Bending of a Circular Beam Resting on an Elastic Foundation

1952 ◽  
Vol 19 (1) ◽  
pp. 1-4
Author(s):  
Enrico Volterra
Keyword(s):  
Explicit Form ◽  
Elastic Foundation ◽  
Initial Curvature ◽  
Elastic Foundations ◽  
Circular Beam

Abstract Saint Venant’s equations (1) for a circular beam, bent out of its plane of initial curvature, are applied to the study of the deflections of beams resting on elastic foundations and loaded by concentrated symmetric forces. The solution of the problem is given in explicit form, and tables for the deflections, angles of twist, bending and twisting moments are presented.

Buckling analysis of functionally graded annular sector plates resting on elastic foundations

2010 ◽  
Vol 225 (2) ◽  
pp. 312-325 ◽  
Author(s):  
A Naderi ◽  
A R Saidi
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Buckling Load ◽  
Critical Buckling Load ◽  
Elastic Foundations ◽  
Annular Sector ◽  
Sector Plate ◽  
Annular Sector Plate

In this study, an analytical solution for the buckling of a functionally graded annular sector plate resting on an elastic foundation is presented. The buckling analysis of the functionally graded annular sector plate is investigated for two typical, Winkler and Pasternak, elastic foundations. The equilibrium and stability equations are derived according to the Kirchhoff's plate theory using the energy method. In order to decouple the highly coupled stability equations, two new functions are introduced. The decoupled equations are solved analytically for a plate having simply supported boundary conditions on two radial edges. Satisfying the boundary conditions on the circular edges of the plate yields an eigenvalue problem for finding the critical buckling load. Extensive results pertaining to critical buckling load are presented and the effects of boundary conditions, volume fraction, annularity, plate thickness, and elastic foundation are studied.

On the Solution of Beam-on-elastic-foundation Problems by Means of a Mechanical Analogue

1950 ◽  
Vol 163 (1) ◽  
pp. 307-310 ◽  
Author(s):  
A. A. Wells
Keyword(s):  
Boundary Conditions ◽  
Elastic Foundation ◽  
Pressure Vessel ◽  
Finite Differences ◽  
Specific Problem ◽  
Elastic Shells ◽  
Elastic Foundations ◽  
Mechanical Analogue ◽  
Method Of Solution ◽  
Direct Use

The equation d4 y/ dx4- f(x)y + g(x) = 0 may be solved by means of the differential analyser, but only straightforwardly when the four boundary conditions are specified at one point. When the equation is associated with beams on elastic foundations, or elastic shells, the boundary conditions are more often specified at two points, and a quicker method of solution is desirable. In the analogue, direct use is made of the beam in the form of an elastic wire, supported at intervals in cradles on which weights may be made to simulate the terms f(x)y and g(x); the wire takes up a transversely deflected form which may be measured, and boundary conditions are imposed where they are required. A specific problem is examined and the results are shown to agree reasonably with the solution by calculation. A disadvantage when d2 y/dx2 is required is the inaccuracy inherent in differentiating by finite differences, but for engineering calculations the simplicity of the method may have its advantages. The solution of a typical pressure-vessel problem, by means of the analogue, is described.

Circular Beam on Elastic Foundation

1979 ◽  
Vol 105 (5) ◽  
pp. 839-853
Author(s):  
Dimitrios E. Panayotounakos ◽  
Pericles S. Theocaris
Keyword(s):  
Elastic Foundation ◽  
Beam On Elastic Foundation ◽  
Circular Beam

Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell

2018 ◽  
Vol 232 (24) ◽  
pp. 4564-4577 ◽  
Author(s):  
Muzamal Hussain ◽  
Muhammad Nawaz Naeem ◽  
Mohammad Reza Isvandzibaei
Keyword(s):  
Cylindrical Shell ◽  
Boundary Conditions ◽  
Elastic Foundation ◽  
Natural Frequencies ◽  
Cylindrical Shells ◽  
Functionally Graded ◽  
Radius Ratio ◽  
Wave Number ◽  
Simply Supported ◽  
Elastic Foundations

In this paper, vibration characteristics of rotating functionally graded cylindrical shell resting on Winkler and Pasternak elastic foundations have been investigated. These shells are fabricated from functionally graded materials. Shell dynamical equations are derived by using the Hamilton variational principle and the Langrangian functional framed from the shell strain and kinetic energy expressions. Elastic foundations, namely Winkler and Pasternak moduli are inducted in the tangential direction of the shell. The rotational motions of the shells are due to the Coriolis and centrifugal acceleration as well as the hoop tension produced in the rotating case. The wave propagation approach in standard eigenvalue form has been employed in order to derive the characteristic frequency equation describing the natural frequencies of vibration in rotating functionally graded cylindrical shell. The complex exponential functions, with the axial modal numbers that depend on the boundary conditions stated at edges of a cylindrical shell, have been used to compute the axial modal dependence. In our new investigation, frequency spectra are obtained for circumferential wave number, length-to-radius ratio, height-to-radius ratio with simply supported–simply supported and clamped–clamped boundary conditions without elastic foundation. Also, the effect of elastic foundation on the rotating cylindrical shells is examined with the simply supported–simply supported edge. To check the validity of the present method, the fundamental natural frequencies of non-rotating isotropic and functionally graded cylindrical shells are compared with the open literature. Also, a comparison is made for infinitely long rotating with the earlier published paper.

Auxetic Plates on Auxetic Foundation

2014 ◽  
Vol 974 ◽  
pp. 398-401 ◽  
Author(s):  
Teik Cheng Lim
Keyword(s):  
Elastic Foundation ◽  
Thick Plate ◽  
Plate Theory ◽  
Materials Selection ◽  
Selection Technique ◽  
Elastic Foundations ◽  
Design Considerations ◽  
Negative Poisson's Ratio ◽  
Plates On Elastic Foundation ◽  
Maximum Stresses

Auxetic solids are materials that exhibit negative Poisson’s ratio. This paper evaluates the maximum stresses in point-loaded (a) auxetic plates on conventional elastic foundation, (b) conventional plates on auxetic elastic foundation, and (c) auxetic plates on auxetic elastic foundation vis-à-vis conventional plates on conventional elastic foundation. Using thick plate theory for infinite plates on elastic foundation, it was found that in most cases the auxetic plates and auxetic foundation play the primary and secondary roles, respectively, in reducing the plate’s maximum stresses. It is herein suggested that, in addition to materials selection technique and other design considerations, the use of auxetic plates and/or auxetic foundation be introduced for reducing stresses in plates on elastic foundations.

scholarly journals Analysis of beams on elastic foundations using Green's functions

2021 ◽  
Vol 37 (2) ◽  
Author(s):  
J. Molina-Villegas ◽  
J. Ortega ◽  
A. Toro
Keyword(s):  
Finite Element Method ◽  
Elastic Foundation ◽  
Low Cost ◽  
Retaining Walls ◽  
Complex Structures ◽  
The Finite Element Method ◽  
Elastic Foundations ◽  
Stiffness Method ◽  
Model Foundation ◽  
Beams On Elastic Foundation

Beams on elastic foundation are basic elements within structural analysis, which are used to model foundation beams, foundation piles, retaining walls, and more complex structures that include some of these elements. For their analysis, the finite element method is usually used [1], which produces an approximate solution of the problem; and the Green's function stiffness method [2], which produces an exact solution. This article presents a methodology 100% based on the use of Green function's (response to a unit point force), to obtain the exact response of beams on elastic foundation. The main advantage of this formulation is its computational low cost compared to the aforementioned alternatives, and even for a large number of problems, it can be expressed only by means of sums and integrals, which can be easily performed numerically. Also, a great variety of Green function's for finite and infinite beams on elastic foundations with different boundary conditions are also presented, as well as some examples with the implementation of the proposed methodology.

Dynamic Behavior of a Moving Belt Supported on Elastic Foundation

1982 ◽  
Vol 104 (1) ◽  
pp. 143-147 ◽  
Author(s):  
R. B. Bhat ◽  
G. D. Xistris ◽  
T. S. Sankar
Keyword(s):  
Differential Equation ◽  
Dynamic Behavior ◽  
Elastic Foundation ◽  
Wave Velocity ◽  
Large Amplitude ◽  
Equation Of Motion ◽  
Numerical Techniques ◽  
Elastic Foundations ◽  
Spatial Response ◽  
The Stability

The dynamic behavior of a belt moving on an elastic foundation and supported on two pulleys at the ends is investigated. The problem is formulated to include the nonlinear terms arising from large amplitude oscillations as well as material damping and the variation in tension along the belt. The differential equation of motion is solved employing numerical techniques, and the spatial response variations with time are presented graphically for different belt velocities. These results indicate that in the absence of damping, the system is unstable for any belt velocity larger than the wave velocity in the belt material. The results are useful in investigating the stability of large continuous conveyor systems supported on elastic foundations.

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