Fundamental Path Research Articles

Direct calculation of buckling strength of imperfect structures

This paper is concerned with the direct calculation of buckling strength of an imperfect structure. The main assumptions are that, for the perfect structure, there is a symmetric or a linear fundamental path and the corresponding first critical point is a simple symmetry-breaking or a simple bifurcation point. An extended system of limit points is proposed for which the newly introduced scaling parameters are regular solution and thus standard methods can be used to compute them. Using the extended system, one can directly obtain the exact buckling loads without tracing the postbuckling paths. An efficient implementation of Newton’s method solving the extended system is presented and numerical examples are given.

SCR of NO x with CH 3OH on H-mordenite: mechanism and reaction intermediates

The selective reduction of NO x over H-mordenite (H-m) was studied using CH 3OH as reducing agent. Results are compared with those obtained with other conventional reducing agents (ethylene and methane), with gas-phase reactions, and with other metal-exchanged mordenites (Cu-mordenite (Cu-m) and Co-mordenite (Co-m)). H-m was found to be an effective catalyst for the SCR of NO x with CH 3OH. When different reducing agents were compared over H-m, CH 3OH > C 2H 4 > CH 4 was the order according to the maximum NO conversion obtained using 1% of oxygen in the feed. Instead, if selectivity is considered, the order results CH 4 > CH 3OH > C 2H 4. In reaction experiments, two distinct zones defined by two maxima with NO to N 2 conversion are obtained at two different temperatures. A correlation exists between the said zones and the CO : CO 2 ratio. At low temperatures, CO prevails whereas at high temperatures CO 2 prevails. These results indicate that there exist different reaction intermediates. Evidence from reaction experiments, FTIR results, and transient experiments suggest that the reaction mechanism involves formaldehyde and dimethyl ether (DME) as intermediates in the 200–500°C temperature range. The surface interaction between CH 3OH (or its decomposition products) and NO is negligible if compared with NO 2, indicating that the oxidation of NO to NO 2 on acid sites is a fundamental path in this system. Different from other non-oxygenated reductants (methane and ethylene), a gas-phase NO x initiation effect on hydrocarbon combustion was not observed.

FORCED VIBRATIONS OF SILOS LEADING TO BUCKLING

The large-amplitude force vibrations of steel thin-walled silos when empty are investigated. The basic geometry configuration modelled is a cylinder clamped at the bottom with a top conical roof. Wind pressure distributions are assumed as non-axisymmetric in the circumferential direction and with a rectangular impulse or step distribution in time. Instability is identified from finite-element computations of the time response of the shell using a criterion due to Budianski and Roth. Results are computed for silos made with plain as well as with corrugated sheets, and the influences of geometric imperfections and the stiffening due to the roof are included in the analysis. The problems are also modelled with static pressures using both continuation techniques and bifurcation analysis from a linear fundamental path. Additional results have been obtained to estimate the dynamic buckling load for step loading using energy procedures. All results are computed using finite-element codes developed by the authors.

Square-ring microstrip antenna with a cross strip for compact circular polarization operation

A novel design of square-ring microstrip antenna with a cross strip for achieving compact circular polarization (CP) operation is proposed and experimentally studied. The cross strip can be placed in the centerlines or diagonals of the square-ring patch. Either design can make its excited fundamental mode patch surface current path much longer than that in a conventional square microstrip antenna with same antenna size. By incorporating a small tuning stub for splitting the fundamental mode into two near-degenerate resonant modes with equal amplitudes and 90/spl deg/ phase-difference compact CP operation for the proposed design can he obtained.

Analysis of ecosystem structure and function: extended path and flow analysis of a steady-state oyster reef model

An analysis of the extended path and flow structure of a six compartment steady-state oyster reef model was conducted. The extended path and flow structure were analyzed in the context of a refined canonical path classification system based on the systems theory methods of environ and network unfolding analyses. A computer implementation of an operational path classification system facilitated investigation of a finite portion (path length ≤17 arcs) of the direct and indirect path structure of the oyster reef model. Important results of the path structure analysis include: (1) few simple paths and large numbers of compound paths enumerated; (2) dominance of path numbers by subsequent passage terminal cycle paths; (3) structural evidence in support of feedback control in ecosystems; (4) results provide evidence by analogy to support the hypothesis of network homogenization first described using the systems analysis methods of environ analysis and network unfolding; (5) constancy of the pattern of origin–destination path counts with increasing path length; (6) importance of nonliving compartments in the extended path structure of ecosystems. Simultaneous path and flow analysis of the oyster reef model assessed the flow contributions of the fundamental path categories for this model using a modification of a path-based network unfolding method. First passage paths contribute most of the flow; however, multiple passage cyclic paths also provide a large (22%) flow contribution. Because of cycling in the system, the numerous long paths in the extended path structure of this ecosystem model are significant in its function as represented by the flows. These results provide microscopic evidence for the macroscopic results of environ analysis that implicate cycling as a key ecosystem attribute in the mechanisms of holistic system determination. The principles enunciated here for a model with a low cycling index (11%) carry over to, and would be even more significant for, models with high cycling indexes. These results also serve to form a link between the extended structure of food webs and their functioning as represented by energy-matter flows. The present analysis demonstrates that extended path structure, and the component articulation from which it is generated, have significant consequences for ecosystem function.

Buckling of short tanks due to hurricanes

Buckling of thin-walled, short tanks, under severe wind conditions, is investigated using numerical methods. The shells are representative of cylindrical tanks that failed during hurricane Marilyn in the Caribbean islands in 1995, with radius/thickness ratio of 1900 and radius/height ratio of 5. Several models are employed to study instability: bifurcation buckling from a linear fundamental path, nonlinear analysis of imperfect tank, and dynamic response. Wind is modeled using several pressure distributions in the circumferential direction, and the results are compared with those due to axisymmetric pressure. The results show that for the present very thin-walled short tanks, bifurcation buckling produces good estimates of the critical state; however, the structure is imperfection sensitive and this load is reduced in the order of 30–50%.

Spatially complex localization after one-twist-per-wave equilibria in twisted circular rods with initial curvature

In experiments on long rubber rods subject to end tension and moment, a onetwistperwave deformation is often observed on the fundamental path prior to the onset of localized buckling. An analysis i...

The primary bending effect and the buckling boundary-value problem in elastic framed structures

This study deals with the stability analysis of elastic framed structures, free from imperfections, which are acted upon by concentrated joint loads. Attention is focused on the conditions under which primary bending is ruled out then the above analysis leads to a classical eigenvalue problem from which one can establish critical buckling loads and corresponding buckling modes. To this end by using simple frames as models the individual and coupling effect of various parameters (e.g. geometric imperfections, elastic supports, etc.) for which these structures may exhibit a trivial fundamental equilibrium path (from the onset of loading up to its critical value) are thoroughly discussed. It is also found that in case of non-existence of primary bending the components of the joint loads in the directions of the bars of the joints, coincide with the corresponding axial (internal) forces.

Qualitative criteria in non-linear dynamic buckling and stability of autonomous dissipative systems

A general qualitative approach for dynamic buckling and stability of autonomous dissipative structural systems is comprehensively presented. Attention is focused on systems which under the same statically applied loading exhibit a limit point instability or an unstable branching point instability with a non-linear fundamental path. Using the total energy equation, the theory of point and periodic attractors of the basin of attraction of a stable equilibrium point, of local and global bifurcations, of the inset and outset manifolds of a saddle and of the geometry of the channel of motion, the stability of the fundamental equilibrium path and the mechanism of dynamic buckling are thoroughly discussed. This allows us to establish useful qualitative criteria leading to exact, approximate and upper/lower bound buckling estimates without integrating the highly non-linear initial-value problem. The individual and coupling effect of geometric and material non-linearities of damping and mass distribution on the dynamic buckling load are also examined. A comparison of the results of the above qualitative analysis with those obtained via numerical simulation is performed on several two- and three-degree-of-freedom models of engineering importance.

NON-LINEAR DYNAMIC BUCKLING OF A SIMPLE MODEL VIA THE LIAPUNOV DIRECT METHOD

The non-linear dynamic response of a simple one-degree-of-freedom dissipative/non dissipative model under the more general case of partial follower loading is considered. The study is confined to imperfection sensitive systems which lose their static stability through a limit point. The analysis proceeds first by employing the inflection point criterion for dynamic buckling which is subsequently confirmed via the Liapunov direct method for global stability (instability). Attention is focused on determining the level of the dynamic buckling load above which the associated equilibria on the fundamental equilibrium path are globally unstable, although they are locally asymptotically stable.

Sensitivity of post-critical states to changes in design parameters

Sensitivity of post-buckling paths is studied in the context of the general theory of elastic stability of discrete structural systems. It is assumed that the sensitivity of the critical state itself has been computed and a formulation is developed to account for sensitivity of the curvature of the post-critical states when there are changes in design parameters. A linear fundamental path is considered. Explicit expressions are obtained for the sensitivities and they take the form of perturbation expansions. Only first order sensitivity of post-critical paths has been developed. A simple example of an angle section column with deformable cross-section illustrates that although a critical state may be insensitive to changes in certain design parameters, the post-critical response may be highly sensitive. In the first example presented (an axially loaded angle section column), the postbuckling response even changes from stable to unstable depending on the values of the design parameter considered.

FINITE ELEMENTS FOR THREE-MODE INTERACTION IN BUCKLING ANALYSIS

SUMMARY This paper presents a finite element formulation for the buckling mode interaction analysis of structures modelled as plate assemblies. The main assumptions are linear elasticity, a linear fundamental path, the existence of distinct critical states that.are coincident or near coincident, and a coupled path arising from a linear combination of modal displacements due to interaction. The formulation adopted is known as the W-formulation, in which the energy is written in terms of a sliding set ofincremental co-ordinates measured with respect to the fundamental path. The energy is then expressed with respect to a reduced modal co-ordinate basis and the coupled solution arising from interaction is computed. An example of a fibre­ reinforced composite I-beam subjected to axial compression illustrates the procedure. The results are compared to a simplified model developed by the authors. Also, an imperfection sensitivity analysis is performed.

Buckling Mode Interaction in Composite Plate Assemblies

The analysis of buckling mode interaction of fiber-reinforced composite columns, modeled as plate assemblies, is presented. The main assumptions are linear elasticity; a linear fundamental equilibrium path; the existence of critical states that are coincident or near coincident; and a coupled path rising from a quadratic combination of modal displacements due to interaction. The formulation adopted is known as the W-formulation, in which the energy is written in terms of a sliding set of incremental coordinates, measured with respect to the fundamental path. The energy is then expressed with respect to a reduced modal coordinate basis, and the coupled solution arising from interaction is computed. An example of a pultruded composite I-column subjected to axial compression illustrates the procedure.

Finite elements for post-buckling analysis. I—The W-formulation

This paper presents a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path, and the existence of distinct critical loads (i.e. no coupling between buckling modes occurs). The formulation developed is known as W-formulation, in which the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the presentation developed here, the only ingredients required to carry out the analysis are the strain-displacement and the constitutive matrices at the element level. The present formulation is compared with the so called V-formulation, in which the displacements refer to the unloaded state. It is shown that under the present assumptions of linear fundamental path, the advantages of the V-formulation are lost and both approaches are similar. An example of a circular plate under in-plane loading illustrates the procedures. Part II of this paper deals with the application to the post buckling analysis of plate assemblies made of composite materials.

Finite elements for post-buckling analysis. II—Application to composite plate assemblies

In a companion paper the authors presented a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path and the existence of distinct critical loads. The formulation developed is known as the W-formulation, where the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the present paper a number of applications of finite elements for post-buckling analysis on composite plate assemblies are presented. Thin-walled composite plates, I-beams, angle sections, and a specially designed box-beam with flanges (unicolumn) are studied in post-buckling when axially loaded. The results are in good agreement with previous studies. Moreover, a parametric study involving critical buckling load and geometry is presented for the case of the unicolumn.

Second‐order sensitivity analysis in vibration and buckling problems

AbstractA perturbation approach for the first‐ and second‐order sensitivity analysis of eigenvalue problems, as they arise in buckling and vibration of structural systems, is presented. A particular feature introduced in the formulation is the dependence of all matrices that model the problem not only on the design parameter, but also on a pre‐critical state. Thus, the sensitivity of the response of the fundamental path in buckling problems, or a pre‐stressed state, in vibration problems, are introduced in the analysis. Two techniques are considered in this work: the direct and the adjoint methods. First‐ and second‐order sensitivity equations for eigenvalues and eigenvectors are obtained, and it is shown that second‐order sensitivity of an eigenvalue can be accomplished by using only one adjoint problem.

Instability of semi-ellipsoidal shells

This paper deals with the analysis of axisymmetric elastic buckling of semi-ellipsoidal shells under external pressure. Reissner's non-linear equations for the axisymmetric deformations of shells of revolution are specialized here for ellipsoidal shells and used in this analysis. The geometric singularity of these equations at the apex of shells is avoided through further modifications by imposing the conditions of continuity of the shell parameters and other variables across the apex. A multisegment method of integration, as developed by Kalnins and Lestingi, has been used to solve the non-linear equations of ellipsoidal shells. The critical pressure of a shell is interpreted from the fact that the mode of primary deformation along the fundamental equilibrium path of a structure cannot change without a change in its status of stability. The appearance of a secondary deformation on the fundamental equilibrium path is always hinted by a substantial increase in the deformation rate with respect to the load parameter. Specifically, for shells at the critical equilibrium, any further increase in loading, however small, causes the shell enormous deformation indicating that the state of deformation of the shell which corresponds to the lowest potential energy is far from that at critical pressure. Extensive numerical results on the buckling of semi-ellipsoidal shells with completely restrained edges and for varying thickness ratios have been obtained. It is observed that the critical pressure increases with an increase in the ratio of minor to major axes of the ellipsoidal shell and decreases with an increase in its thickness ratio.

A QUALITATIVE ANALYSIS FOR THE LOCAL AND GLOBAL DYNAMIC BUCKLING AND STABILITY OF AUTONOMOUS DISCRETE SYSTEMS

A general analytic approach is presented for the local and global dynamic buckling and stability of discrete dissipative systems described by the most general form of autonomous ordinary differential equations. Attention is focused on nonlinear dynamical gradient systems which under static loading exhibit all types of simple branching points emanating from a trivial fundamental equilibrium path. The relationship between the static stability and the corresponding dynamic stability associated with the Jacobian eigenvalue is thoroughly discussed

"Rapid Learning" Techniques for Discrete Event Systems: Some Recent Results and Applications to Traffic Smoothing

We present a unified framework based on the fundamental sample path constructabilitY problem for Discrete Event Systems (DES): for a finite discrete parameter set Θ = {θ1,…., θm}. given a sample path under θ1, the problem is to simultaneously construct sample paths under all remaining parameter values. We review and compare two approaches for the solution of this problem. the Standard Clock (SC) approach and Augmented System Analysls (ASA). Next, we present a new area of potential use of the ASA approach: smoothing arbitrary, highly bursty, and possibly nonstationary traffic processes which are encountered in many applications (e.g., high-speed networks). We show how ASA can be used to (a) derive some basic structural properties of a simple smoothing scheme known as the "Leaky Bucket" mechanism under very general conditions. and (b) control a parameter of the scheme on line.

部分荷重を受ける球形シェルの非保存的安定性解析

The purpose of the-present study is to show theoretically that dynamic instability takes place in a clamped spherical shell under a partially distributed normal follower force. Large deflection and stability of shallow and deep spherical shells are studied. The axisymmetric equilibrium states are solved through a Newton-Raphson technique on discretized nonlinear shell equations by means of finite difference method. The stability of the respective equilibrium states on the axisymmetric fundamental equilibrium path is examined by the dynamic method. In this study, it is shown that transition from stability to instability takes place not only in the form of static instability, but also in the form of dynamic instability.