scholarly journals SERIES SOLUTION OF LARGE DEFORMATION OF A BEAM WITH ARBITRARY VARIABLE CROSS SECTION UNDER AN AXIAL LOAD

2009 ◽  
Vol 51 (1) ◽  
pp. 10-33 ◽  
Author(s):  
SHIJUN LIAO
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Axial Load ◽  
Nonlinear Differential Equations ◽  
Variable Cross Section ◽  
Variable Coefficients ◽  
Buckling Load ◽  
Moment Of Inertia ◽  
Variable Cross ◽  
Critical Buckling Load

AbstractA general analytic approach is proposed for nonlinear eigenvalue problems governed by nonlinear differential equations with variable coefficients. This approach is based on the homotopy analysis method for strongly nonlinear problems. As an example, a beam with arbitrary variable cross section acted on by a compressive axial load is used to show its validity and effectiveness. This approach provides us with great freedom to transfer the original nonlinear buckling equation with variable coefficients into an infinite number of linear differential equations with constant coefficients that are much easier to solve. More importantly, it provides us with a convenient way to guarantee the convergence of solution series. As an example, the beam displacement and the critical buckling load can be obtained for arbitrary variable cross sections. The influence of nonuniformity of moment of inertia is investigated in detail and the optimal distributions of moment of inertia are studied. It is found that the critical buckling load of a beam with the optimal distribution of moment of inertia can be approximately 21–22% larger than that of a uniform beam with the same average moment of inertia. Mathematically, this approach is rather general and thus can be used to solve many other linear/nonlinear differential equations with variable coefficients.

Nonlinear Coupled Transverse and Axial Vibration of Variable Cross-Section Beam Flexures Interconnecting Rigid Body

2016 ◽  
Author(s):  
Hasan Malaeke ◽  
Hamid Moeenfard ◽  
Amir H. Ghasemi
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Cross Section ◽  
Multiple Scales ◽  
Nonlinear Behavior ◽  
Single Mode ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Closed Form Solutions ◽  
Dynamic Performances

The objective of this paper is to analytically study the nonlinear behavior of variable cross-section beam flexures interconnecting an eccentric rigid body. Hamilton’s principle is utilized to obtain the partial differential equations governing the nonlinear vibration of the system as well as the corresponding boundary conditions. Using a single mode approximation, the governing equations are reduced to a set of two nonlinear ordinary differential equations in terms of end displacement components of the beam which are coupled due to the presence of the transverse eccentricity. The method of multiple scales are employed to obtain parametric closed-form solutions. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. These analytical expressions provide design insights for modeling and optimization of more complex flexure mechanisms for improved dynamic performances.

scholarly journals INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS

2019 ◽  
Vol 81 (4) ◽  
pp. 449-460
Author(s):  
V.V. Saurin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Cross Section ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
New Variables ◽  
Differential Relations

Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.

Buckling Load of Columns with Same Volume and Length but Variable Cross-section along the Length

2015 ◽  
Vol 6 (3) ◽  
pp. 77-85
Author(s):  
Hong-Kyu Lee ◽  
◽  
Jong-Ho Yoo ◽  
Seung-Won Lee ◽  
Sun-Hee Kim ◽  
...  
Keyword(s):  
Cross Section ◽  
Variable Cross Section ◽  
Buckling Load ◽  
Variable Cross

scholarly journals Analysis of thin-walled beams with variable monosymmetric cross section by means of Legendre polynomials

2019 ◽  
Vol 41 (1) ◽  
pp. 1-12
Author(s):  
Józef Szybiński ◽  
Piotr Ruta
Keyword(s):  
Cross Section ◽  
Legendre Polynomials ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Coefficients ◽  
Beam Axis ◽  
Geometrical Parameters ◽  
Beam Model ◽  
Variable Cross ◽  
Thin Walled

AbstractThis article deals with the vibrations of a nonprismatic thin-walled beam with an open cross section and any geometrical parameters. The thin-walled beam model presented in this article was described using the membrane shell theory, whilst the equations were derived based on the Vlasov theory assumptions. The model is a generalisation of the model presented by Wilde (1968) in ‘The torsion of thin-walled bars with variable cross-section’, Archives of Mechanics, 4, 20, pp. 431–443. The Hamilton principle was used to derive equations describing the vibrations of the beam. The equations were derived relative to an arbitrary rectilinear reference axis, taking into account the curving of the beam axis and the axis formed by the shear centres of the beam cross sections. In most works known to the present authors, the equations describing the nonprismatic thin-walled beam vibration problem do not take into account the effects stemming from the curving (the inclination of the walls of the thin-walledcross section towards to the beam axis) of the analysed systems. The recurrence algorithm described in Lewanowicz’s work (1976) ‘Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series’, Applicationes Mathematicae, XV(3), pp. 345–396, was used to solve the derived equations with variable coefficients. The obtained solutions of the equations have the form of series relative to Legendre polynomials. A numerical example dealing with the free vibrations of the beam was solved to verify the model and the effectiveness of the presented solution method. The results were compared with the results yielded by finite elements method (FEM).

scholarly journals INTEGRAL-DIFFERENTIAL RELATIONS IN THE PROBLEM OF FREE BENDING VIBRATIONS OF VARIABLE CROSS-SECTION BEAMS

2019 ◽  
Vol 81 (4) ◽  
pp. 449-461
Author(s):  
V.V. Saurin
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Cross Section ◽  
Boundary Value ◽  
Approximate Solutions ◽  
Variable Cross Section ◽  
Free Vibrations ◽  
Variable Cross ◽  
New Variables ◽  
Differential Relations

Issues related to eigen-vibrations of elastic beams of variable cross-section are discussed. It is noted that one of the common features characteristic of boundary-value problems of mathematical physics is certain ambiguity of their formulations. A boundary-value problem of determining eigen-frequencies of a variable cross-section beam in displacements is formulated. By introducing new variables characterizing the behavior of the system, the boundary-value problem is reduced to three ordinary differential equations with variable coefficients. The new variables have a distinct physical meaning. One of the functions is linear density of the pulse and the other is bending moment in the cross-section of the beam. Such a formulation of the problem of free vibrations of a variable cross-section beam makes it possible to reduce the system of differential equations to a single fourth-order equation written in terms of pulse functions. This equation is equivalent to the initial one, formulated in displacements, but has a different form. A method of integral-differential relations, alternative to classical numerical approaches, is described. The possibility of constructing various bilateral energy-based evaluations of the accuracy of approximate solutions resulting from the method of integral-differential relations is studied. The projection approach to analyzing spectral problems of nonlinear beam theory is considered. The efficiency of the method of integral-differential equations is demonstrated, using the problem of free vibrations of a rectangular beam with a constructional depth quadratically varying along its length. Energy-based evaluations of the accuracy of the approximate solutions constructed using polynomial approximations of the sought functions are presented. It is shown that applying standard Bubnov-Galerkin's method to the problem of free vibrations leads to the appearance of complex eigen-frequencies. At the same time, the ratio of the imaginary component to the real one of the eigen-value is a relative inaccuracy of the solution of the boundary-value problem. The introduced numerical algorithm makes it possible to evaluate unambiguously the local and integral quality of numerical solutions obtained.

scholarly journals Lateral-torsional buckling of compressed and highly variable cross section beams

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Ida Mascolo ◽  
Mario Pasquino
Keyword(s):  
Differential Equations ◽  
Cross Section ◽  
Equilibrium Shape ◽  
Variable Cross Section ◽  
Thin Wall ◽  
Variable Cross ◽  
Total Potential ◽  
Energetic Formulation ◽  
Rayleigh Method ◽  
Minimum Total Potential Energy

AbstractIn the critical state of a beam under central compression a flexural-torsional equilibrium shape becomes possible in addition to the fundamental straight equilibrium shape and the Euler bending. Particularly, torsional configuration takes place in all cases where the line of shear centres does not correspond with the line of centres of mass. This condition is obtained here about a z-axis highly variable section beam; with the assumptions that shear centres are aligned and line of centres is bound to not deform. For the purpose, let us evaluate an open thin wall C-cross section with flanges width and web height linearly variables along z-axis in order to have shear centres axis approximately aligned with gravity centres axis. Thus, differential equations that govern the problem are obtained. Because of the section variability, the numerical integration of differential equations that gives the true critical load is complex and lengthy. For this reason, it is given an energetic formulation of the problem by the theorem of minimum total potential energy (Ritz-Rayleigh method). It is expected an experimental validation that proposes the model studied.

scholarly journals Elastic Buckling Behaviour of a Four-Lobed Cross Section Cylindrical Shell with Variable Thickness under Non-Uniform Axial Loads

2009 ◽  
Vol 2009 ◽  
pp. 1-17 ◽  
Author(s):  
Mousa Khalifa Ahmed
Keyword(s):  
Cylindrical Shell ◽  
Differential Equations ◽  
Transfer Matrix ◽  
Cross Section ◽  
Variable Thickness ◽  
Nonlinear Differential Equations ◽  
Longitudinal Direction ◽  
Variable Coefficients ◽  
Thickness Variation ◽  
Matrix Differential

The static buckling of a cylindrical shell of a four-lobed cross section of variable thickness subjected to non-uniform circumferentially compressive loads is investigated based on the thin-shell theory. Modal displacements of the shell can be described by trigonometric functions, and Fourier's approach is used to separate the variables. The governing equations of the shell are reduced to eight first-order differential equations with variable coefficients in the circumferential coordinate, and by using the transfer matrix of the shell, these equations can be written in a matrix differential equation. The transfer matrix is derived from the nonlinear differential equations of the cylindrical shells by introducing the trigonometric series in the longitudinal direction and applying a numerical integration in the circumferential direction. The transfer matrix approach is used to get the critical buckling loads and the buckling deformations for symmetrical and antisymmetrical shells. Computed results indicate the sensitivity of the critical loads and corresponding buckling modes to the thickness variation of cross section and the radius variation at lobed corners of the shell.

scholarly journals Work of a railway sleeper as a structure with variable cross-section - the issue of an equivalent cross-section

2018 ◽  
Vol 2018 (6) ◽  
pp. 1-12
Author(s):  
Włodzimierz Czyczuła ◽  
Dorota Błaszkiewicz ◽  
Małgorzata Urbanek
Keyword(s):  
Numerical Model ◽  
Cross Section ◽  
Real Variable ◽  
Variable Cross Section ◽  
Moment Of Inertia ◽  
Variable Cross ◽  
Bending Stress ◽  
Variable Section ◽  
Vertical Displacements ◽  
Railway Sleeper

Abstract: The article presents an analysis of the work of a sleeper as a construction with variable section, and of the method of determining an equivalent section, constant throughout the length, the utilisation of which would have similar shapes of deflection and bending stress lines in relation to the real, variable cross section. Using an analytical and a numerical model, vertical displacements and stresses for two types of sleepers – PS-94 and PS-08 – were determined. The comparison of the methods allows for calculating an equivalent moment of inertia for analytical calculations, specifically the dynamic ones.

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