Conical shell edge disturbance - An engineer’s derivation

J. Blaauwendraad¹, J.H. Hoefakker²

¹ Emeritus-Professor in Structural Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands
² Former lecturer in Structural Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands

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, 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.

Key words: Rotationally symmetric shell, edge disturbance zone, engineer’s approach

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	Conical shell edge disturbance - An engineer’s derivation J. Blaauwendraad¹, J.H. Hoefakker² ¹ Emeritus-Professor in Structural Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands ² Former lecturer in Structural Mechanics, Faculty of Civil Engineering and Geosciences, Delft University of Technology, the Netherlands 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, 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. Key words: Rotationally symmetric shell, edge disturbance zone, engineer’s approach