The curved beam/deep arch/finite ring element revisited

An attempt is made to understand the errors arising in curved finite elements which undergo both flexural and membrane deformation. It is shown that with elements of finite size (i. e. a practical level of discretization at which reasonably accurate results can be expected), there can be errors of a special nature that arise because the membrane strain fields are not consistently interpolated with terms from the two independent field functions that characterize such a problem. These lead to errors, described here as of the 'second kind' and a physical phenomenon called 'membrane locking'. It seems possible to determine optimal integration rules that will allow the extensional deformation of a curved beam/deep arch/finite ring element to be modelled by independently chosen low order polynomial functions and which will recover the inextensional case in the penalty limit of extreme thinness without spurious locking constraints. What is emphasized is that the choice of shape functions, or subsequent operations to determine the discretized functionals, must consistently model the physical requirements the problem imposes on the field variables.

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