Gilbert N. Lewis

Large deflections of cantilever beams of non-linear materials of the Ludwick type subjected to an end moment

Abstract This paper deals with the large deflections (finite) of thin cantilever beams of non-linear materials of the Ludwick type. The beam is subjected to an end constant moment. Large deflections of beams induce geometrical non-linearity. Therefore, in formulating the analysis, the exact expression of the curvature is used in the Euler-Bernoulli law. A closed-form solution is presented for the resulting second-order non-linear differential equation. This solution is compared to previous results assuming linear elastic materials. Deflections at the free end of beams of aluminum alloy and annealed copper are obtained.

Large deflections of cantilever beams of nonlinear materials

Abstract This paper deals with the large deflections (finite) of thin cantilever beams of nonlinear materials, subjected to a concentrated load at the free end. The stress-strain relationships of the materials are represented by the Ludwick relation. Because of the large deflections, geometrical nonlinearity arises and, therefore, the analysis is formulated according to the nonlinear bending theory. Consequently, the exact expression of the curvature is used in the moment-curvature relationship. The resulting second-order nonlinear differential equation is solved numerically using fourth-order Runge-Kutta method. For comparison purposes, the differential equation is solved for linear material and the results are compared to the exact solution which uses elliptic integrals. Deflections and rotations along the central axis of beams of nonlinear materials are obtained. The numerical algorithm was performed on the UNIVAC 1110.

Large deflections of point loaded cantilevers with nonlinear behaviour

SummaryThin beams, being flexible, form a curve with large deflections when subjected to sufficiently large transverse loads. Therefore, geometrical nonlinearity occurs, and the problem must be formulated in terms of the nonlinear theory of bending. In this paper, the beam is constructed from nonlinear elastic material, and subjected to several transverse concentrated loads. Due to the large deflection of the beam, the exact expression of the curvature of the deflected shape is used in the Bernoulli-Euler relationship. Therefore, this leads to a second order nonlinear differential equation for the transverse deflection. The solution of this equation is obtained by using the fourth-order Runge-Kutta method, and the arc length is evaluated using Simpsons Rule. The results obtained from this procedure are compared with previously published results for thin beams of linear elastic materials in order to verify the theory and the method of analysis.ZusammenfassungDünne, flexible Träger formen eine stark gekrümmte Kurve, wenn sie genügend großen Querkräften ausgesetzt sind. Deshalb tritt geometrische Nichtlinearität auf, und das Problem muß mit Hilfe nichtlinearer Biegetheorie formuliert werden. Dieser Aufsatz handelt von einem Träger aus einem nichtlinearen, elastischen Material, der mehreren konzentrierten Querkräften ausgesetzt ist. Wegen der großen Durchbiegung des Trägers wird der exakte Ausdruck für dessen Krümmung in dem Euler-Bernoullischen Gesetz benutzt. Dies führt daher zu einer nicht-linearen Differentialgleichung zweiter Ordnung für die Durchbiegung. Die Lösung dieser Gleichung erhält man, indem man die Runge-Kutta Methode vierter Ordnung benutzt; die Bogenlänge wird mit Simpsons Regel bestimmt. Die auf diese Weise erhaltenen Resultate werden mit bereits veröffentlichten Resultaten für dünne Träger aus linearen, elastischen Materialien verglichen, um die Theorie und Berechnungsmethode zu prüfen.

A variational approach for the deflections and stability behavior of postbuckled elastic-plastic slender struts

Abstract In this paper, we present the large deflection behavior of a postbuckled vertical slender strut made of elastic-plastic material and subjected to uniformly distributed loads, w , including their own weight. Such struts will buckle at the critical buckling load. w cr . As the load is increased beyond w cr , the deflections of the strut also increase. Eventually the strut may undergo a second instability, where a small increase in the load will cause a large jump in the deflections, and the strut will move to another position of equilibrium. The problem is formulated in terms of a variational approach, to which the Ritz method of solution is applied. In addition, we use determinants to develop an inequality for stability behavior of the strut in the postbuckled region.

A Contact Problem for a Convection-diffusion Equation

We have proposed an efficient iterative domain decomposition method that solves general convection-diffusion singular perturbation problems. Our specific application involves a piecewise constant diffusion coefficient. We have established sufficient conditions for the convergence of the method, identified suitable values of the relaxation parameter ϑ, and started investigations into the rate of convergence. The method was implemented to O(e2) to solve test problems.

A mathematical model for decompression

Abstract A mathematical model for decompression is presented. It includes a first-order linear differential equation, which models the diffusion of gases within the body, as well as initial conditions and a nonlinear constraint inequality which insures safety from decompression sickness. The model is solved (numerically and/or analytically) under various hypotheses, and the results are compared with the U.S. Navy Dive Table.