The Use of Mixed Eulerian-Lagrangian Displacement in Geometrically Nonlinear Analysis of Structural System

Binsar Hariandja

Abstract


Abstract


The paper deals with the use of mixed Eulerian-Lagrangian displacement in geometrically nonlinear analysis of structural system, in which displacement and deformation are observed from a selected referential configuration, i.e., a configuration once occupied by the system along the loading process. The displacement measured from initial configuration into referential configuration is referred to as Eulerian displacement, and the displacement measured from referential configuration into current configuration is referred to as Lagrangian displacement. Geometrical nonlinearity of structure occurs when the displacement primarily consists of rigid body displacement, in which the choice in referential configuration is of great concern. The same deformation may be observed differently according to the choice in referential configuration. Analysis of continuum system is cast in finite element method and written in matrix formulation. The geometrical nonlinearity is approached by successive incremental steps in which the total loading is divided into several incremental loadings. The process is then linearized and incremental global stiffness matrix is used at every iteration step. The proposed mixed displacement is cast in a computer package program using Fortran language. The program is applied in several structural analysis, in which the conventional Lagrangian displacement may not be appropriate to model the analysis. 

Abstrak


Makalah membahas penerapan perpindahan campuran Euler-Lagrange dalam analisis nonlinier geometri sistem struktur, dalam mana perpindahan dan deformasi diamati dari konfigurasi referensi yang dipilih, yaitu konfigurasi yang pernah dilalui oleh sistem selama proses pembebanan. Perpindahan yang diukur dari konfigurasi awal ke konfigurasi referensi dinamakan perpindahan Euler, dan perpindahan yang diukur dari konfigurasi referensi ke konfigurasi akhir dinamakan perpindahan Lagrange. Nonlinieritas geometri sistem struktur terjadi dalam kasus di mana perpindahan terutama mencakup perpindahan badan kaku, dalam mana pemilihan konfigurasi referensi menjadi suatu langkah penting. Deformasi yang sama dapat diamati berlainan seturut dengan pilihan konfigurasi referensi. Analisis sitem kontinu didekati dengan langkah inkremental berturutan dalam mana beban total dibagi atas beberapa beban inkremental. Proses kemudian dilinierisasi dan matriks kekakuan global inkremental digunakan pada setiap langkah iterasi. Perpindahan campuran yang diusulkan dituangkan dalam program paket komputer yang dituliskan dalam bahasa Fortran. Program diterapkan dalam analisis beberapa sistem struktur, dalam mana perpindahan Lagrange konvensional tidak cukup untuk memodelkan perpindahan dalam analisis.


Keywords


Finitesimal displacement, geometrical nonlinearity, finite element method, successive incremental loading steps

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References


Fung, Y.C., 1965, Foundations of Solid Mechanics, Prentice Hall, Inc., Engelwood Cliffs, New Jersey.

Gantmacher, F., 1975, Lectures in Analytical Mechanics, MIR Publishers, Moscow.

Hariandja, B., 1985, Adaptive Finite Element Analysis Nonlinear Frictional Contact with Mixed Eulerian-Lagrangian Coordinates, Ph.D dissertation, University of Illinois at Urbana-Champaign.

Malvern, L.E., 1969, Introduction to the Mechanics of a Continuous Medium, Prentice-Hall, Inc., Engelwood Cliffs, New Jersey.




DOI: http://dx.doi.org/10.5614%2Fjts.2017.24.2.1

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