Dynamic Response of an Infinite Beam

1975 ◽  
Vol 97 (2) ◽  
pp. 107-109 ◽  
Author(s):  
H. Durlofsky
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Dynamic Response ◽  
Elastic Foundation ◽  
Fourier Transforms ◽  
Euler Beam ◽  
Concentrated Load ◽  
Conservation Of Energy ◽  
Infinite Beam ◽  
Good Agreement

Both the exact and an approximate solution for the dynamic response of an infinite Bernoulli-Euler beam under an instantaneously applied, concentrated load are presented in this paper. The exact solution is obtained by means of complex Fourier transforms. The approximate solution is obtained by assuming the dynamic response has the form of a deflected infinite beam on an elastic foundation, with wavelength a function of time. This assumption is motivated by the similarity between the dynamic response problem and the problem of an infinite beam on an elastic foundation. A governing equation for the wavelength in the assumed response is derived by application of the principle of conservation of energy, and solved by straightforward methods. A comparison of the two solutions shows good agreement near the point of loading. Results applicable to pipe whip problems are presented.

Bending of an Infinite Beam on an Elastic Foundation

1937 ◽  
Vol 4 (1) ◽  
pp. A1-A7 ◽  
Author(s):  
M. A. Biot
Keyword(s):  
Elastic Foundation ◽  
Poisson Ratio ◽  
Elementary Theory ◽  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Concentrated Load ◽  
Elastic Continuum ◽  
Infinite Beam ◽  
A Value

Abstract The elementary theory of the bending of a beam on an elastic foundation is based on the assumption that the beam is resting on a continuously distributed set of springs the stiffness of which is defined by a “modulus of the foundation” k. Very seldom, however, does it happen that the foundation is actually constituted this way. Generally, the foundation is an elastic continuum characterized by two elastic constants, a modulus of elasticity E, and a Poisson ratio ν. The problem of the bending of a beam resting on such a foundation has been approached already by various authors. The author attempts to give in this paper a more exact solution of one aspect of this problem, i.e., the case of an infinite beam under a concentrated load. A notable difference exists between the results obtained from the assumptions of a two-dimensional foundation and of a three-dimensional foundation. Bending-moment and deflection curves for the two-dimensional case are shown in Figs. 4 and 5. A value of the modulus k is given for both cases by which the elementary theory can be used and leads to results which are fairly acceptable. These values depend on the stiffness of the beam and on the elasticity of the foundation.

scholarly journals Computational Methods for Solving Linear Fuzzy Volterra Integral Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Jihan Hamaydi ◽  
Naji Qatanani
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Numerical Methods ◽  
Volterra Integral Equation ◽  
Taylor Expansion ◽  
Variational Iteration Method ◽  
Variational Iteration ◽  
Iteration Methods ◽  
Good Agreement

Two numerical schemes, namely, the Taylor expansion and the variational iteration methods, have been implemented to give an approximate solution of the fuzzy linear Volterra integral equation of the second kind. To display the validity and applicability of the numerical methods, one illustrative example with known exact solution is presented. Numerical results show that the convergence and accuracy of these methods were in a good agreement with the exact solution. However, according to comparison of these methods, we conclude that the variational iteration method provides more accurate results.

scholarly journals The elastic stability of a thin twisted strip—II

1937 ◽  
Vol 161 (905) ◽  
pp. 197-220 ◽  
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Main Part ◽  
Fourier Expansion ◽  
Elastic Stability ◽  
Numerical Work ◽  
The Stability ◽  
Good Agreement

I—In a previous paper the present writer discussed both theoretically and experimentally the equilibrium and elastic stability of a thin twisted strip, and the results obtained by the theory were found to be in good agreement with observation. It has, however, been pointed out by Professor Southwell, F. R. S., that the solution of the stability equations which was given in that paper may only be regarded as an approximate solution for, although it satisfies exactly the differential equations and two boundary conditions along the edge of the strip, it only satisfies the two remaining boundary conditions approximately. The author has also noticed that the coefficients n a m in the Fourier expansion of θ 2 cos mθ which were used in A are incorrect when m = 0, and this has led to errors in the numerical work so that the values of ᴛb 2 / π 2 h which are given in Table I of A are wrong. In the present paper a solution of the stability equations is obtained which satisfies all the boundary conditions. This solution is very much more complicated than the approximate solution and much greater labour is required for the numerical work. The numerical work for the approximate solution of A has also been revised and the corrected results are given in 9, 10. It is found that the results for the approximate solution are in good agreement with those obtained from the exact solution and that both agree moderately well with the experimental results which are given in A. The main part of this paper is an extension of the previous work and is concerned with the stability of a thin twisted strip when it is subjected to a tension along its length. The theory has been compared with experiment and satisfactorily good agreement between them was found.

Shear Deformation in Beams on Elastic Foundations

1962 ◽  
Vol 29 (2) ◽  
pp. 313-317 ◽  
Author(s):  
F. Essenburg
Keyword(s):  
Shear Deformation ◽  
Elastic Foundation ◽  
Cross Section ◽  
Transverse Shear ◽  
Elastic Beam ◽  
Practical Significance ◽  
Transverse Shear Deformation ◽  
Concentrated Load ◽  
Infinite Beam ◽  
Concentrated Couple

The importance of the effect of transverse shear deformation in the flexure of an elastic beam of symmetric cross section, constrained by a Winkler-type elastic foundation, is found to depend upon both the elastic properties of the beam and the foundation and the geometry of the beam cross section. Under certain conditions the form of the solution is substantially altered and the periodic character predicted by the classical treatment is not present. The practical significance of these modifications is illustrated by means of the specific examples of an infinite beam under concentrated load and an infinite beam under concentrated couple.

The Analysis Method of Highline Cable of Alongside Replenishment System Based on Suspended Cable Theory

2012 ◽  
Vol 490-495 ◽  
pp. 633-637 ◽  
Author(s):  
Kai Wei ◽  
Liang Xin Zhang ◽  
Ai Di Ren
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Static Analysis ◽  
Mechanical Characteristics ◽  
Analysis Method ◽  
Mathematical Methods ◽  
Concentrated Load ◽  
Suspended Cable ◽  
Cable Theory

According to mechanical characteristics of highline cable of alongside replenishment system at sea, two mathematical methods for static analysis of the highline cable under its own weight are investigated: catenary method, which is the exact solution, and parabolic method, which is the approximate solution but simpler to calculate. A model consisting of two sections of catenary is established for the analysis of the highline cable under a concentrated load. The expressions of the length and the tension of the highline cable are derived, and the result of each section is obtained.

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