Pengantar Metode Elemen HIngga untuk Analisis Struktur- Teori, Perumusan, Implementasi Komputer, dan Aplikasi

Metode elemen hingga (MEH) adalah metode perhitungan yang paling banyak dipakai saat ini untuk menyelesaikan masalah analisis struktur. Metode ini digunakan dalam banyak software komersial analisis struktur yang digunakan sehari-hari dalam pekerjaan desain para insinyur dalam bidang teknik sipil, teknik mesin, teknik perkapalan, dll. Di lain pihak, bagi matematikawan terapan MEH merupakan metode numerik yang ampuh untuk menyelesaikan boundary value problems dalam ruang berdimensi dua dan tiga. Buku ini bertujuan memperkenalkan MEH secara lengkap, mulai dari teori model yang akan diselesaikan, perumusan MEH untuk menyelesaikan masalah model itu, implementasi komputer dengan menggunakan MATLAB, dan sampai dengan aplikasi MEH. Karena tujuan buku ini adalah memperkenalkan MEH, model-model yang dibahas dipilih yang sederhana dan umum digunakan untuk memperkenalkan MEH, yaitu model deformasi aksial batang dan model tegangan dan regangan bidang. Bab 1 mulai dengan memperkenalkan MEH secara menyeluruh. Setelah membaca Bab ini, pembaca diharapkan memiliki gambaran umum apa itu MEH dan termotivasi untuk mempelajarinya lebih lanjut. Setelah itu Bagian I buku ini membahas langkah-langkah perhitungan MEH yang ditujukan untuk implementasi komputer, yaitu metode kekakuan langsung (direct stiffness method, disingkat DSM). Pembahasan DSM diberikan dalam konteks masalah sistem pegas (Bab 2), struktur rangka batang dalam ruang 1D dan 2D (Bab 3), dan rangka batang dalam ruang 3D (Bab 4). Pembahasan DSM di dalam Bab 2 s/d 4 ini ditekankan kepada prosedur perhitungan dan implementasi komputer. Perumusan elemen-elemen hingga dilakukan secara sederhana berdasarkan penerapan langsung hukum-hukum fisika. Setiap Bab disertai dengan contoh-contoh soal untuk memperjelas konsep dan penyelesaiannya, yang disajikan secara cukup detail. Program komputer yang lengkap untuk mengaplikasikan prosedur DSM secara efisien diberikan pada setiap Bab disertai dengan penjelasannya. Bagian II membahas MEH untuk masalah model dalam ruang berdimensi satu. Model yang dibahas adalah batang berdeformasi aksial. Pembahasan mencakup teori dasar model batang berdeformasi aksial, perumusan MEH secara matematis berdasarkan metode Galerkin dan prinsip perpindahan virtual, implementasi komputer untuk batang 1D, dan pengembangannya untuk struktur rangka batang 2D dan 3D. Setiap konsep yang dibahas disertai dengan contoh-contoh soal dan penyelesaiannya secara detail sehingga diharapkan dapat memperkuat pemahaman pembaca. Pembaca diperkenalkan kepada prosedur perhitungan MEH dan implementasi komputernya (Bagian I) sebelum mempelajari perumusan formal MEH (Bagian II) dengan tujuan supaya pembaca, khususnya yang berlatar belakang pendidikan bidang teknik sipil atau teknik mesin, dapat lebih siap untuk mempelajari perumusan MEH di bagian-bagian buku ini selanjutnya. Namun bagi pembaca yang berlatar belakang pendidikan matematika terapan, mungkin akan lebih nyaman untuk mempelajari terlebih dahulu perumusan MEH dalam ruang 1D di Bagian II buku ini kemudian mempelajari Bagian I. Bagian III buku ini membahas MEH untuk masalah dalam ruang dua dimensi. Masalah yang dibahas adalah model tegangan/ regangan bidang. Bagian ini dimulai dengan pembahasan model tegangan/ regangan bidang secara detail (Bab 6). Selanjutnya, penyelesaian masalah tegangan/ regangan bidang dengan menggunakan elemen dua dimensi yang paling sederhana, yaitu elemen segitiga regangan konstan (CST), dibahas dalam Bab 7. Setiap bab buku ini disertai dengan contoh-contoh soal yang memadai dan MATLAB codes yang lengkap untuk mengaplikasikan MEH

Adaptive mesh refinements for analyses of 2D linear elasticity problems using the Kriging-based finite element method

Finite element analyses of irregular structures require adaptive mesh refinement to achieve more accurate results in an efficient manner. This is also true for a non-conventional finite element method with Kriging interpolation, called the Kriging-based finite element method (K-FEM). This paper presents a study of automatic adaptive meshing procedures for analyses of two-dimensional linear elasticity problems using the K-FEM. The Matlab Partial Differential Equation Toolbox was utilized for generating meshes with Delaunay triangulation. Three error indicators, namely, the strain energy error, the gradient of effective stresses, and the element-free Galerkin strain energy error, were employed for estimating the element errors. To find the most effective error indicator, the resulting total number of elements and configurations of the final meshes were compared. The results show that the resulting final meshes were affected by the initial mesh configurations, the refinement criteria, and the termination criteria. The gradient of effective stresses indicator was found to be the most effective error indicator for the K-FEM, as it can accurately estimate the element errors

Development of the DKMQ Element for Analysis of Composite Laminated Folded Plate Structures

The discrete-Kirchhoff Mindlin quadrilateral (DKMQ) element has recently been developed for analysis of composite laminated plates. This paper presents further development of the DKMQ for analysis of composite laminated folded plates. In this development, a local coordinate system is set up for each element at its centroid. The DKMQ stiffness matrix is superimposed with that of the standard four-node plane stress quadrilateral element to obtain a 24-by-24 folded plate stiffness matrix in the local coordinate system. To avoid singularity of the stiffness matrix, a small stiffness coefficient is added in the entries corresponding to the drilling degrees of freedom. The local stiffness matrix and force vector are then transformed to the global ones and assembled. The accuracy and convergence of the folded plate element are assessed using a number of numerical examples. The results show that the element is accurate and converge well to the reference solutions

Study of the Discrete Shear Gap Technique in Timoshenko Beam Elements

A major difficulty in formulating a finite element for shear-deformable beams, plates, and shells is the shear locking phenomenon. A recently proposed general technique to overcome this difficulty is the discrete shear gap (DSG) technique. In this study, the DSG technique was applied to the linear, quadratic, and cubic Timoshenko beam elements. With this technique, the displacement-based shear strain field was replaced with a substitute shear strain field obtained from the derivative of the interpolated shear gap. A series of numerical tests were conducted to assess the elements performance. The results showed that the DSG technique works perfectly to eliminate the shear locking. The resulting deflection, rotation, bending moment, and shear force distributions were very accurate and converged optimally to the corresponding analytical solutions. Thus the beam elements with the DSG technique are better alternatives than those with the classical selective-reduced integration

Kriging-Based Timoshenko Beam Elements with the Discrete Shear Gap Technique

Kriging-based finite element method (K-FEM) is an enhancement of the FEM through the use of Kriging interpolation in place of the conventional polynomial interpolation. In this paper, the K-FEM is developed for static, free vibration, and buckling analyses of Timoshenko beams. The discrete shear gap technique is employed to eliminate shear locking. The numerical tests show that a Kriging-Based beam element with cubic basis and three element-layer domain of influencing nodes is free from shear locking. Exceptionally accurate displacements, bending moments, natural frequencies, and buckling loads and reasonably accurate shear force can be achieved using a relatively course mesh

Enhanced Symbiotic Organisms Search (ESOS) for Global Numerical Optimization

Symbiotic organisms search (SOS) is a simple yet effective metaheuristic algorithm to solve a wide variety of optimization problems. Many studies have been carried out to improve the performance of the SOS algorithm. This research proposes an improved version of the SOS algorithm called the “enhanced symbiotic organisms search” (ESOS) for global numerical optimization. The conventional SOS is modified by implementing a new searching formula into the parasitism phase to produce a better searching capability. The performance of the ESOS is verified using 26 benchmark functions and one structural engineering design problem. The results are then compared with existing metaheuristic optimization methods. The obtained results show that the ESOS gives a competitive and effective performance for global numerical optimization

A Locking-free Kriging-based Timoshenko Beam Element with the Discrete Shear Gap Technique

The Kriging-based FEM (K-FEM) is an enhancement of the FEM through the use of Kriging interpolation in place of the conventional polynomial interpolation. The key advantage of the K-FEM is that the polynomial refinement can be performed without adding nodes or changing the element connectivity. In the development of the K-FEM for analyses of shear deformable beams, plates and shells, the well-known difficulty of shear locking also presents. This paper presents the K-FEM with the discrete shear gap (DSG) technique to eliminate the shear locking in the Timoshenko beam. The numerical tests show that the DSG technique can completely eliminate the shear locking for the Kriging-based beam element with cubic basis and three element-layer domain of influencing nodes. However, the DSG technique does not work well for the K-beam with linear (except with one element layer) and quadratic bases

Adaptive mesh refinements for analyses of 2D linear elasticity problems using the Kriging-based finite element method

Finite element analyses of irregular structures require adaptive mesh refinement to achieve more accurate results in an efficient manner. This is also true for a non-conventional finite element method with Kriging interpolation, called the Kriging-based finite element method (K-FEM). This paper presents a study of automatic adaptive meshing procedures for analyses of two-dimensional linear elasticity problems using the K-FEM. The Matlab Partial Differential Equation Toolbox was utilized for generating meshes with Delaunay triangulation. Three error indicators, namely, the strain energy error, the gradient of effective stresses, and the element-free Galerkin strain energy error, were employed for estimating the element errors. To find the most effective error indicator, the resulting total number of elements and configurations of the final meshes were compared. The results show that the resulting final meshes were affected by the initial mesh configurations, the refinement criteria, and the termination criteria. The gradient of effective stresses indicator was found to be the most effective error indicator for the K-FEM, as it can accurately estimate the element errors

Development of the DKMQ element for buckling analysis of shear-deformable plate bending

In this paper the discrete-Kirchhoff Mindlin quadrilateral (DKMQ) element was developed for buckling analysis of plate bending including the shear deformation. In this development the potential energy corresponding to membrane stresses was incorporated in the Hu-Washizu functional. The bilinear approximations for the deflection and normal rotations were used for the membrane stress term in the functional, while the approximations for the remaining terms remain the same as in static analysis. Numerical tests showed that the element has good predictive capability for thin plates. For thick plates, however, the element tends to give a slightly lower solution

Displacement and Stress Function-based Linear and Quadratic Triangular Elements for Saint-Venant Torsional Problems

Torsional problems commonly arise in frame structural members subjected to unsym-metrical loading. Saint-Venant proposed a semi inverse method to develop the exact theory of torsional bars of general cross sections. However, the solution to the problem using an analytical method for a complicated cross section is cumbersome. This paper presents the adoption of the Saint-Venant theory to develop a simple finite element program based on the displacement and stress function approaches using the standard linear and quadratic triangular elements. The displacement based approach is capable of evaluating torsional rigidity and shear stress distribution of homogeneous and nonhomogeneous; isotropic, orthotropic, and anisotropic materials; in singly and multiply-connected sections. On the other hand, applications of the stress function approach are limited to the case of singly-connected isotropic sections only, due to the complexity on the boundary conditions. The results show that both approaches converge to exact solutions with high degree of accuracy