A Sequel to Technical Note 15: The SHEBA Family of Shell Elements for the Matrix Displacement Method

Extract

The large displacement analysis of shells of arbitrary form is rightly considered to be a formidable undertaking. A purely analytical approach to the solution of such problems involving also initial buckling, post-buckling and secondary instability or snap-through is at the present state of knowledge impossible. However, with the advent of the matrix displacement method initiated in 1954 it became increasingly apparent that this method would prove in time the ideal tool for the solution of such complex structural phenomena. Nevertheless, all efforts in this direction were bedevilled until recently by the lack of suitable elements, satisfying all kinematic compatibility conditions. The major breakthrough came in this direction through the invention of the triangular shell element SHEBA, which admits an arbitrary variation of curvature. This again evolved from a natural extension of the plate element TUBA of TN 14. At the same time the SHEBA theory of ref. 1 was restricted to small displacements or linear behaviour. To generalise it to large displacements was not possible without an additional major conceptual progress. This was achieved within the Technical Notes 17 to 20, which introduced the idea of a local sub-element and showed how its geometrical stiffness could be used to deduce the geometrical stiffness of the complete element by a physically evident argument. The attentive reader will have noticed that the sub-element of an arbitrary beam in space discussed in ref. 3 is the obvious component block for our shell element. As a matter of fact, the technique is simply based on the representation of the SHEBA element by a grid of beam sub-elements running along the natural α, β, γ directions.