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 A hybrid analytical-numerical solution for a circular pile under lateral load in multilayered soil

2020 ◽  
Vol 14 (1) ◽  
pp. 1-14
Author(s):  
Nghiem Xuan Hien
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Numerical Solution ◽  
Elastic Foundation ◽  
Three Dimensional ◽  
Bending Moment ◽  
Lateral Load ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element

A hybrid analytical-numerical solution is proposed to solve the problem of a laterally loaded pile with a circular cross-section in multilayered soils. In the pile-soil model, the lateral load is located at the pile head including both lateral force and bending moment. The single pile is considered as a beam on elastic foundation while shear beams model the soil column below the pile toe. The differential equations governing pile deflections are derived based on the energy principles and variational approaches. The differential equations are solved iteratively by using the finite element method that provides results of pile deflection, rotation angle, shear force, and bending moment along the pile and equivalent stiffness of the pile-soil system. The modulus reduction equation is also developed to match the proposed results well to the three-dimensional finite element analyses. Several examples are conducted to validate the proposed method by comparing the analysis results with those of existing analytical solutions, the three-dimensional finite element solutions. Keywords: beam on elastic foundation; finite element method; pile; energy principle; lateral load.

Exact, Hierarchical Solutions for Localized Loadings in Isotropic, Laminated, and Sandwich Shells

2009 ◽  
Vol 131 (4) ◽  
Author(s):  
E. Carrera ◽  
G. Giunta
Keyword(s):  
Three Dimensional ◽  
Bending Moment ◽  
Two Dimensional ◽  
Application Area ◽  
Spherical Shells ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
Transverse Pressure ◽  
Shell Models ◽  
Stress And Deformation

This paper presents closed form solutions for simply supported cylindrical and spherical shells subjected to uniform localized distributions of transverse pressure and bending moment. These distributions have been expanded in terms of Fourier’s series for which Navier type “exact” solutions have been found for the governing differential equations of the employed shell theories. Shells made of isotropic materials, composites laminates, and sandwich have been analyzed. Carrera’s unified formulation has been adopted in order to implement a large variety of two-dimensional theories. Classical, refined, zigzag, layerwise, and mixed theories are compared in order to evaluate the stress and deformation variables. Conclusions are drawn with respect to the accuracy of the various theories for the considered loadings and layouts. The importance of the refined shell models in order to describe accurately the three-dimensional stress state in the neighborhood of the localized loading application area is outlined.

Three-dimensional and multicellular steady and unsteady convection in fluid-saturated porous media at high Rayleigh numbers

1979 ◽  
Vol 94 (1) ◽  
pp. 25-38 ◽  
Author(s):  
Gerald Schubert ◽  
Joe M. Straus
Keyword(s):  
Single Cell ◽  
Cross Section ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Arbitrary Length ◽  
Dimensional Flow ◽  
A Value ◽  
Square Cylinders ◽  
Two Dimensional Flow ◽  
Rayleigh Numbers

In an effort to determine the characteristics of the various types of convection that can occur in a fluid-saturated porous medium heated from below, a Galerkin approach is used to investigate three-dimensional convection in a cube and two-dimensional convection in a square cross-section. Strictly two-dimensional, single-cell flow in a square cross-section is steady for Rayleigh numbers R between 4π2 and a critical value which lies between 300 and 320; it is unsteady at higher values of R. Double-cell, two-dimensional flow in a square cross-section becomes unsteady when R exceeds a value between 650 and 700, and triple-cell motion is unsteady for R larger than a value between 800 and 1000. Considerable caution must be exercised in attributing physical reality to these flows. Strictly two-dimensional, steady, multicellular convection may not be realizable in a three-dimensional geometry because of instability to perturbations in the orthogonal dimension. For example, even though single-cell, two-dimensional convection in a square cross-section is steady at R = 200, it cannot exist in either an infinitely long square cylinder or in a cube. It could exist, however, in a cylinder whose length is smaller than 0.38 times the dimension of its square cross-section. Three-dimensional convection in a cube becomes unsteady when R exceeds a value between 300 and 320, similar to the unicellular two-dimensional flow in a square cross-section. Nusselt numbers Nu, generally accurate to 1%, are given for the strictly two-dimensional flows up to R = 1000 and for three-dimensional convection in cubes up to R = 500. Single-cell, two-dimensional, steady convection in a square cross-section transports the most heat for R < 97; this mode of convection is also stable in square cylinders of arbitrary length including the cube for R < 97. Steady three-dimensional convection in cubes transports more heat for 97 [lsim ] R [lsim ] 300 than do any of the realizable two-dimensional modes. At R [gsim ] 300 the unsteady modes of convection in both square cylinders and cubes involve wide variations in Nu.

High-resolution RHEED analysis of dynamics of low-temperature superstructure transitions in Ge/Si(001) epitaxial system

2021 ◽  
Vol 33 (11) ◽  
pp. 115603
Author(s):  
Vladimir V Dirko ◽  
Kirill A Lozovoy ◽  
Andrey P Kokhanenko ◽  
Alexander V Voitsekhovskii
Keyword(s):  
Critical Thickness ◽  
Three Dimensional ◽  
Diffraction Method ◽  
High Energy ◽  
Two Dimensional ◽  
A Value ◽  
Wide Range ◽  
Dimensional Growth ◽  
Analysis Of Dynamics ◽  
First Time

Abstract In this paper, we analyze superstructural transitions during epitaxial growth of two-dimensional layers and the formation of quantum dots by the Stranski–Krastanov mechanism in elastically stressed systems by the reflection high-energy electron diffraction method. Detailed dependences of the periodicity parameter N of the 2 × N reconstruction on the effective thickness of the deposited material in a wide range of growth temperatures during epitaxy of germanium on a silicon surface with a crystallographic orientation (001) are obtained. Superstructural transitions and the change in the value of the parameter N at low temperatures of epitaxy in this system have been investigated for the first time. It is shown that the length of dimer rows in such a reconstruction during the growth of pure germanium on silicon can reach a value of no less than N = 11. A relationship is found between the value of the parameter N, determined by elastic strains in the system, and the critical thickness of the transition from two-dimensional to three-dimensional growth. Based on this relationship, a physical mechanism is proposed that explains the nature of the temperature dependence of the critical thickness of the Stranski–Krastanov transition, which has been the subject of constant scientific disputes until now.

scholarly journals Equilibrium of Two-Dimensional Cycloidal Pantographic Metamaterials in Three-Dimensional Deformations

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1523 ◽  
Author(s):  
Daria Scerrato ◽  
Ivan Giorgio
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Variational Formulation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Element Analysis ◽  
Elastic Continuum ◽  
Bias Extension ◽  
Three Dimensional Space

A particular pantographic sheet, modeled as a two-dimensional elastic continuum consisting of an orthogonal lattice of continuously distributed fibers with a cycloidal texture, is introduced and investigated. These fibers conceived as embedded beams on the surface are allowed to be deformed in a three-dimensional space and are endowed with resistance to stretching, shearing, bending, and twisting. A finite element analysis directly derived from a variational formulation was performed for some explanatory tests to illustrate the behavior of the newly introduced material. Specifically, we considered tests on: (1) bias extension; (2) compressive; (3) shear; and (4) torsion. The numerical results are discussed to some extent. Finally, attention is drawn to a comparison with other kinds of orthogonal lattices, namely straight, parabolic, and oscillatory, to show the differences in the behavior of the samples due to the diverse arrangements of the fibers.

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.

An Interferometer Based Experimental Technique to Evaluate Large Strains and Springback on Sheet Metal

2013 ◽  
Author(s):  
Luis Rafael Sanchez ◽  
Shannon Peterson ◽  
Carl G. Simonsen ◽  
Abrar Satar
Keyword(s):  
Sheet Metal ◽  
Three Dimensional ◽  
Bending Moment ◽  
Large Strains ◽  
Two Dimensional ◽  
Strain Gages ◽  
Additional Information ◽  
Software Algorithms ◽  
Dimensional Measurements ◽  
Vertical Scanning Interferometry

A technique was successfully developed to measure large tensile, compressive strains, springback and strain reversal effects on sheet metal bent to small radii. Vertical Scanning Interferometry (VSI) was used to measure three dimensional data from surfaces with sides varying from 160 nm to 2 mm. Software algorithms were utilized to determine surface topography maps from three-dimensional curved locations and to represent them in a two dimensional plane. Fine reference marks were engraved on both sides of sample. The sample was bent /unbent to small radii under a pure bending moment. Outer strains were calculated from VSI two-dimensional measurements of the original and final lengths between the reference marks. Strain gages, applied at locations close to the reference marks, gave additional information at the elasto-plastic range. Experimental data collected included bending moment as a function of strain, 3-D curvature profiles, springback and reverse bending effects. The technique was proved useful for the experimental evaluation and theoretical validation of bending and springback properties of sheet metal. Experimental results for aluminum and steel alloys are presented.

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.

Service Factor Assessment of a Great Lakes Bulk Carrier Incorporating the Effects of Hydroelasticity

2009 ◽  
Vol 46 (02) ◽  
pp. 116-121
Author(s):  
Spyridon E. Hirdaris ◽  
Norbert Bakkers ◽  
Nigel White ◽  
Pandeli Temarel
Keyword(s):  
Rigid Body ◽  
Great Lakes ◽  
Body Wave ◽  
Three Dimensional ◽  
Bending Moment ◽  
Suez Canal ◽  
Two Dimensional ◽  
Bulk Carrier ◽  
Service Factor

This paper presents a summary of an investigation into the effects of hull flexibility when deriving an equivalent service factor for a single passage of a Great Lakes Bulk Carrier from the Canadian Great Lakes to China. induced bending moment predicted using traditional three-dimensional rigid body hydrodynamic methods is augmented due to the effects of springing and whipping by including allowances based on two-dimensional hydroelasticity predictions across a range of headings and sea states. The analysis results are correlated with full scale measurements that are available for this ship. By combining the long term "rigid body" wave-bending moment with the effects of hydroelasticity, a suitable service factor is derived for a Great Lakes Bulk Carrier traveling from the Canadian Great Lakes to China via the Suez Canal.

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