Dhananjaya, H R
(2003)
*A family of plate bending finite elements using Integrated Force Method.*
Phd. Thesis thesis, Indian Institute of Science, Banglore.

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## Abstract

The Integrated Force Method(IFM) is a novel matrix formulation developed in recent years for analyzing civil, mechanical and aerospace structures. The IFM is a modified classical force method for systematic computerization and is independent of redundant forces and basis determinate structures. In this method all internal forces are treated as unknown variables which are calculated by simultaneously imposing equations of equilibrium and compatibility conditions. IFM has yielded superior results in many situations as compared to conventional finite element method. The present research work is focussed on the development of plate bending finite elements for analyzing thin, moderately thick and laminated composite plates using the Integrated Force M Method. Both the Kirchhoff plate theory and the Mindlin-Reissner theory form the basis for element formulation. A family of elements developed here include two elements based 011 on the Kirchhoff plate theory, four elements based 011 the Mindlin-Reissner plate theory for for analyziug thin plate and thin/moderately thick plate problems respectively. one more elements based on the Mindlin-Reissner plate theory have been developed for analyzing laminated composite plate problems using the IFM. For each element, eigen value test is carried out to check for the spurious energy modes. All these elements are free from spurious energy modes except the four-node laminated composite element which is having two extra energy modes. Various types of patch tests like constant curvature and shear tests are conducted. The performances of these elements are examined by solving several benchmark problems using the IFM and comparing the results with those of similar displacement method based elements, as well as with the exact solutions. The main contributions in this research work is documented in various chapters as follows. Chapters 1 presents introduction and, chapter 2 presents historical back ground, a brief review of the Integrated Force Method(IFM) and the scope of the present investigations. Chapter 3 gives the basic theory of the Integrated Force Method, variational functional for IFM, equilibrium equations and compatibility conditions. It also explains general solution procedure for IFM, displacement and stress-resultant fields, equilibrium and flexibility matrices, and exact and numerical integrations. Chapter 4 presents the formulation of element equilibrium and flexibilit.y matrices required in the IFM formulation using the Kirchhoff plate theory. Based on this theory, two elements KRP4( four-node Kirchhof] Rectangular Plate element with twelve decrees of freedom) and KRP16( four-node Kirchhot] Rectangular Plate element with sixteen deqrees of freedom)have been developed. The element KRP4 considers three degrees of freedom at each node while KRP16 considers four degrees of freedom at each node. This chapter also gives numerical results relating to deflections and moments for variousbenchmark problems solved using these elements and discusses their performances. These elements have yielded satisfactory results in all example problems considered. Chapter 5 covers the Mindlin-Reissner theory based rectangular plate bending elements. Formulation of element equilibrium and flexibility matrices using the Mindlin-Rcissner plate theory required for IFfvI are given here. There are three rectangular plate bending elements MRP 4 (Mindlin- Reissner Rectangular Plate element), MRP 8 ( Mirulliu- Reissner Rectangular Plate element) and MRP12( Mindlin-Reissner Rectangular Plate element) respectively 4-node, 8-node and 12-nodes are developed using the Mindlin-Reissncr theory. These three elements consider three degrees of freedom at each nocle. A set of benchmark problems are analyzcd by these elements via IF'M. Results are compared with i.lioso of displacement method based plate bending elements and also with the exact solutions In comparison, MRP4 has yielded satisfactory results while MRP8 and MRP12 cxcc-llout results. Further these elements do not lock in thin plate bending situations. Therefore these elements can be used for analyzing both thin and moderately thick plate bending problems. Chapter 6 presents quadrilateral plate bending elements. Skew plate and circular plate bending problems arc analyzod using the quadrilateral plate brnding CICIlH'11Ls MQP4 ( fournode Mindlin- Reissncr Quadrilateral Plate element], 1\11QP 8 ( eiqlii-n ot!r: Mirulliu- Rcissner Quadrilateral Plate element) and MQP 12 (twelve-node Mnuiliu- Rcissiier (j1J.acir"ilateml Plate element). Another efficient four-node Mindlin-Reissner Quadrilateral Plate bending element MQP41 has also been developed here. Performances of these elements arc studied by considering standard plate benchmark problems inculding Morley and Razzaque's skew plate and circular plate bending problems. Results are compared with those of displacement method based plate bending elements and also with the exact solutions. In comparison with displacement method based elements these elements have yielded better results. The MQP41 element is also do not lock in thin plate bending situations and yielded excellent results. Chapter 7 is dedicated to composite plates and gives the brief review of composite plates, constitutive behavior of laminated composite plates. Using the Mindlin-Reissner theory, a 4-node (MCP4) Mindlin-Reissner rectangular laminated composite plate bending element has been developed. Performances of these elements are studied by solving a set of benchmark problems using the IFM. This elements has yielded satisfactory results as compared to displacement based elements. Totally there are SEVEN plate bending elements developed for the Integrated Force Method. Chapter 8 gives the overall conclusions of the thesis along with scope for the feature work. It is followed by list of references. There are SIX appendices kept in the thesis. Features and fiowchart of the code for automatic generation of element equilibrium and flexibility matrices for the Mindlin-Reissner theory and the Kirchhoff theory based elements are given in the Appendix A. For comparison of the procedures of IFM and FEM, flowcharts of both If'M and FEM are given in the appendix B. Appendix C gives the stepwise procedure of IFM through the solution for a truss problem. Element equilibrium and flexibility matrices for the Mindlin-Reissner theory based -l-node and 8-node rectangular plate bending elements are given ill closed-form in the appendix D. Variational formulation of IFM for thin plate problems has been given in the appendix E. Constitutive relations of laminated composite plates arc gi\'ell ill the appendix F.

Item Type: | Thesis (Phd. Thesis) |
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Subjects: | Engineering > MIT Manipal > Civil Engineering |

Depositing User: | MIT Library |

Date Deposited: | 30 May 2014 06:05 |

Last Modified: | 07 Nov 2014 09:42 |

URI: | http://eprints.manipal.edu/id/eprint/138856 |

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