On the Nonlinear Strain-Curvature-Displacement Relationships for Thin Elastic Shells

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T N RECENT YEARS a large number of investigations have been made in problem areas involving the nonlinear, large-deflection theory for the static deflection, vibration, and buckling of shells. These investigations require the use of the nonlinear relationships between the strains and curvatures and the displacements. For example, a very good approach to these problems is by means of the Rayleigh-Ritz technique. Using this technique one can readily avoid the solution of the difficult nonlinear differential equations by choosing combinations of deflection functions with undetermined coefficients which satisfy the boundary conditions exactly. The coefficients are then determined so as to make the total energy of the elastic system a minimum, thereby giving the best approximation—e.g., deflection, vibration frequency, buckling load—to the exact solution as is obtainable with the functions chosen. The strain energy stored in a shell is given by. 2