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**1 - 3**of**3**### ABSTRACT Title of Thesis: Analysis and Evaluation of a Shell Finite Element with Drilling Degree of Freedom Name of degree candidate:

"... A at shell nite element is obtained by superposing plate bending and membrane components. Normally, shellelements of this type possess ve degrees of freedom (DOF), three displacement DOF, u, v and w, andtwo in-plane rotation DOF, x and y, ateachnode.A sixth degree of freedom, z, is associated with t ..."

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A at shell nite element is obtained by superposing plate bending and membrane components. Normally, shellelements of this type possess ve degrees of freedom (DOF), three displacement DOF, u, v and w, andtwo in-plane rotation DOF, x and y, ateachnode.A sixth degree of freedom, z, is associated with the shell normal rotation, and is not usually required by the theory. In practice, however, computational and modeling problems can be caused by a failure to include this degree of freedom in nite element models. This paper presents the formulation and testing of a four node quadrilateralthin at shell nite element, which has six DOF per node. The sixth DOF is obtained by combining by a membrane element with a normal rotation z, the so-called the drilling degree of freedom, and a discrete Kirchho plate element. The at shell has a 24 24 element sti ness matrix. Numerical examples are given for (a) shear-loaded cantilever beam, (b) square plate, (c) cantilever I-beam and (d) folded plate. Performance of the at shell nite element is also compared to a four node at shell element in ANSYS-5.0 in case studies (a)-(d), and a quadrilateral at shell element from SAP-90 in case study (c).DEDICATION For my parents and Aoyasuly. ii ACKNOWLEDGMENTS I would like to sincerely thank my advisor, Professor Mark A. Austin, for his vision, guidance and patience during this project. This work would not have been possible without the assistance of Professor R. L. Taylor, University of California at Berkeley, who provided help and encouragement to pursue this project. My appreciations go also to Professor Peter Chang, who reviewed the thesis carefully and o ered important comments. In addition, I wish to take this opportunity to thank all of the students, faculty and sta

### Simulation of Elastic Membranes

, 1997

"... Spring meshes have been used to model elastic material by numerous researchers, with skin, textiles, and soft tissue being typical applications. However, given a speci ed set of elastic material properties, the question of whether a particular spring mesh accurately simulates those properties, has b ..."

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Spring meshes have been used to model elastic material by numerous researchers, with skin, textiles, and soft tissue being typical applications. However, given a speci ed set of elastic material properties, the question of whether a particular spring mesh accurately simulates those properties, has been largely ignored in the literature. In two dimensions, given a discretization of a membrane as a triangle mesh, the standard nite element method analyzes each triangle approximately as a membrane with speci ed elastic properties, computing stresses and strains. An alternative is to regard each edge as a spring, assuming the springs are connected by \pin-joints &quot; at the vertices of the discretization. This alternative, called a \spring mesh&quot;, is computationally more attractive. Previous reports on the technique are silent on the subject of assigning sti ness to the various springs. This paper shows that assigning the same sti ness to all springs badly fails to simulate a uniform elastic membrane, for equilibrium calculations. A formula for spring sti ness that provides a more accurate simulation is then derived. Its accuracy is demonstrated on test and practical mesh examples. It is also shown that an exact simulation is not possible, in general.

### MATHEMATICAL AND HISTORICAL REFLECTIONS ON THE LOWEST-ORDER FINITE ELEMENT MODELS FOR THIN STRUCTURES

"... Juhani PitkÄaranta: Mathematical and historical re°ections on the lowest-order ¯nite ..."

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Juhani PitkÄaranta: Mathematical and historical re°ections on the lowest-order ¯nite