Series solution of large deformation of a beam with arbitrary variable cross section under an axial load

2010 ◽  
Vol 48 ◽  
pp. 10
Author(s):  
Shijun Liao
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Axial Load ◽  
Series Solution ◽  
Variable Cross Section ◽  
Variable Cross

scholarly journals Series solution of beams with variable cross-section

2020 ◽  
Vol 44 ◽  
pp. 489-496
Author(s):  
Valerio De Biagi ◽  
Bernardino Chiaia ◽  
Giuseppe Carlo Marano ◽  
Alessandra Fiore ◽  
Rita Greco ◽  
...  
Keyword(s):  
Cross Section ◽  
Series Solution ◽  
Variable Cross Section ◽  
Variable Cross

A Higher-Order Variable Cross-Section Viscoelastic Beam Element Via ANCF for Kinematic and Dynamic Analyses of Two-Link Flexible Manipulators

2017 ◽  
Vol 09 (08) ◽  
pp. 1750116 ◽  
Author(s):  
Haidong Yu ◽  
Chunzhang Zhao ◽  
Hui Zheng
Keyword(s):  
Large Deformation ◽  
Cross Section ◽  
Variable Cross Section ◽  
Beam Element ◽  
Higher Order ◽  
Flexible Manipulator ◽  
Variable Cross ◽  
Viscoelastic Beam ◽  
Shear Locking ◽  
Large Deformation Problems

A new viscoelastic beam element with variable cross-sections is developed based on the absolute nodal coordinate formulation, in which the higher-order slope coordinates are used to describe the variable geometric boundaries and circumvent possible shear-locking problem. The mass and stiffness matrices of the new element are derived by considering the variable geometrical boundary in the integration functions. The modified Kelvin–Voigt viscoelastic constitutive model for large deformation problems is introduced into the stiffness matrix. The dynamic model of a typical two-link manipulator with variable cross-section links is established where the constraint equations of revolute joints are considered with Lagrange multipliers. The kinematic trajectories of the manipulator with various materials and geometrical parameters are numerically studied. It is shown that the new element could circumvent shear-locking problem and yield improved accuracy and convergence compared with the conventional beam elements for solving large deformation problems. Also, the viscosity of the structural material helps to reduce the deformation of the links and improve the kinematic precision of the manipulator, hence the trajectory of the flexible manipulator could be controlled by changing the geometrical shape of the cross-section of links under the constraint of same mass.

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.

The Bent Strut with Variable Cross Section

1941 ◽  
Vol 45 (362) ◽  
pp. 51-66 ◽  
Author(s):  
Jean Drymael
Keyword(s):  
Cross Section ◽  
Axial Load ◽  
Graphical Method ◽  
Variable Cross Section ◽  
Transverse Load ◽  
Variable Cross ◽  
Transverse Loads ◽  
Variable Section

SummaryA graphical method is developed for solving the problem of the beam with variable section, bent by transverse loads and an axial load. The latter may be either tensile or compressive. The beam on statically determinate supports is first dealt with, successively as regards the stress and the deformation. Then a similar procedure is followed for a beam with redundant supports. Finally, the case is studied when there is no transverse load at all, that is buckling only.

Boundary element method in numerical study of three-dimensional Stokes flows in channels of arbitrary shape

2012 ◽  
Vol 9 (1) ◽  
pp. 94-97
Author(s):  
Yu.A. Itkulova
Keyword(s):  
Boundary Element Method ◽  
Boundary Element ◽  
Cross Section ◽  
Numerical Study ◽  
Three Dimensional ◽  
Cylindrical Tube ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Periodic Flows ◽  
Element Method

In the present work creeping three-dimensional flows of a viscous liquid in a cylindrical tube and a channel of variable cross-section are studied. A qualitative triangulation of the surface of a cylindrical tube, a smoothed and experimental channel of a variable cross section is constructed. The problem is solved numerically using boundary element method in several modifications for a periodic and non-periodic flows. The obtained numerical results are compared with the analytical solution for the Poiseuille flow.

Longitudinal oscillation of a rod with a variable cross section

2019 ◽  
Vol 14 (2) ◽  
pp. 138-141
Author(s):  
I.M. Utyashev
Keyword(s):  
Cross Section ◽  
Natural Frequencies ◽  
Liouville Problem ◽  
Variable Cross Section ◽  
Variable Cross ◽  
Longitudinal Vibrations ◽  
Cross Sectional ◽  
Independent Functions ◽  
The Cross ◽  
The Right

Variable cross-section rods are used in many parts and mechanisms. For example, conical rods are widely used in percussion mechanisms. The strength of such parts directly depends on the natural frequencies of longitudinal vibrations. The paper presents a method that allows numerically finding the natural frequencies of longitudinal vibrations of an elastic rod with a variable cross section. This method is based on representing the cross-sectional area as an exponential function of a polynomial of degree n. Based on this idea, it was possible to formulate the Sturm-Liouville problem with boundary conditions of the third kind. The linearly independent functions of the general solution have the form of a power series in the variables x and λ, as a result of which the order of the characteristic equation depends on the choice of the number of terms in the series. The presented approach differs from the works of other authors both in the formulation and in the solution method. In the work, a rod with a rigidly fixed left end is considered, fixing on the right end can be either free, or elastic or rigid. The first three natural frequencies for various cross-sectional profiles are given. From the analysis of the numerical results it follows that in a rigidly fixed rod with thinning in the middle part, the first natural frequency is noticeably higher than that of a conical rod. It is shown that with an increase in the rigidity of fixation at the right end, the natural frequencies increase for all cross section profiles. The results of the study can be used to solve inverse problems of restoring the cross-sectional profile from a finite set of natural frequencies.

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