Limit point buckling of a finite beam on a nonlinear foundation

In this paper, we consider an imperfect finite beam lying on a nonlinear foundation, whose dimensionless stiffness is reduced from 1 to k as the beam deflection increases. Periodic equilibrium solutions are found analytically and are in good agreement with a numerical resolution, suggesting that localized buckling does not appear for a finite beam. The equilibrium paths may exhibit a limit point whose existence is related to the imperfection size and the stiffness parameter k through an explicit condition. The limit point decreases with the imperfection size while it increases with the stiffness parameter. We show that the decay/growth rate is sensitive to the restoring force model. The analytical results on the limit load may be of particular interest for engineers in structural mechanics.

IMPROVEMENTS ON THE ARC-LENGTH-TYPE METHOD

Arc-length-type and energy-type methods are two main strategies used in structural nonlinear tracing analysis, but the former is widely used due to the explicitness and clarity in conception, as well as the convenience and reliability in calculation. It is very important to trace the complete load-deflection path in order to know comprehensively the characteristics of structures subjected to loads. Unfortunately, the nonlinear analysis techniques are only workable for tracing the limit-point-type equilibrium path. For the bifurcation-point-type path, most of them cannot secure a satisfactory result. In this paper, main arc-length-type methods are reviewed and compared, and the possible reasons of failures in tracing analysis are briefly discussed. Some improvements are proposed, a displacement perturbation method and a force perturbation method are presented for tracing the bifurcation-point-type paths. Finally, two examples are analyzed to verify the ideas, and some conclusions are drawn with respect to the arc-length-type methods.

Some Problems in Nonlinear Dynamic Instability and Bifurcation Theory for Engineering Structures

In civil engineering, the nonlinear dynamic instability of structures occurs at a bifurcation point or a limit point. The instability at a bifurcation point can be analyzed with the theory of nonlinear dynamics, and that at a limit point can be discussed with the theory of elastoplasticity. In this paper, the nonlinear dynamic instability of structures was treated with mathematical and mechanical theories. The research methods for the problems of structural nonlinear dynamic stability were discussed first, and then the criterion of stability or instability of structures, the method to obtain the bifurcation point and the limit point, and the formulae of the directions of the branch solutions at a bifurcation point were elucidated. These methods can be applied to the problems of nonlinear dynamic instability of structures such as reticulated shells, space grid structures, and so on. Key words nonlinear dynamic instability - engineering structures - non-stationary nonlinear system - bifurcation point - instability at a bifurcation point - limit point MSC 2000 74K25 Project supported by the Science Foundation of Shanghai Municipal Commission of Education (Grant No. 02AK04), the Science Foundation of Shanghai Municipal Commission of Science and Technology (Grant No. 02ZA14034)

NONLINEAR STABILITY ANALYSIS OF A CLAMPED ROD CARRYING ELECTRIC CURRENT

This paper is devoted to the analysis of the nonlinear Stability of a clamped rodcarrying electric current in the magnetic field which is produced by the current frowingin a pair of inifinitely long parallel rigid wires. The natural State of the rod is in theplane of the wires and is equidistant from them.Firstly under the assumption of apatial deformation, the governing equations of the problem are derived, and the linearizedproblem and critical currents are discussed. Secondly, it ls proved that the buckledstates of the rod are always in planes. Finally. the global responses of the bifurcationproblem of the rod are compuled numerically and the distributions of the deflections.axial forces and bending monents are obtained. The results show that the buckledslates of the rod may be either supercritical or Subcritical. depending on the distancebetween the rod and the wires. Furthermore, it is found that -there exists a limit pointon the branch solution of the supercritical buckled State. This is distinctively differentfrom the buckled slate of the elastic compressive rods.

NUMERICAL IMPLEMENTATION OF PSEUDO-ARCLENGTH ALGORITHM IN NONLINEAR FINITE ELEMENT ANALYSIS

A numerical procedure to calculate the pre-buckling and post- buckling response of general structures is presented. This procedure is based on the pseudo-arclength algorithm suggested by E. Riks et al., which has some numerical difficulties during implementation of large applied analysis programs. To overcome these difficulties, a scheme based on rank-1 modification of the matrix is proposed. Some examples show this procedure behaves well in passing through the limit point and is rather efficient.

Compression-induced stiffness in the buckling of a one fiber composite

We study the buckling of a one fiber composite whose matrix stiffness is slightly dependent on the compressive force. We show that the equilibrium curves of the system exhibit a limit load when the induced stiffness parameter gets bigger than a threshold. This limit load increases when the stiffness parameter is increasing and it is related to a possible localized path in the post-buckling domain. Such a change in the maximum load may be very desirable from a structural stand point.

Fluctuation Limits of the Single Point Catalytic Super-stable Processes

We prove a fluctuating limit theorem of a sequence of super-stable processes overR with a single point catalyst.The weak convergence of the processes on the space of Schwartz distributions is established.The limiting process is an Ornstein–Uhlenbeck type process solving a Langevin type equation driven by a one-dimensional stable process.

CENTER CONDITIONS AND BIFURCATION OF LIMIT CYCLES FOR A CLASS OF FIFTH DEGREE SYSTEMS

The center conditions and bifurcation of limit cycles for a class of fifth degree systems are investigated.Two recursive formulas to compute singular quantities at infinity and at the origin are given.The first nine singular point quantities at infinity and first seven singular point quantities at the origin for the system are given in order to get center conditions and study bifurcation of limit cycles.Two fifth degree systems are constructed.One allows the appearance of eight limit cycles in the neighborhood of infinity,which is the first example that a polynomial differential system bifurcates eight limit cycles at infinity.The other perturbs six limit cycles at the origin.

STRONG LIMIT POINT FOR LINEAR HAMILTONIAN DIFFERENCE SYSTEM

The definition of strong limit-point for singular Hamiltonian difference expressions with complex coefficients are given, and some strong limit-point criteria are established.

The Limit of Finite Sample Breakdown Point of Tukey’s Halfspace Median for General Data

Under special conditions on data set and underlying distribution, the limit of finite sample breakdown point of Tukey＇s halfspace median （1） has been obtained in the literature. In this paper, we establish the result under weaker assumptions imposed on underlying distribution （weak smoothness） and on data set （not necessary in general position）. The refined representation of Tukey＇s sample depth regions for data set not necessary in general position is also obtained, as a by-product of our derivation.

Characterization of pore volume of cumulative water injection distribution

Pore volume of Cumulative water injection is one of the factors for evaluating water flood effect in a water flood oil field.In previous study,there were limited lab studies for evaluating oil displacement efficiency.A method to characterize the distribution of pore volume of cumulative water injection is proposed in this paper,and it is verified by a five-spot water flooding streamline simulation model.The logarithmic relation between pore volume of cumulative water injection and water saturation is established by regression.An inflection point and limit point of cumulative water injection pore volume are identified.Current simulation model indicates inflection point appears after 2e5 pore volume(PV)injection,and limit point appears after 15e25 PV injection.Both inflection and limit point vary in different regions of reservoir.