http://link.springer.com/content/pdf/10.1007/978-3-642-69569-8_2.pdf

Theory of Elastic Stability

An extended Kantorovich method for the solution of eigenvalue problems

The 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.

An application of the extended Kantorovich method to the stress analysis of a clamped rectangular plate

Computation of elastic buckling loads of rectangular thin plates using the extended Kantorovich method

Vibration of curved plates

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An extension of the Kantorovich method

Using the cxact solutiom of a shallow elastic arch clamped at both ends and subjected to a uniform lateral load, it is shown that, in addition to the equilibrium states usually discussed in the literature, also other equilibrium states, which correspond to higher modes of deformation, do exist. It is shown that there exist only a finite number of equilibrium branches which correspond to these higher modes, the number depending upon the shallowness of the arch. A number of graphs are presented which demonstrate these findings. The paper concludes with the proof that all states of equilibrium which correspond to the branches of higher order are unstable. Introduction. The elastic shallow arch or infinitely long shallow panel, clamped or simply supported, and subjected to a uniform lateral load as shown in Fig. 1, was analyzed in the past by a number of investigators [1]—[8]. Whereas in references [1], [3], [5] and [6] approximate methods were used to solve the resulting nonlinear formulations, in references [2], [4], [7] and [8] the resulting two nonlinear differentia] equations were reduced to a linear one, which is amenable to an exact analysis. Recently, during the preparation of reference [8], it became apparent that the obtained exact solution contains, in addition to the usual equilibrium states shown in Fig. 2, also other equilibrium branches. A similar finding, in connection with the buckling of spherical shells, was reported recently in [9]. The purpose of the present paper is to determine, by means of the exact solution derived in [8], the additional equilibrium branches and the corresponding stresses and deformations of the shallow clamped arch, and then discuss their stability. The equilibrium branches. With the usual assumptions for thin shallow arches and the notation of Fig. 1, the principle of stationary total potential energy yields the two simultaneous nonlinear differential equations [8, 10]:

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On the nonunique equilibrium states of a shallow arch subjected to a uniform lateral load

: Using the exact solution of a shallow elastic arch clamped at both ends and subjected to a uniform lateral load, it is shown that, in addition to the equilibrium states usually discussed in the literature, also other equilibrium states, which correspond to higher modes of deformation, do exist. It is shown that there exist only a finite number of higher modes, the number depending upon the shallowness of the arch. A number of graphs are presented which demonstrate these findings. The paper concludes with a discussion of the stability of the determined equilibrium branches. The stability of the main branch is discussed first. It is then shown that all states of equilibrium which correspond to the branches of higher order, are unstable. (Author)

Arnold D. Kerr

Papers

An extension of the Kantorovich method

On the nonunique equilibrium states of a shallow arch subjected to a uniform lateral load

Equilibrium and stability of a shallow arch subjected to a uniform lateral load.