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Finite element based solutions of thin-shell problems with a small strain

Phusakulkajorn, Wassamon

[Thesis]. Manchester, UK: The University of Manchester; 2013.

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Abstract

In this thesis, we consider the deformation of shell structures defined as thin three-dimensional elastic bodies. These can be modelled using a lower-dimensional theory but the governing partial differential equation of thin shells contains fourth-order derivatives which require C1-continuity in their solutions. Consequently, both the unknown and its first derivatives have to be continuous.Employing a finite element method in our study suggests that the C1-finite element representation of the shell solution has to be employed. Therefore, appropriate interpolation functions defined on a typical finite element are studied on both straight and curvilinear boundary domains. Our study of C1-finite element representations shows that the Bell triangular finite element which is derived from the quintic polynomials is more appropriate than the bi-cubic Hermite rectangular element as it converges faster and provides higher accuracy on domains with straight boundaries. However, when the physical boundary is curved, a straight-line approximation is not exact and the performance of the Bell triangular element decreases in terms of both accuracy and convergence rate. To retain a convergence rate and accuracy of the solution of a C1-problem on a curvilinear boundary domain, the C1-curved triangular finite element is introduced. It is proved to show superiority in both convergence rate and accuracy when solving the C1-problem on a curved boundary domain.Furthermore, numerical comparisons between the solutions obtained from the linear and nonlinear governing equations with the linear constitutive law are also reported here. These comparisons confirm that the solutions obtained from the linearised governing equation agree with those of the nonlinear when a loading is small and they start to disagree when the loading becomes larger.