scholarly journals Bending of a ship's skin panel loaded along the axis of symmetry

2021 ◽  
pp. 14-19
Author(s):  
Т.П. Кныш ◽  
М.В. Сухотерин ◽  
С.О. Барышников
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Double Fourier Series ◽  
Transverse Load ◽  
Bending Moments ◽  
Approximate Methods ◽  
Loaded Part ◽  
Axis Of Symmetry ◽  
Square Plate ◽  
3D Shapes

Задача изгиба прямоугольной панели обшивки от действия распределенной по оси симметрии поперечной нагрузки не имеет точного решения в конечном виде в виду сложности краевых условий и вида нагрузки. Использование другими авторами различных приближенных методов оставляет открытым вопрос о точности полученных результатов. Целью исследования является получение точного решения с помощью гиперболо-тригонометрических рядов по двум координатам. Для этого используется метод бесконечной суперпозиции указанных рядов, которые в отдельности удовлетворят лишь части граничных условий. Порождаемые ими невязки взаимно компенсируются в ходе итерационного процесса и стремятся к нулю. Частное решения представлено двойным рядом Фурье. Точное решение достигается увеличением количества членов в рядах и числа итераций. При достижении заданной точности процесс прекращается. Получены численные результаты для прогибов и изгибающих моментов для квадратной пластины при различной длине загруженной части оси пластины. Представлены 3D-формы изогнутой поверхности пластины и эпюры изгибающих моментов. The problem of bending a rectangular skin panel from the action of a transverse load distributed along the axis of symmetry does not have an exact solution in the final form due to the complexity of the boundary conditions and the type of load. The use of various approximate methods by other authors leaves open the question of the accuracy of the results obtained. The aim of the study is to obtain an exact solution using hyperbolo-trigonometric series in two coordinates. To do this, we use the method of infinite superposition of these series, which individually satisfy only part of the boundary conditions. The residuals generated by them are mutually compensated during the iterative process and tend to zero. The quotient of the solution is represented by a double Fourier series. The exact solution is achieved by increasing the number of terms in the series and the number of iterations. When the specified accuracy is reached, the process stops. Numerical results are obtained for deflections and bending moments for a square plate with different lengths of the loaded part of the plate axis. 3D shapes of the curved surface of the plate and diagrams of bending moments are presented.

Exact analysis of simply supported rhombic plates under uniform pressure

1974 ◽  
Vol 76 (1) ◽  
pp. 381-388 ◽  
Author(s):  
K. Rajaiah ◽  
Akella Kameswara Rao
Keyword(s):  
Exact Solution ◽  
Complex Variables ◽  
Transverse Load ◽  
Uniform Pressure ◽  
Approximate Methods ◽  
Simply Supported ◽  
Series Form ◽  
Exact Analysis ◽  
Present Solution

AbstractThe simply supported rhombic plate under transverse load has received extensive attention from elasticians, applied mathematicians and engineers. All known solutions are based on approximate procedures. Now, an exact solution in a fast converging explicit series form is derived for this problem, by applying Stevenson's tentative approach with complex variables. Numerical values for the central deflexion and moments are obtained for various corner angles. The present solution provides a basis for assessing the accuracy of approximate methods for analysing problems of skew plates or domains.

On the Thermal Buckling Response of Shear-Flexible All-Edge Clamped Rectangular Plates

2003 ◽  
Vol 9 (5) ◽  
pp. 495-506 ◽  
Author(s):  
Abdulateef M. Al-Khaleefi ◽  
Humayun R. H. Kabir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Shear Deformation Theory ◽  
Double Fourier Series ◽  
Transverse Displacement ◽  
Rectangular Plates ◽  
Partial Differential ◽  
Approximate Methods ◽  
Flexible Response

Using an analytical approach, we investigate a thermal stability response for a rectangular plate with all-edge clamped boundary conditions. We consider the first-order shear deformation theory that utilizes shear flexible response, in order to obtain three highly coupled governing partial differential equations in three unknowns: one transverse displacement, and two independent rotations of the normal. The solution functions are assumed in the form of double Fourier series that satisfy the boundary conditions, as well as the partial differential equations. The results obtained from the analytical solution are compared with available finite element solutions. These analytically obtained results can be capitalized to check the accuracy of various approximate methods.

Exact solution for frequency response of sandwich microbeams with functionally graded cores

2019 ◽  
Vol 25 (19-20) ◽  
pp. 2641-2655 ◽  
Author(s):  
Ehsan Taati ◽  
Famida Fallah
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Frequency Response ◽  
Functionally Graded ◽  
Strain Gradient Theory ◽  
Transverse Load ◽  
Gradient Theory ◽  
Beam Model ◽  
Various Boundary Conditions ◽  
Size Dependent

Based on the Euler–Bernoulli beam model and the modified strain gradient theory, the size-dependent forced vibration of sandwich microbeams with a functionally graded (FG) core is presented. The equation of motion and the corresponding classical and nonclassical boundary conditions are derived using the Hamilton’s principle. An exact solution of the governing equation is developed for sandwich beams with various boundary conditions and subjected to an arbitrarily distributed harmonic transverse load. Finally, parametric studies are presented to investigate the effects of geometric ratios, length scale parameters, power index, boundary conditions, layup, and thickness of the FG layer on the frequency response of clamped and simply supported microbeams. Numerical results show that in the case of clamped microbeams, the essential and natural size-dependent boundary conditions have a significant effect on the resonance frequency and transverse deflection of microbeams. Also, it is seen that an optimal layup (without change in total volume of each material) can significantly improve the frequency characteristics of sandwich microbeams.

Solution of heat and mass transfer problems with non-linear coefficients

2019 ◽  
Vol 5 (4) ◽  
pp. 10-20
Author(s):  
Boris G. Aksenov ◽  
Yuri E. Karyakin ◽  
Svetlana V. Karyakina
Keyword(s):  
Mass Transfer ◽  
Exact Solution ◽  
Numerical Methods ◽  
Heat And Mass Transfer ◽  
Unknown Function ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Maximum Deviation ◽  
Approximate Methods ◽  
Nonmonotonic Dependence

Equations, which have nonlinear nonmonotonic dependence of one of the coefficients on an unknown function, can describe processes of heat and mass transfer. As a rule, existing approximate methods do not provide solutions with acceptable accuracy. Numerical methods do not involve obtaining an analytical expression for the unknown function and require studying the convergence of the algorithm used. The value of absolute error is uncertain. The authors propose an approximate method for solving such problems based on Westphal comparison theorems. The comparison theorems allow finding upper and lower bounds of the unknown exact solution. A special procedure developed for the stepwise improvement of these bounds provide solutions with a given accuracy. There are only a few problems for equations with nonlinear nonmonotonic coefficients for which the exact solution has been obtained. One of such problems, presented in this article, shows the efficiency of the proposed method. The results prove that the proposed method for obtaining bounds of the solution of a nonlinear nonmonotonic equation of parabolic type can be considered as a new method of the approximate analytical solution having guaranteed accuracy. In addition, the proposed here method allows calculating the maximum deviation from the unknown exact solution of the results of other approximate and numerical methods.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

Large Deflection of a Rectangular Plate Resting on a Pasternak-Type Elastic Foundation

1977 ◽  
Vol 44 (3) ◽  
pp. 509-511 ◽  
Author(s):  
P. K. Ghosh
Keyword(s):  
Rectangular Plate ◽  
Elastic Foundation ◽  
Large Deflection ◽  
Lateral Load ◽  
Bending Moments ◽  
Shear Forces ◽  
Simply Supported ◽  
Base Parameters ◽  
Square Plate

The problem of large deflection of a rectangular plate resting on a Pasternak-type foundation and subjected to a uniform lateral load has been investigated by utilizing the linearized equation of plates due to H. M. Berger. The solutions derived and based on the effect of the two base parameters have been carried to practical conclusions by presenting graphs for bending moments and shear forces for a square plate with all edges simply supported.

Marine Pipeline Analysis Based on Newton’s Method With an Arctic Application

1970 ◽  
Vol 92 (4) ◽  
pp. 827-833 ◽  
Author(s):  
D. W. Dareing ◽  
R. F. Neathery
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Newton’S Method ◽  
Finite Differences ◽  
Newton's Method ◽  
Nonlinear Differential Equations ◽  
Bending Moments ◽  
Iterative Solutions ◽  
Linear Differential ◽  
Pipe Deflection

Newton’s method is used to solve the nonlinear differential equations of bending for marine pipelines suspended between a lay-barge and the ocean floor. Newton’s method leads to linear differential equations, which are expressed in terms of finite differences and solved numerically. The success of Newton’s method depends on initial trial solutions, which in this paper are catenaries. Iterative solutions converge rapidly toward the exact solution (pipe deflection) even though large bending moments exist in the pipe. Example calculations are given for a 48-in. pipeline suspended in 300 ft of water.

Torsional oscillations of a fluid sphere with a rigid boundary in a uniform magnetic field

1966 ◽  
Vol 24 (2) ◽  
pp. 275-284
Author(s):  
R. A. Wentzell
Keyword(s):  
Magnetic Field ◽  
Exact Solution ◽  
Uniform Magnetic Field ◽  
Perfect Conductor ◽  
Rigid Boundary ◽  
Torsional Oscillations ◽  
Magnetic Viscosity ◽  
Eigen Values ◽  
Infinite Conductivity ◽  
Axis Of Symmetry

Plumpton & Ferraro (1955) considered the torsional oscillations of an infinitely conducting sphere in a uniform magnetic field. They showed that if the fluid and magnetic viscosity were assumed to be zero in the governing differential equations, then a continuous spectrum of eigenvalues could be obtained. This novel feature was clarified by Stewartson (1957) when he obtained the exact solution and showed that in the correct limit of a perfect conductor the eigen-values are discrete. Furthermore, in the limit of infinite conductivity the oscillations occur only on the axis of symmetry (figure 1).

Clamped Semicircular Plate Under Uniform Bending Load

1955 ◽  
Vol 22 (1) ◽  
pp. 129
Author(s):  
S. Woinowsky-Krieger
Keyword(s):  
Exact Solution ◽  
Approximate Methods ◽  
Bending Load ◽  
Distribution Of Stresses

Abstract The semicircular plate subjected to bending usually is considered as a particular case of a sectorial plate and one introduces polar co-ordinates to discuss the deflection of the plate and the corresponding distribution of stresses. If the semicircular plate is clamped along the boundary the application of approximate methods becomes necessary to this end. It is worthy of note that a rather simple exact solution can be given in this latter case by making use of bipolar instead of polar co-ordinates.

Standardization of Three Efficient Footbridges: Von Mises, Monocontentio and Bicontentio Prototypes.

2021 ◽  
Author(s):  
Mario Guisasola
Keyword(s):  
Boundary Conditions ◽  
Structural Characteristics ◽  
Structural Behavior ◽  
Bending Moments ◽  
Variable Depth ◽  
Constant Depth ◽  
Simply Supported ◽  
Vertical Walls ◽  
Basic Concepts ◽  
Von Mises

<p>The Von Mises, Monocontentio and Bicontentio footbridges are three parameterized metal bridge whose main structural characteristics are their variable depth depending on the applied stress and the embedding of abutments. Its use is considered suitable for symmetrical or asymmetrical topographies with slopes or vertical walls on one or both edges. The footbridges include spans spaced apart by 20 to 66 meters, and are between 2 to 4.5 meters wide.</p><p>Its design is based on five basic concepts: integration in the geometry of the environment; continuous search for simplicity; design based on a geometry that emanates from structural behavior; unitary and round forms; and long- lasting details.</p><p>The structural behavior of these prototypes has been compared with three types of constant-depth metal beams: the bridge simply supported, and the bridge embedded on one or both sides.</p><p>The embedding of abutments, and the adoption of a variation of depth adapted to the bending moments diagrams, allow for more efficient and elegant forms which are well-adapted to the boundary conditions.</p>

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