tested on a seismic table is finally presented as a preliminary incursion into the field of modeling and identification of real structures.

A dynamic programming approach to the formulation and solution of finite element equations

Abstract A method for formulating and algorithmically solving the equations of finite element problems is presented. The method starts with a parametric partition of the domain in juxtaposed strips that permits sweeping the whole region by a sequential addition (or removal) of adjacent strips. The solution of the difference equations constructed over that grid proceeds along with the addition removal of strips in a manner resembling the transfer matrix approach, except that different rules of composition that lead to numerically stable algorithms are used for the stiffness matrices of the strips. Dynamic programming and invariant imbedding ideas underlie the construction of such rules of composition. Among other features of interest, the present methodology provides to some extent the analysts control over the type and quantity of data to be computed. In particular, the one-sweep method presented in Section 9, with no apparent counterpart in standard methods, appears to be very efficient insofar as time and storage is concerned. The paper ends with the presentation of a numerical example.

Dynamic programming and a max-min problem in the theory of structures

Abstract A max—min problem in the realm of optimum beam design is formulated and thoroughly investigated from a dynamic programming point of view. It is shown that the conditions of optimality can be directly derived from the Hamilton—Jacobi—Bellman equation of the process. The classical Euler—Lagrange equations for the beam are derived from the fundamental partial differential equation. It is shown that the conditions of optimality associated with the minimum operation are local expressions of the theorem of Castigliano. An analytical solution for the unconstrained optimum cantilever laying on elastic foundation is presented, and a method of successive approximations consisting in a stable, two-sweep iterative procedure, is developed. Numerical examples are given.

A quasilinearization approach to the solution of elastic beams on nonlinear foundations

Abstract In this paper, the solution of a beam on nonlinear elastic foundation whose deflection satisfies the nonlinear boundary value problem (1, 2), is studied by means of the theory of quasilinearization. The problem is formulated in Section 2 where conditions for the existence and uniqueness of the solution are stated. In Section 3, the idea of quasilinearization is introduced and the positivity of an associated linear differential operator is investigated. In Section 4 the usual version of quasilinearization, i.e. The Newton-Raphson-Kantorovich sequence, is presented and conditions under which this sequence is monotonically convergent, are established. In Section 5, an alternative successive approximation scheme whose derivation relies on ideas of quasilinearization, is presented. Finally, an example is solved by numerical procedures based in the methods discussed in previous sections.

Identification of the viscoelastic characteristics of an artificial neck

Abstract This paper is devoted to the identification of the viscoelastic characteristics of an artificial neck employed to conduct experiments on dummies in automobile crash simulation tests. The neck is assumed to be a linear viscoelastic cantilever beam subject to a bending moment and a shear force at the head-neck junction. Time histories of these forces and the associated rotation obtained in a calibration test simulating an automobile crash condition are available and furnished by the manufacturer of the neck. Using these data, an identification procedure in the Laplace transform space is developed. The analytical relationships between forces and displacements at the head-neck junction are presented in Sec. 2, and the viscoelastic model is introduced in Sec. 3. In Sec. 4, the identification problem is formulated, and in Sec. 5 an example displaying experimental versus predicted results is presented. Finally in Sec. 6 an extension to consider higher dimensional representations is outlined.

Quasilinearization and the solution of nonlinear design problems in structures undergoing creep deformations

Abstract A class of multipoint-value problems involving generally nonlinear integro-differential equations of Volterra type, arising in applications of structural design in the presence of nonlinear creep is formulated and thoroughly investigated. First, a nonlinear multipoint-value problem is formulated and solved algorithmically by quasilinearization. This is done in Sections 2 and 3. Relaxation of design specifications yields a class of optimization problems whose solution is then outlined in Section 4. Sections 5–7 are devoted to computational aspects. In Section 6, a method to overcome some of the computational drawbacks of the classical Newton-Raphson-Kantorovich sequence in function space, is presented and applied to the problem under consideration. Section 7 deals with the reduction of large systems of Volterra integral equations to initial-value differential systems, while in Section 8 a numerical example is presented to illustrate the application and feasibility of the method.

Invariant imbedding and optimum beam design with displacement constraints

Abstract In the present paper, the condition of optimality for a beam under elastic foundation subject to a displacement constraint are derived from the calculus of variations and subsequently employed as the starting point in the development of a stable method for the numerical solution of the optimization problem. Using ideas of invariant imbedding and the method of successive approximations, the pertinent nonlinear boundary value problem is reduced to a two sweep iterative procedure in terms of a system of Riccati differential equations subject to initial values, exhibiting favorable stability properties. Two examples thoroughly developed, are finally presented to illustrate the application and the accuracy of the method.

Invariant imbedding and the effects of changes of poisson's ratio in thin plate theory

Abstract The effect of changes of Poissons ratio ν in the solutions of equilibrium problems in thin plate theory is investigated from an invariant imbedding point of view. The fundamental variable of the imbedding is ν, the physical variable for which the functional perturbation is desired. Using this imbedding, a Cauchy system is then formulated, using as initial conditions the corresponding solutions of the initial-value problem previously developed in [1], [3], to study the equilibrium of a plate for a fixed value of ν. Integration of the Cauchy system furnishes the desired results.

On alternative representations of time varying viscoelastic materials

Abstract Conditions for equivalence between linear differential and a class of integral operators are given and the results are devoted to applications to alternative representations of linear, time varying viscoelastic materials. Several examples are presented, including a discussion on sufficient conditions under which time varying linear differential operators can be used to analytically represent asymptotically stable materials, and an example on the inversion of integral equations appearing in the treatment of thermorheologically simple materials.