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Equilibrium Path (equilibrium + path)
Selected AbstractsAnalysis of single rock blocks for general failure modes under conservative and non-conservative forcesINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 14 2007F. Tonon Abstract After describing the kinematics of a generic rigid block subjected to large rotations and displacements, the Udwadia's General Principle of Mechanics is applied to the dynamics of a rigid block with frictional constraints to show that the reaction forces and moments are indeterminate. Thus, the paper presents an incremental-iterative algorithm for analysing general failure modes of rock blocks subject to generic forces, including non-conservative forces such as water forces. Consistent stiffness matrices have been developed that fully exploit the quadratic convergence of the adopted Newton,Raphson iterative scheme. The algorithm takes into account large block displacements and rotations, which together with non-conservative forces make the stiffness matrix non-symmetric. Also included in the algorithm are in situ stress and fracture dilatancy, which introduces non-symmetric rank-one modifications to the stiffness matrix. Progressive failure is captured by the algorithm, which has proven capable of detecting numerically challenging failure modes, such as rotations about only one point. Failure modes may originate from a limit point or from dynamic instability (divergence or flutter); equilibrium paths emanating from bifurcation points are followed by the algorithm. The algorithm identifies both static and dynamic failure modes. The calculation of the factor of safety comes with no overhead. Examples show the equilibrium path of a rock block that undergoes slumping failure must first pass through a bifurcation point, unless the block is laterally constrained. Rock blocks subjected to water forces (or other non-conservative forces) may undergo flutter failure before reaching a limit point. Copyright © 2007 John Wiley & Sons, Ltd. [source] Displacement-controlled method and its applications to material non-linearityINTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 3 2005H. Zheng Abstract For the analysis of non-linear problems, the displacement-controlled method (DCM) has a more extensive application scope and more powerful abilities than the load-controlled method (LCM). However, difficulties of the DCM's procedure not amenable to most finite element implementations of the conventional LCM have restricted its applications in geomechanics. By means of Sherman,Morrison's theorem, the solution of DCM is improved. The improved procedure is characterized by high efficiency, good numerical stability and a programme structure similar to LCM. Two aspects of applications of DCM are illustrated. The first application is to compute the response of a structure under a given load level like the conventional finite element analysis. The second application is to trace the equilibrium path of a structure under a given load distribution type. A simple but effective algorithm is presented for automatically adjusting the step length in tracing the equilibrium path. Examples illustrate that the proposed procedures are suited for modelling complicated non-linear problems in geomechanics. Copyright © 2005 John Wiley & Sons, Ltd. [source] Heterogeneous transport costs and spatial sorting in a model of New Economic Geography,PAPERS IN REGIONAL SCIENCE, Issue 1 2010Corey Lang Economic geography; transport costs; sorting; agglomeration Abstract Transportation costs are of central importance in the New Economic Geography literature, though assumptions about transportation costs continue to be simplistic. This paper begins to address these simplifications by assuming that transportation costs for manufactured goods are heterogeneous. Basic results are consistent with standard models showing dispersion of economic activity for high transport costs and eventual agglomeration as transport costs decline. However, several novel features arise too. Many unstable, dispersed equilibria exist for high average transport costs, but converge to a stable equilibrium path as transport costs decrease. Equilibrium paths smoothly transition from dispersion to agglomeration and do so at an increasing rate. Additionally, transport costs directly influence firms' location decisions and firms spatially sort by transport cost. Resumen Los costos de transporte tienen una importancia capital en la literatura sobre la Nueva Geografía Económica, aunque se siguen asumiendo los costos de transporte de manera simplista. Este artículo empieza a tratar estas simplificaciones, asumiendo que los costos de transporte de bienes manufacturados son heterogéneos. Los resultados básicos son consecuentes con los modelos estándar que muestran la dispersión de la actividad económica para costos de transporte elevados y una eventual aglomeración a medida que los costos de transporte disminuyen. Sin embargo, también aparecen varias características novedosas. Existen muchos equilibrios dispersos, inestables, para costos de transporte promedio elevados, pero convergen en una línea de equilibrio a medida que los costos disminuyen. Las líneas de equilibrio tienen una transición fluida de la dispersión a la aglomeración y lo hacen a una tasa en aumento. Además, los costos de transporte influyen directamente las decisiones de localización de las empresas y estas se ordenan espacialmente de acuerdo a los costos de transporte. [source] Imperfection Sensitivity or Insensitivity of Zero-stiffness Postbuckling , that is the QuestionPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009Xin Jia Zero-stiffness postbuckling of a structure is characterized by a secondary load-displacement path along which the load remains constant. In sensitivity analysis of the (initial) postbuckling path it is usually considered as a borderline case between imperfection sensitivity and imperfection insensitivity. However, it is unclear whether zero-stiffness postbuckling as such is imperfection sensitive or insensitive. In this paper, Koiter's initial postbuckling analysis is used as a tool for sensitivity analysis. Distinction between two kinds of imperfections is made on the basis of the behavior of the equilibrium path of the imperfect structure. New definitions of imperfection insensitivity of the postbuckling behavior are provided according to the classification of imperfections. A structure with two degrees of freedom with a zero-stiffness postbuckling path is studied, considering four different imperfections. The results from this example show that zero-stiffness postbuckling is a case of transition from imperfection sensitivity to imperfection insensitivity for imperfections of the first kind and that it is imperfection insensitive for imperfections of the second kind. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] Minorities and majorities: a dynamic model of assimilationCANADIAN JOURNAL OF ECONOMICS, Issue 4 2005István Kónya Depending on the minority's size, the long-run outcome can be full or no assimilation. Under certain parameter values multiple equilibria exist, including the two extreme cases. The paper demonstrates that both the long-run outcome and the equilibrium path may be inefficient. Two extensions to the basic model are considered. The first one allows for a comparison between a multicultural and a ,melting pot' society. The second one introduces population growth and studies the interplay between exogenous and endogenous changes in the minority's size. JEL classification: J110, J150, J180 Minorités et majorités : un modèle dynamique d'assimilation., Ce mémoire analyse la dynamique de la population d'un pays composé de deux groupes ethniques , une majorité et une minorité, où il y a possibilité pour les membres de la minorité de s'assimiler à la majorité. Selon la taille de la minorité, le résultat à long terme peut être une assimilation complète ou pas d'assimilation du tout. Pour certaines valeurs des paramètres, des équilibres multiples existent incluant les deux cas limites. On montre que à la fois le résultat à long terme et le sentier d'équilibre qui y mène peuvent ne pas être efficaces. On considère deux extensions du modèle: la première permet une comparaison entre une société multiculturelle et une société de «melting pot»; la seconde injecte un élément de croissance démographique et étudie l'interaction entre les changements exogènes et endogènes dans la taille de la minorité. [source] | ||||||||||