scholarly journals Stability of orthotropic plates

2020 ◽  
Vol 166 ◽  
pp. 06004
Author(s):  
Mykola Surianinov ◽  
Dina Lazarieva ◽  
Iryna Kurhan
Keyword(s):  
Differential Equation ◽  
Finite Element ◽  
Boundary Conditions ◽  
Critical Loads ◽  
Orthotropic Plate ◽  
Algebraic Equations ◽  
Stability Equation ◽  
Element Analysis ◽  
Linear Algebraic Equations ◽  
The Stability

The solution to the problem of the stability of a rectangular orthotropic plate is described by the numerical-analytical method of boundary elements. As is known, the basis of this method is the analytical construction of the fundamental system of solutions and Green’s functions for the differential equation (or their system) for the problem under consideration. To account for certain boundary conditions, or contact conditions between the individual elements of the system, a small system of linear algebraic equations is compiled, which is then solved numerically. It is shown that four combinations of the roots of the characteristic equation corresponding to the differential equation of the problem are possible, which leads to the need to determine sixty-four analytical expressions of fundamental functions. The matrix of fundamental functions, which is the basis of the transcendental stability equation, is very sparse, which significantly improves the stability of numerical operations and ensures high accuracy of the results. An analysis of the numerical results obtained by the author’s method shows very good convergence with the results of finite element analysis. For both variants of the boundary conditions, the discrepancy for the corresponding critical loads is almost the same, and increases slightly with increasing critical load. Moreover, this discrepancy does not exceed one percent. It is noted that under both variants of the boundary conditions, the critical loads calculated by the boundary element method are less than in the finite element calculations. The obtained transcendental stability equation allows to determine critical forces both by the static method and by the dynamic one. From this equation it is possible to obtain a spectrum of critical forces for a fixed number of half-waves in the direction of one of the coordinate axes. The proposed approach allows us to obtain a solution to the stability problem of an orthotropic plate under any homogeneous and inhomogeneous boundary conditions.

scholarly journals ВІЛЬНІ КОЛИВАННЯ ОРТОТРОПНИХ ПЛАСТИН

2020 ◽  
Vol 138 (5) ◽  
pp. 53-61
Author(s):  
Д. В. Лазарєва ◽  
І. В. Курган
Keyword(s):  
Differential Equation ◽  
Finite Element Analysis ◽  
Finite Element ◽  
Boundary Conditions ◽  
Orthotropic Plate ◽  
Free Vibrations ◽  
Element Analysis ◽  
Method Of Boundary Elements ◽  
Rectangular Orthotropic Plate ◽  
Numerical Analytical Method

The solution of the problem of free vibrations of a rectangular orthotropic plate by the methods of boundary and finite elements under any boundary conditions. Transformation of the two-dimensional differential equation of free vibrations of an orthotropic rectangular plate to one-dimensional. Determination of the complete system of its fundamental solutions using the numerical-analytical method of boundary elements. Implementation of the algorithm on the example of a specific plate and comparison with the results of finite element analysis in ANSYS. The solution to the problem of natural vibrations of a rectangular orthotropic plate is obtained without any restrictions on the nature of the fixing of its sides. A transcendental frequency equation is obtained whose roots give the full spectrum of natural frequencies. The modeling and calculations of the orthotropic plate by the finite element method are performed. An analysis of the numerical results obtained by the author's method shows a very good convergence with the results of finite element analysis. For a plate with rigid fastening of three sides with a free fourth side, the discrepancy is slightly higher than for a plate with a hinged support along the contour. Under both variants of the boundary conditions, the frequency spectrum calculated by the boundary element method is lower than in the finite element calculations. Analytical expressions of fundamental functions are obtained that correspond to all possible solutions to the differential equation of free oscillations. For the first time, a solution to the problem of free vibrations of a rectangular orthotropic plate is presented by the numerical-analytical method of boundary elements. The results allow us to solve the problem of free vibrations of a rectangular orthotropic plate by two methods under any boundary conditions, including inhomogeneous ones.

The Influence of Different Restrained Boundary Conditions on the Stability of Steel-Concrete Composite Ribbed Shell

2011 ◽  
Vol 243-249 ◽  
pp. 7005-7008
Author(s):  
Yu Zhen Chang ◽  
Ling Ling Wang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Bearing Capacity ◽  
Ultimate Bearing Capacity ◽  
Element Analysis ◽  
Boundary Constraints ◽  
Ribbed Shell ◽  
Post Buckling ◽  
The Stability ◽  
Section Dimension

The steel-concrete composite ribbed shell is a new type of spatial structure. Different restrained boundary conditions have a considerably influence on the ultimate bearing capacity and stability. Based on the nonlinear finite element method, a numerical model is made by finite element analysis software ANSYS, in which material and geometrical nonlinear are considered. A spherical composite ribbed shell with 40m span, three different section dimensions and two different vector heights is used as an example, in which 4 different restrained boundary conditions are considered, including all fixed, all hinged, node fixed and node hinged. The results show that when the section dimension and span height are the same, the ultimate bearing capacity will be greater as the boundary becoming rigid, and when the section dimension is larger, the ratio of ultimate bearing capacity under different restrained boundary conditions is increasing, while as the span height is greater, the ratio is decreasing. To the instable shape, the influence of different restrained boundary is minor, all the instable modes are extreme point instability, but the trend of load-displacement curves are almost similar, and when the cross-section dimension of composite rib increases, the composite ribbed shell under different boundary constraints has shown higher post-buckling strength.

scholarly journals IMPLEMENTATION OF A FINITE ELEMENT CLASS LIBRARY USING GENERALIZED PROGRAMMING

2021 ◽  
pp. 164-173
Author(s):  
S. V. Choporov ◽  
M. S. Ihnatchenko ◽  
O. V. Kudin ◽  
A. G. Kryvokhata ◽  
S. I. Homeniuk
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
General Purpose ◽  
Open Architecture ◽  
Algebraic Equations ◽  
Complex Objects ◽  
Element Analysis ◽  
Class Library ◽  
Linear Algebraic Equations ◽  
Numerical Finite Element

Context. For computer modeling of complex objects and phenomena of various nature, in practice, the numerical finite element method is often used. Its software implementation (especially for the study of new classes of problems) is a rather laborious process. The high cost of software development makes the development of new approaches to improving the efficiency of programming and maintenance (including the addition of new functions) urgent. Objective. The aim of the work is to create a new effective architecture of programs for finite element analysis of problems in mathematical physics, which makes it easy to expand their functionality to solve new classes of problems. Method. A method for developing programs for finite element analysis using generalized programming is proposed, which makes it possible to significantly simplify the architecture of the software and make it more convenient for maintenance and modification by separating algorithms and data structures. A new architecture of classes that implement finite element calculation is proposed, which makes it possible to easily expand the functionality of programs by adding new types of finite elements, methods for solving systems of linear algebraic equations, parallel computations, etc. Results. The proposed approach was implemented in software as a class library in C ++. A number of computational experiments have been carried out, which have confirmed its efficiency in solving practical problems. Conclusions. The developed approach can be used both to create general-purpose finite element analysis systems with an open architecture, and to implement specialized software packages focused on solving specific classes of problems (fracture mechanics, elastomers, contact interaction, etc.).

Finite‐element analysis of controlled‐source electromagnetic induction using Coulomb‐gauged potentials

Geophysics ◽  
2001 ◽  
Vol 66 (3) ◽  
pp. 786-799 ◽  
Author(s):  
Eugene A. Badea ◽  
Mark E. Everett ◽  
Gregory A. Newman ◽  
Oszkar Biro
Keyword(s):  
Finite Element ◽  
Heterogeneous Media ◽  
Petroleum Exploration ◽  
Algebraic Equations ◽  
Weak Formulation ◽  
Element Analysis ◽  
Controlled Source ◽  
Linear Algebraic Equations ◽  
Em Induction ◽  
Log Interpretation

A 3-D finite‐element solution has been used to solve controlled‐source electromagnetic (EM) induction problems in heterogeneous electrically conducting media. The solution is based on a weak formulation of the governing Maxwell equations using Coulomb‐gauged EM potentials. The resulting sparse system of linear algebraic equations is solved efficiently using the quasi‐minimal residual method with simple Jacobi scaling as a preconditioner. The main aspects of this work include the implementation of a 3-D cylindrical mesh generator with high‐quality local mesh refinement and a formulation in terms of secondary EM potentials that eliminates singularities introduced by the source. These new aspects provide quantitative induction‐log interpretation for petroleum exploration applications. Examples are given for 1-D, 2-D, and 3-D problems, and favorable comparisons are presented against other, previously published multidimensional EM induction codes. The method is general and can also be adapted for controlled‐source EM modeling in mining, groundwater, and environmental geophysics in addition to fundamental studies of EM induction in heterogeneous media.

scholarly journals Buckling of stabilizers deep-sea vehicles

2020 ◽  
Author(s):  
С.О. Барышников ◽  
М.В. Сухотерин ◽  
Т.П. Кныш ◽  
Н.Ф. Пижурина
Keyword(s):  
Deep Sea ◽  
Critical Loads ◽  
Water Pressure ◽  
High Water ◽  
Homogeneous System ◽  
Order Partial Differential Equation ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Compressive Loads ◽  
The Stability

В данной работе исследуется устойчивость прямоугольной консольной панели как приближенной расчетной модели стабилизаторов глубоководных аппаратов. Вследствие высокого давления воды сжимающие усилия в плоскости стабилизатора, приложенные к свободным граням, могут быть значительными и приводить к потере устойчивости. Целью настоящей работы является разработка эффективного метода численного моделирования устойчивости стабилизаторов принципиально новых судов и кораблей, в том числе из новых материалов. Задачей исследования является определение спектра критических сжимающих нагрузок, а также соответствующих форм закритического равновесия для этих элементов. Краевая задача устойчивости прямоугольной консольной панели описывается дифференциальным уравнением четвертого порядка в частных производных по двум переменным для искомой функции прогибов и системой граничных условий, содержащих частные производные этой функции до третьего порядка включительно. В качестве параметра основное уравнение изгиба содержит интенсивность равномерно распределенного давления на свободные края панели. Функция прогибов выбирается в виде суммы двух гиперболо-тригонометрических рядов по двум координатам и дополняется затем специальными компенсирующими членами. Проблема сводится к исследованию бесконечной однородной системы линейных алгебраических уравнений относительно неизвестных коэффициентов рядов. Поиск критических нагрузок осуществляется перебором величины давления и анализом бесконечной системы. Получен спектр нескольких первых критических нагрузок, при которых появляется новая форма равновесия. In this paper, we study the stability of a rectangular console panel as an approximate computational model of deep-sea vehicle stabilizers. Due to high water pressure, compressive forces in the stabilizer plane applied to free faces can be significant and lead to loss of stability. The purpose of this work is to develop an effective method for numerical modeling of stability of stabilizers of fundamentally new vessels and ships, including those made of new materials. The aim of the study is to determine the spectrum of critical compressive loads, as well as the corresponding forms of supercritical equilibrium for these elements. The boundary value problem of stability of a rectangular console panel is described by a fourth-order partial differential equation for two variables for the desired deflection function and a system of boundary conditions containing partial derivatives of this function up to and including the third order. As a parameter, the basic bending equation contains the intensity of evenly distributed pressure on the free edges of the panel. The deflection function is selected as the sum of two hyperbolic-trigonometric series over two coordinates and then supplemented with special compensating terms. The problem is reduced to the study of an infinite homogeneous system of linear algebraic equations with respect to unknown series coefficients. The search for critical loads is performed by searching the pressure value and analyzing the infinite system. The spectrum of the first few critical loads at which a new form of equilibrium appears is obtained.

scholarly journals Generalized Neumann Expansion and Its Application in Stochastic Finite Element Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Xiangyu Wang ◽  
Song Cen ◽  
Chenfeng Li
Keyword(s):  
Finite Element ◽  
Perturbation Method ◽  
High Efficiency ◽  
Expansion Method ◽  
Error Estimator ◽  
Algebraic Equations ◽  
Stochastic Finite Element ◽  
Stochastic Field ◽  
Element Analysis ◽  
Linear Algebraic Equations

An acceleration technique, termed generalized Neumann expansion (GNE), is presented for evaluating the responses of uncertain systems. The GNE method, which solves stochastic linear algebraic equations arising in stochastic finite element analysis, is easy to implement and is of high efficiency. The convergence condition of the new method is studied, and a rigorous error estimator is proposed to evaluate the upper bound of the relative error of a given GNE solution. It is found that the third-order GNE solution is sufficient to achieve a good accuracy even when the variation of the source stochastic field is relatively high. The relationship between the GNE method, the perturbation method, and the standard Neumann expansion method is also discussed. Based on the links between these three methods, quantitative error estimations for the perturbation method and the standard Neumann method are obtained for the first time in the probability context.

scholarly journals Shape Sensing of a Complex Aeronautical Structure with Inverse Finite Element Method

Sensors ◽  
2021 ◽  
Vol 21 (4) ◽  
pp. 1388
Author(s):  
Daniele Oboe ◽  
Luca Colombo ◽  
Claudio Sbarufatti ◽  
Marco Giglio
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Strain Field ◽  
Element Analysis ◽  
Inverse Finite Element Method ◽  
Shape Sensing ◽  
Definition Of ◽  
High Level ◽  
Element Method

The inverse Finite Element Method (iFEM) is receiving more attention for shape sensing due to its independence from the material properties and the external load. However, a proper definition of the model geometry with its boundary conditions is required, together with the acquisition of the structure’s strain field with optimized sensor networks. The iFEM model definition is not trivial in the case of complex structures, in particular, if sensors are not applied on the whole structure allowing just a partial definition of the input strain field. To overcome this issue, this research proposes a simplified iFEM model in which the geometrical complexity is reduced and boundary conditions are tuned with the superimposition of the effects to behave as the real structure. The procedure is assessed for a complex aeronautical structure, where the reference displacement field is first computed in a numerical framework with input strains coming from a direct finite element analysis, confirming the effectiveness of the iFEM based on a simplified geometry. Finally, the model is fed with experimentally acquired strain measurements and the performance of the method is assessed in presence of a high level of uncertainty.

scholarly journals Inlay-Retained Dental Bridges—A Finite Element Analysis

2021 ◽  
Vol 11 (9) ◽  
pp. 3770
Author(s):  
Monica Tatarciuc ◽  
George Alexandru Maftei ◽  
Anca Vitalariu ◽  
Ionut Luchian ◽  
Ioana Martu ◽  
...  
Keyword(s):  
Finite Element ◽  
Young Patients ◽  
Maximum Pressure ◽  
Class Iii ◽  
Element Analysis ◽  
Dental Prostheses ◽  
Abutment Teeth ◽  
Fixed Dental Prostheses ◽  
Implant Therapy ◽  
The Stability

Inlay-retained dental bridges can be a viable minimally invasive alternative when patients reject the idea of implant therapy or conventional retained full-coverage fixed dental prostheses, which require more tooth preparation. Inlay-retained dental bridges are indicated in patients with good oral hygiene, low susceptibility to caries, and a minimum coronal tooth height of 5 mm. The present study aims to evaluate, through the finite element method (FEM), the stability of these types of dental bridges and the stresses on the supporting teeth, under the action of masticatory forces. The analysis revealed the distribution of the load on the bridge elements and on the retainers, highlighting the areas of maximum pressure. The results of our study demonstrate that the stress determined by the loading force cannot cause damage to the prosthetic device or to abutment teeth. Thus, it can be considered an optimal economical solution for treating class III Kennedy edentation in young patients or as a provisional pre-implant rehabilitation option. However, special attention must be paid to its design, especially in the connection area between the bridge elements, because the connectors and the retainers represent the weakest parts.

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