JH
JH Hoefakker
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Conical shell edge disturbance
An engineer's derivation
Because a rigorous bending theory for thin shells of revolution is complicated, attempts have been made for reliable approximations of the edge disturbance problem under axisymmetric loading. A well-known one was published by Geckeler [1, 2], who obtained his approximation by mathematical considerations. He started from kinematic, constitutive and equilibrium equations for the rotationally symmetric thin shell without approximations. Herein he introduced mathematical simplifications. Each time when derivatives of a function of different orders appeared, he just kept the highest order derivative and neglected all lower ones. This is permitted if the function varies rapidly, as is the case for edge disturbances. Here we will present Geckeler's result in an alternate way, which illustrates the physical background of his mathematical approximation. Said in another way, we offer a derivation in the language of structural engineers.
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Because a rigorous bending theory for thin shells of revolution is complicated, attempts have been made for reliable approximations of the edge disturbance problem under axisymmetric loading. A well-known one was published by Geckeler [1, 2], who obtained his approximation by mathematical considerations. He started from kinematic, constitutive and equilibrium equations for the rotationally symmetric thin shell without approximations. Herein he introduced mathematical simplifications. Each time when derivatives of a function of different orders appeared, he just kept the highest order derivative and neglected all lower ones. This is permitted if the function varies rapidly, as is the case for edge disturbances. Here we will present Geckeler's result in an alternate way, which illustrates the physical background of his mathematical approximation. Said in another way, we offer a derivation in the language of structural engineers.
Donnell published in 1933 an elegant and rather simple differential equation for the stress analysis in circular cylindrical shells. The civil engineering structural mechanics group in Delft has a long and strong experience in the fifties and sixties of last century in working with this shell equation in close cooperation with the then building institute of TNO. Shallow roof shells, tanks and chimneys were examined. It is known that the equation has restricted reliability for shells in which wavelengths in circumferential direction are relatively large. In 1959 Morley replaced Donnell's equation by a slightly extended equation which retained the essential simplicity of the original but the accuracy does not decrease as the wave length of circumferential distortion increases. The impact on longitudinal bending stresses at the base of chimneys due to wind is not negligible as appears from here summarized research. Application of Morley's theory is easily extended to ring-stiffened and elastically supported chimneys. This note demonstrates how subsequent research in time of individual persons provides stepping stones in the process of discovering.
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Donnell published in 1933 an elegant and rather simple differential equation for the stress analysis in circular cylindrical shells. The civil engineering structural mechanics group in Delft has a long and strong experience in the fifties and sixties of last century in working with this shell equation in close cooperation with the then building institute of TNO. Shallow roof shells, tanks and chimneys were examined. It is known that the equation has restricted reliability for shells in which wavelengths in circumferential direction are relatively large. In 1959 Morley replaced Donnell's equation by a slightly extended equation which retained the essential simplicity of the original but the accuracy does not decrease as the wave length of circumferential distortion increases. The impact on longitudinal bending stresses at the base of chimneys due to wind is not negligible as appears from here summarized research. Application of Morley's theory is easily extended to ring-stiffened and elastically supported chimneys. This note demonstrates how subsequent research in time of individual persons provides stepping stones in the process of discovering.