Optimal design of elastic columns subject to the general conservative behaviour of loading

Zusammenfassung Optimale Formen eines gedrückten elastischen Stabes werden bei allgemeinem Verhalten der Druckkraft nach dem Stabilitätsverlust untersucht. Als Zielfunktion wird das minimale Volumen angenommen unter der Nebenbedingung einer konstanten kritischen Belastung. Das konservative Verhalten der Kraft ist mit einer Beziehung verbunden, welche die allgemeine Euler-Lagrangesche Gleichung vereinfacht. Es werden räumlich sowie eben verjüngte Stäbe untersucht und die Resultate bei einigen Beispielen diskutiert.     

Optimum design methods for multiple loading

Zusammenfassung Es wird gezeigt, dass die früher für die optimale Bemessung von gewöhnlichen Tragwerken entwickelten Verfahren auf Mehrzweck-Tragwerke erweitert werden können, welche unabhängig voneinander zwei oder mehr Lastsysteme aufzunehmen haben. Als Beispiel wird die Bemessung auf minimales Volumen einer kreisförmigen Sandwich-Platte behandelt.

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The results presented in this paper were obtained in the course of research sponsored by the Office of Ordnance Research, Department of the Army under Contract DA-ARO-(D)-31-124-G2.     

Design sensitivity for arch structures with respect to midsurface shape under static loading

We consider in this paper an example of structural optimization in which the structure is a loaded arch and the design variable is the shape of the arch. We concentrate on differentiability of static response with respect to shape changes. After recalling the arch equation with its functional spaces and the optimization problem, we state a differentiability theorem and provide a detailed proof. Numerical use of this result is finally discussed.     

A general theory for the equilibrium and stability of discrete conservative systems

Zusammenfassung Eine vom Verfasser kürzlich entwickelte allgemeine Theorie der elastischen Stabilität wird in einheitlicher und moderner Form dargestellt. Die Theorie behandelt Gleichgewicht und Stabilität diskreter kontinuierlicher Systeme und ist somit ein Beitrag zur klassischen Mechanik, mit Anwendungen in der numerischen Untersuchung von in der Praxis auftretenden Körpern und Tragwerken.Normales und kritisches Verhalten eines allgemeinen Systems werden durch einen natürlichen Störungsansatz klassifiziert und berechnet; beides mit oder ohne Diagonalisierungsschema. Das anfängliche überkritische Verhalten eines Systems bei einem diskreten kritischen Punkt wird ausführlich untersucht; besondere Aufmerksamkeit wird der wichtigen Frage der Imperfektionenanfälligkeit gewidmet.Einige neue Theoreme werden aufgestellt.     

Effect of the end conditions on the critical loads for filament-wound glass-reinforced plastic columns

The behavior of glass-reinforced plastic columns with various end conditions has been experimentally investigated. The cross-wound columns had a length to mean diameter ratio of 18–20. Four types of support with different degrees of restraint on end rotation ranging from almost free ends to fixed ends were employed. The possibilities of increasing the critical loads by manipulating the boundary conditions are estimated. Numerical estimates are obtained for the various methods of support. Random imperfections and the number of loadings were not found to have any effect on the critical loads. It is shown that the use of Southwell's method makes it possible to estimate the critical loads for columns with different end conditions from the subcritical loading data.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Moscow. Translated from Mekhanika Polimerov, No. 1, pp. 54–62, January–February, 1976.     

Hearing the shape of a general doubly-connected domain inR 3 with mixed boundary conditions

The basic problem in this paper is that of determining the geometry of an arbitrary doubly-connected region inR ³ with mixed boundary conditions, from the complete knowledge of the eigenvalues { _n } _n=1 for the three-dimensional Laplacian, using the asymptotic expansion of the spectral function ast0.     

A general mixed covolume framework for constructing conservative schemes for elliptic problems

We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.

     

An analytical formula for an elastic thin rod shape under composite loading

The Euler problem on the stability of a thin elastic rod under compressing forces is widely known. In a number of papers this problem is generalized to finding the shape of such a rod under the simultaneous action of a compressing force and a torsional moment. The shape is determined by solving a complex nonlinear boundary value problem by numerical methods. In this paper, an approach providing a full analytical solution of the problem for a broad range of external conditions is discussed. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 7, pp. 35–42, 2005.     

Designing a neutral elliptic inhomogeneity in the case of a general non-uniform loading

We derive a general expression for an interface parameter which makes possible the design of a neutral elliptic inhomogeneity when the stress field in the surrounding matrix is a polynomial function of $n\mathrm{th}$ order and the composite is subjected to antiplane shear deformations.     

Amplification of surface ground motion by an inclusion of arbitrary shape for harmonic surface line loading

The plane strain model for the Lamb's problem with an elastic inclusion of arbitrary shape embedded completely within an elastic half space is investigated by using an indirect boundary integral equation method for steady-state elastodynamics. The surface of the half space is subjected to vertical or horizontal harmonic line loads. The displacement field is evaluated throughout the elastic medium so that the continuity of the displacement and traction fields along the interface between the half space and the inclusion is satisfied in a least-square sense. The numerical results demonstrate that the presence of the inclusion may cause locally very large amplification of the surface ground motion and that the amplification pattern depends upon the frequency and the type of the input load, the impedance contrast between the half space and the inclusion, the type of the inclusion, and the location of the observation point at the surface of the half space.     

On instability of conservative systems with gyroscopic forces

Theorems on equilibrium instability of conservative systems with gyroscopic forces are proved. The theorems obtained are nonlinear analogs of the Kelvin theorem. The equilibrium instability of the Chaplygin nonholonomic systems is considered. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 10, pp. 1422–1428, October, 1997.     

Periodic motions of conservative systems with singular potentials

The existence of closed orbits with prescribed energy for second order Hamiltonian Systems with singular potentials is studied.The first author was supported by M.U.R.S.T.