On the Solution of Beam-on-elastic-foundation Problems by Means of a Mechanical Analogue

Abstract

The equation d⁴y/dx⁴-f(x)y + g(x) = 0 may be solved by means of the differential analyser, but only straightforwardly when the four boundary conditions are specified at one point. When the equation is associated with beams on elastic foundations, or elastic shells, the boundary conditions are more often specified at two points, and a quicker method of solution is desirable. In the analogue, direct use is made of the beam in the form of an elastic wire, supported at intervals in cradles on which weights may be made to simulate the terms f(x)y and g(x); the wire takes up a transversely deflected form which may be measured, and boundary conditions are imposed where they are required.

A specific problem is examined and the results are shown to agree reasonably with the solution by calculation. A disadvantage when d²y/dx² is required is the inaccuracy inherent in differentiating by finite differences, but for engineering calculations the simplicity of the method may have its advantages.

The solution of a typical pressure-vessel problem, by means of the analogue, is described.

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References

Coates

W. M.

1930 Trans. A. S. M. E., vol. 52, p. 117, “The State of Stress in Full Heads of Pressure Vessels”.

Hetényi

1946 “Beams on Elastic Foundation” (University of Michigan Press and Oxford University Press).

Timoshenko

1940 “Theory of Plates and Shells” (McGraw-Hill Publishing Company, New York and London).

1941 “Strength of Materials”, Part 2, p. 2 (Van Nostrand, New York).