THE NON-CLASSICAL PROBLEM OF AN ELASTICALLY CLAMPED ORTHOTROPIC BEAM OF VARIABLE THICKNESS UNDER THE COMBINED ACTION OF COMPRESSIVE FORCES AND TRANSVERSE LOAD

2018 ◽  
Vol 52 (2 (246)) ◽  
pp. 101-108
Author(s):  
R.M. Kirakosyan ◽  
S.P. Stepanyan
Keyword(s):  
Variable Thickness ◽  
Classical Problem ◽  
Compressive Force ◽  
Transverse Load ◽  
Simultaneous Action ◽  
Refined Theory ◽  
Orthotropic Plates ◽  
Combined Action ◽  
Beam Bending ◽  
The Stability

On the basis of the refined theory of orthotropic plates of variable thickness, the equations of the beam bending problem are obtained with the simultaneous action of compressive forces and transverse load. It is accepted that the edges of the beam have an elastically clamped support and the reduction of the compressive force by the support due to friction is taking into account. Passing to dimensionless quantities, a certain problem is solved. The stability of a beam is discussed. Based on the results obtained, conclusions are drawn.

ON THE NUMERICAL SOLUTION TO A NON-CLASSICAL PROBLEM OF BENDING AND STABILITY FOR AN ORTHOTROPIC BEAM OF VARIABLE THICKNESS

2021 ◽  
pp. 111-120
Author(s):  
S.P. Stepanyan ◽  
Keyword(s):  
Variable Thickness ◽  
Classical Problem ◽  
Compressive Force ◽  
Variable Coefficients ◽  
Simultaneous Action ◽  
Refined Theory ◽  
Orthotropic Plates ◽  
Axial Compressive Force ◽  
Axial Forces ◽  
The Stability

The mathematical model of the problem of bending of an elastically clamped beam is constructed on the basis of the refined theory of orthotropic plates of variable thickness. To solve the problem in the case of simultaneous action of its own weight and compressive axial forces, a system of differential equations with variable coefficients is obtained. The effects of transverse shear and the effect of reducing compressive force of the support are also taken into account. Passing on to dimensionless quantities, the specific problem for a beam of linearly varying thickness is solved by the collocation method. The stability of the beam is discussed. The critical values of forces are obtained by varying the axial compressive force. Results are presented in both tabular and graphical styles. Based on the results obtained, appropriate conclusions are drawn.

THE NON-CLASSICAL PROBLEM OF AN ORTHOTROPIC BEAM OF VARIABLE THICKNESS WITH THE SIMULTANEOUS ACTION OF ITS OWN WEIGHT AND COMPRESSIVE AXIAL FORCES

2019 ◽  
Vol 53 (3 (250)) ◽  
pp. 183-190
Author(s):  
R.M. Kirakosyan ◽  
S.P. Stepanyan
Keyword(s):  
Collocation Method ◽  
Variable Thickness ◽  
Transverse Shear ◽  
Specific Problem ◽  
Classical Problem ◽  
Compressive Force ◽  
Simultaneous Action ◽  
Axial Forces ◽  
The Stability

thickness, the equations of the problem of bending of an elastically clamped beam in the case of simultaneous action of its own weight and axial compressive forces are obtained. The effects of transverse shear and the effect of reducing the compressive force of the support are taken into account. Turning to dimensionless quantities, the specific problem for a beam of linearly variable thickness is solved by the collocation method. The question of the stability of the beam is discussed.

Nonlinear Analysis of Orthotropic Plates

1975 ◽  
Vol 17 (3) ◽  
pp. 133-138 ◽  
Author(s):  
C. Y. Chia ◽  
M. K. Prabhakara
Keyword(s):  
Large Deflection ◽  
Composite Plates ◽  
Double Fourier Series ◽  
Combined Loading ◽  
Transverse Load ◽  
Orthotropic Plates ◽  
Aspect Ratios ◽  
Patch Load ◽  
Combined Action ◽  
Transverse Deflection

The large deflection of a rectangular orthotropic plate subjected to the combined action of edge compression and transverse load is investigated on the basis of von Kármán-type large-deflection equations. The edges of the plate are assumed to be either all clamped or all simply supported. A solution is obtained in the form of double Fourier series consisting of beam eigenfunctions for both transverse deflection and force function. The postbuckling of the plate is treated as a special case. Taking the first nine terms in each truncated series, numerical results in load-deflection relations and bending moments are graphically presented for three types of fibre-reinforced composite plates with various aspect ratios. The three types of transverse load considered in the combined loading are central patch load, eccentric patch load and hydrostatic pressure. The present results for postbuckling and large deflection of isotropic and orthotropic plates are in good agreement with available data.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

Stability of a Rigid Body With an Oscillating Particle: An Application of MACSYMA

1985 ◽  
Vol 52 (3) ◽  
pp. 686-692 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Angular Velocity ◽  
Rigid Body ◽  
Classical Problem ◽  
Moment Of Inertia ◽  
Model Parameters ◽  
Stability Chart ◽  
The Stability ◽  
Transition Curves ◽  
Minimum Moment ◽  
Small Angular Velocity

This problem is a generalization of the classical problem of the stability of a spinning rigid body. We obtain the stability chart by using: (i) the computer algebra system MACSYMA in conjunction with a perturbation method, and (ii) numerical integration based on Floquet theory. We show that the form of the stability chart is different for each of the three cases in which the spin axis is the minimum, maximum, or middle principal moment of inertia axis. In particular, a rotation with arbitrarily small angular velocity about the maximum moment of inertia axis can be made unstable by appropriately choosing the model parameters. In contrast, a rotation about the minimum moment of inertia axis is always stable for a sufficiently small angular velocity. The MACSYMA program, which we used to obtain the transition curves, is included in the Appendix.

Bending tests for the structural safety assessment of space truss members

2018 ◽  
Vol 33 (3-4) ◽  
pp. 138-149 ◽  
Author(s):  
Marco Bonopera ◽  
Kuo-Chun Chang ◽  
Chun-Chung Chen ◽  
Tzu-Kang Lin ◽  
Nerio Tullini
Keyword(s):  
Cross Sections ◽  
Flexural Rigidity ◽  
Compressive Force ◽  
Scale Space ◽  
Transverse Load ◽  
Structural Safety ◽  
Small Scale ◽  
Order Effects ◽  
Beam Columns ◽  
Space Truss

This article compares two nondestructive static methods used for the axial load assessment in prismatic beam-columns of space trusses. Examples include the struts and ties or the tension chords and diagonal braces of steel pipe racks or roof trusses. The first method requires knowledge of the beam-column’s flexural rigidity under investigation, whereas the second requires knowledge of the corresponding Euler buckling load. In both procedures, short-term flexural displacements must be measured at the given cross sections along the beam-column under examination and subjected to an additional transverse load. The proposed methods were verified by numerical and laboratory tests on beams of a small-scale space truss prototype made from aluminum alloy and rigid connections. In general, if the higher second-order effects are induced during testing and the corresponding total displacements are accurately measured, it would be easy to obtain tensile and compressive force estimations.

Vibration Modeling of Machine Tool Spindles: A Calibrated Dynamic Stiffness Matrix Method

2013 ◽  
Vol 651 ◽  
pp. 710-716 ◽  
Author(s):  
Omar Gaber ◽  
Seyed M. Hashemi
Keyword(s):  
Stiffness Matrix ◽  
Natural Frequencies ◽  
Dynamic Stiffness ◽  
Dynamic Stiffness Matrix ◽  
Spring Element ◽  
Linear Spring ◽  
Beam Bending ◽  
The Stability ◽  
Spinning Speed ◽  
Vibration Modeling

The effects of spindles vibrational behavior on the stability lobes and the Chatter behavior of machine tools have been established. The service life has been observed to reducethe system natural frequencies. An analytical model of a multi-segment spinning spindle, based on the Dynamic Stiffness Matrix (DSM) formulation, exact within the limits of the Euler-Bernoulli beam bending theory, is developed. The system exhibits coupled Bending-Bending (B-B) vibration and its natural frequencies are found to decrease with increasing spinning speed. The bearings were included in the model usingboth rigid, simply supported, frictionless pins and flexible linear spring elements. The linear spring element stiffness is then calibrated so that the fundamental frequency of the system matches the nominal value.

Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach

2021 ◽  
Vol 24 (3) ◽  
pp. 895-922
Author(s):  
Platon G. Surkov
Keyword(s):  
Control Theory ◽  
Fractional Derivative ◽  
Approximate Calculation ◽  
Classical Problem ◽  
Tikhonov Regularization Method ◽  
Inaccurate Data ◽  
Local Modification ◽  
Ill Posed ◽  
Computational Errors ◽  
The Stability

Abstract A specific formulation of the “classical” problem of mathematical analysis is considered. This is the problem of calculating the derivative of a function. The purpose of this work is to construct an algorithm for the approximate calculation of the Caputo-type fractional derivative based on the methods of control theory. The input data of the algorithm is represented by inaccurate measured function values at discrete, frequently enough, times. The proposed algorithm is based on two aspects: a local modification of the Tikhonov regularization method from the theory of ill-posed problems and the Krasovskii extremal shift method from the guaranteed control theory, both of which ensure the stability to informational noises and computational errors. Numerical experiments were carried out to illustrate the operation of the algorithm.

A Refined Theory of Elastic, Orthotropic Plates

1958 ◽  
Vol 25 (4) ◽  
pp. 437-443 ◽  
Author(s):  
S. J. Medwadowski
Keyword(s):  
Body Force ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Refined Theory ◽  
System Of Equations ◽  
Order Partial Differential Equation ◽  
Type Solution ◽  
Orthotropic Plates ◽  
Dimensional Theory ◽  
Nonlinear System Of Equations

Abstract A refined theory of elastic, orthotropic plates is presented. The theory includes the effect of transverse shear deformation and normal stress and may be considered a generalization of the classical theory of von Karman modified by the refinements of the Levy-Reissner-Mindlin theories. A nonlinear system of equations is derived directly from the corresponding equations of the three-dimensional theory of elasticity in which body-force terms have been retained. Next, the system of equations is linearized and reduced to a single sixth-order partial differential equation in a stress function. A Levy-type solution of this equation is discussed.

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