Arnold D. Kerr

A study of a new foundation model

SummaryThe characteristics of a new foundation model, consisting of two spring layers interconnected by a shear layer, are studied. The study is conducted on the classical problem of a foundation subjected to a rigid stamp. In order to reduce the number of foundation constants to an absolutely necessary minimum, special attention is given to a possible dependence of the constants of the upper and lower spring layer, particularly to the spring constant ratio three which is suggested byReissners foundation model. A comparison of the obtained pressure distributions with relevant experimental data seem to support the adoption of this value, thus reducing the number of foundation constants to two. Advantages of the presented model over other foundation models are pointed out.ZusammenfassungDie Eigenschaften eines neuen Fundamentmodells, das aus zwei Federschichten besteht, die durch eine Schubschichte verbunden sind, werden studiert. Die Untersuchung folgt dem klassischen Problem des starren Stempels auf elastischer Unterlage. Um die Zahl der Fundamentkonstanten auf das absolut Notwendige zu reduzieren, wird einer möglichen Abhängigkeit zwischen den Konstanten der oberen und der unteren Federschichte besondere Aufmerksamkeit gewidmet, wobei vor allem das Konstantenverhältnis 3 betrachtet wird, das demReissnerschen Modell entspricht. Ein Vergleich der errechneten Druckverteilung mit Versuchsergebnissen scheint diesen Wert zu bestätigen, womit sich die Zahl der Konstanten auf 2 erniedrigt. Die Vorteile des vorgeschlagenen Modells werden hervorgehoben.

An extended Kantorovich method for the solution of eigenvalue problems

Abstract The paper presents an extended Kantorovich method for the solution of eigenvalue problems in partial differential equations. The specific examples treated are: the vibrations of a rectangular membrane and the stability of an elastic rectangular plate compressed in its plane. It is shown that for the membrane problems, the generated expressions for the eigenvalues and eigenfunctions are identical with the corresponding exact solution. For the clamped plate compressed uni-axially or bi-axially, problems which are not separable and for which no exact solutions are available, the generated eigenvalues, based on a one term expression for the eigenfunction, are shown to agree very closely with the relevant results found by other investigators. For plate problems which are separable, the method generates the exact eigenvalues and eigenfunctions. It was found that in all treated cases the final results are independent of the initial choice of the functions and that the iterative procedure converges very rapidly.

The continuously supported rail subjected to an axial force and a moving load

The recent practice of welding railroad rails to each other suggests that considerable axial compression forces may be induced in the rails because of a rise in temperature. This in turn may reduce the critical velocity for the track to the range of operational velocities of modern high-speed trains. The purpose of the paper is to demonstrate that this is indeed a possibility.

AN APPLICATION OF THE EXTENDED KANTOROVICH METHOD TO THE STRESS ANALYSIS OF A CLAMPED RECTANGULAR PLATE

SummaryThe extendedKantorovich method discussed recently, byA. D. Kerr, is used to analyze a clamped rectangular plate subjected to a uniform lateral load. It was found that the generated one term solution approximates very closely, throughout the plate region, not only the deflections but also the bending moments and shearing forces. It is shown that the final form of the solution is independent of the initial choice, and that the convergence of the iterative procedure is very rapid. Because of the lack of a closed form exact solution in the technical literature, the coefficients occurring in the obtained solution were evaluated for various plate side ratios and are presented in graphs in order to simplify the utilization of the obtained results in engineering practice.ZusammenfassungDie erweiterte methode vonKantorowitsch, die kürzlich vonA. D. Kerr diskutiert wurde wird benützt, um die eingespannte Rechteckplatte unter Gleichlast zu untersuchen. Es ergab sich, daß die erzielte eingliedrige Lösung nicht nur die Durchbiegung, sondern auch die Biegemomente und Querkräfte im Bereich der Platte sehr gut approximiert. Es wird gezeigt, daß die Endgestalt der Lösung unabhängig ist von der anfangs getroffenen Wahl und daß die Konvergenz des Iterationsverfahrens sehr rasch fortschreitet. Da in der technischen Literatur eine exakte Lösung in geschlossener Form fehlt, wurden die in der erhaltenen Lösung auftretenden Koeffizienten für verschiedene Seitenverhältnisse der Platte ausgerechnet und in Schaubildern dargestellt, um die Benützung der erhaltenen Resultate in der Ingenieurpraxis zu erleichtern.

The instability of a rectangular elastic solid

SummaryThe paper starts with a critical survey of recent investigations of related problems. The title problem shown in Fig. 2 is then solved whenq is constant directional and whenq is hydrostatic, using the complete formulation given byNovozhilov with the only assumption being that the elongations and shears are small compared to unity. Since the angles of rotation were retained in the formulation, and the obtained expressions did reduce for thin plates to the correspondingEuler-type expressions, it is concluded thatLegenyas suggestion, that the classic stability theory of plates and shells should be reconsidered taking into consideration the angles of rotation, is based on erroneous results. The results of the present paper confirm that whenq=p the solid can become unstable which contradicts the findings of other investigators. The paper concludes with a comparison of “buckling pressure versus slenderness ratio” curves which are based on results of a number of investigators. These graphs demonstrate that the classical theory of buckling of thin plates (and beams) yields accurate results up toh/L=0.15, thus verifying the validity of a usual assumption of structural mechanics.ZusammenfassungDie Arbeit beginnt mit einer kritischen Übersicht über die neueren Untersuchungen verwandter Probleme. Das gestellte Problem, in Fig. 2 erläutert, wird im folgenden für richtungstreue und für hydrostatische Querlastq gelöst, wobei die vonNovozhilov angegebene Formulierung benützt wird und die Dehnungen sowie die Schubverformungen klein gegen eins angenommen werden. Da die Drehwinkel in der Rechnung behalten werden und sich die erhaltenen Ausdrücke für dünne Platten auf entsprechendeEulersche zurückführen lassen, wird gefolgert, daß derLegenyasche Vorschlag, die klassische Stabilitätstheorie sollte durch Hineinnahme des Drehwinkeleinflusses ergänzt werden, auf fehlerhaften Ergebnissen beruht. Das hier gefundene Ergebnis, daß der Quader fürq=p instabil werden kann widerspricht den Ergebnissen anderer Autoren. Die Arbeit schließt mit einem Vergleich von Knicklast-Schlankheitsgrad-Kurven, die aus den Resultaten verschiedener Autoren gewonnen wurden. Die Schaubilder zeigen, daß die klassische Beul-(Knick)-Theorie dünner Platten (und Stäbe) genaue Ergebnisse bis zum Schlankheitsgradh/L=0,15 liefert und bestätigen so die Gültigkeit einer in der Baumechanik üblichen Annahme.

The stress analysis of circular plates sealing a compressible liquid.

Abstract The stress analysis of plates sealing an incompressible liquid was treated recently in a number of publications. The present paper generalizes the results presented previously by assuming that the sealed liquid is compressible. The problem is demonstrated on a clamped circular plate subjected at an arbitrary point to a lateral concentrated force P . The solution is obtained in closed form. Graphs are presented which show the effect of the compressibility of the sealed liquid upon the pressure in the sealed liquid as well as upon the plate deflections, bending moments and shearing forces. Using as an illustrative example a steel plate which seals oil, the effect of the sealed volume upon the plate behavior is shown.

The stability of core-filled long cylinders subjected to uniform outside pressure

Abstract The title problem is solved for the cases where pressure is (a) hydrostatic, (b) centrally directed and (c) directionally constant. It is assumed that the core responds like a Pasternak foundation. Assuming that the instability phenomena of the title problem are of the non-snapping bifurcation type, the buckling pressures are obtained by means of the perturbation technique. The problems are formulated going out from the geometrically non-linear equations of elasticity using a variational approach. It was found that each of the obtained expressions for the buckling pressures consists of three terms. The first term represents the classical solution of a long empty cylindrical shell, the second term the effect of the spring layer (Winkler foundation) and the third term the effect of the shear interactions. The paper concludes with remarks about the determination of the foundation constants.

On the stability of a shallow circular arch

SummaryThe equilibrium configurations of a shallow circular arch were determined in the past by a number of investigators. Although recently it was proven that some of the determined equilibrium branches are not stable, these proofs do not imply that the remaining parts of these branches correspond to stable equilibrium states. The purpose of the present paper is to present a necessary and sufficient proof of stability for these equilibrium branches.ZusammenfassungDie Gleichgewichtslagen eines flachen Kreisbogenträgers wurden von einer Reihe von Forschern bestimmt. Kürzlich wurde gezeigt, daß einige dieser Gleichgewichtslagen nicht stabil sind; aber diese Beweise besagen nicht, daß die übrigen Gleichgewichtslagen stabil sind. Zweck dieser Arbeit ist Angabe eines notwendigen und hinreichenden Beweises der Stabilität dieser Gleichgewichtslagen.