Active earth thrust by backfills subject to a line surcharge

2005 ◽  
Vol 42 (5) ◽  
pp. 1255-1263 ◽  
Author(s):  
Venanzio R Greco
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Lateral Pressure ◽  
Analytical Form ◽  
Active State ◽  
Active Condition ◽  
Simple Analytical Form

The current analytical solution for the thrust computation when a line of vertical surcharge acts on the backfill behind a wall assumes the soils to be simultaneously elastic, for calculating part of the thrust, and in the active state of failure, for calculating the remaining part of the thrust. This paper gives a coherent analytical solution based on Coulomb's approach. The position of the active thrust is also given in a simple analytical form and compared with the approximate solution of Terzaghi.Key words: active earth force, cantilever walls, lateral pressure, active condition.

Modified Lanczos Method for Optimal Polynomial Approximation of the Solution of Differential Equations with Polynomial Coefficients

1991 ◽  
Vol 02 (01) ◽  
pp. 243-245
Author(s):  
A.S. BERDNICOV
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Approximate Analytical Solution ◽  
Analytical Form ◽  
Lanczos Method ◽  
Physical Models ◽  
Strict Solution ◽  
Polynomial Coefficients ◽  
Optimal Polynomial

The construction of an approximate analytical solution of a differential equation is an important task for a variety of physical models. A solution in analytical form allows to investigate the properties of a model, and the use of numerical methods allows to construct a parametrized approximate solution when the strict solution is absent.

Zur Lösung der BETHE-GOLDSTONE-Gleichung bei nicht-verschwindendem Gesamtimpuls I

1963 ◽  
Vol 18 (4) ◽  
pp. 531-538
Author(s):  
Dallas T. Hayes
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Nuclear Matter ◽  
Analytical Solutions ◽  
Projection Operator ◽  
Center Of Mass ◽  
Separable Potential ◽  
Localized Solutions ◽  
Non Local ◽  
Square Well

Localized solutions of the BETHE—GOLDSTONE equation for two nucleons in nuclear matter are examined as a function of the center-of-mass momentum (c. m. m.) of the two nucleons. The equation depends upon the c. m. m. as parameter due to the dependence upon the c. m. m. of the projection operator appearing in the equation. An analytical solution of the equation is obtained for a non-local but separable potential, whereby a numerical solution is also obtained. An approximate solution for small c. m. m. is calculated for a square-well potential. In the range of the approximation the two analytical solutions agree exactly.

scholarly journals Analytical Solution of a Circular Opening considering Nonuniform Pressure and Its Engineering Application

2020 ◽  
Vol 2020 ◽  
pp. 1-13
Author(s):  
Peng Wu ◽  
Yanlong Chen ◽  
Liang Chen ◽  
Xianbiao Mao ◽  
Wei Zhang
Keyword(s):  
Analytical Solution ◽  
Plastic Zone ◽  
Young’S Modulus ◽  
Lateral Pressure ◽  
Young's Modulus ◽  
Pressure Coefficient ◽  
Surrounding Rock ◽  
Engineering Practice ◽  
Circular Opening ◽  
Lateral Pressure Coefficient

Based on the Mohr–Coulomb criterion, a new analytical solution of a circular opening under nonuniform pressure was presented, which considered rock dilatancy effect and elastic-brittle-plastic failure characteristics. In the plastic zone, the attenuation of Young’s modulus was considered using a radius-dependent model (RDM), and solution of the radius and radial displacement of plastic zone was obtained. The results show that many factors have important impact on the response of the surrounding rock, including lateral pressure coefficient, dilation coefficient, buried depth, and Young’s modulus attenuation. Under nonuniform pressure condition, the distribution of plastic zone and deformation around the opening show obvious nonuniform characteristic: with the increasing of lateral pressure coefficient, the range of plastic zone and deformation decrease gradually at side, while they increase at roof and floor, and the location of the maximum value of support and surrounding rock response curve transfers from side to roof. Based on the analytical results and engineering practice, an optimization method of support design was proposed for the circular opening under nonuniform pressure.

scholarly journals ANALYTICAL SOLUTION OF THE FREQUENCY OF THE LOAD OSCILLATION AT AN ARBITRARY GIRDER NODE IN THE SYSTEM MAPLE

2019 ◽  
pp. 3-3
Author(s):  
Mikhail N. Kirsanov ◽  
Dmitriy V. Tinkov
Keyword(s):  
Analytical Solution ◽  
Arbitrary Number ◽  
Analytical Form ◽  
Labor Costs ◽  
Induction Method ◽  
Regular Type ◽  
Final Formula ◽  
The Common ◽  
Two Stages ◽  
Elastic Lattice

Introduction. We study the oscillations of a massive load on a planar statically definable symmetric truss of a regular type with parallel belts. Truss weight is not included. Free vertical oscillations are considered. The stiffness of the truss rods is assumed to be the same, the deformations are elastic. Lattice of the truss is double with descending braces and racks. New in the formulation and solution of the problem is the analytical form of the solution, which makes it possible in practice to easily evaluate the frequency characteristics of the structure depending on an arbitrary number of truss panels and the location of the load. Materials and methods. The operators and methods of the system of computer mathematics Maple are used. To determine the forces in the rods, the knotting method is used. The common terms of the sequence of coefficients of solutions for different numbers of panels are obtained from solving linear homogeneous recurrent equations of various order, obtained by special operators of the Maple system. Dependence on two arbitrary natural parameters is revealed in two stages. First, solutions for fixed load positions are found, then these solutions are summarized into one final formula for frequency. Results. By a series of individual solutions to the problem of load oscillation using the double induction method, it was possible to find common members of all sequences. The solution is polynomial in both natural parameters. Graphs constructed for particular cases, showed the adequacy of the approach. The discontinuous non-monotonic nature of the intermittent change depending on the number of truss panels and some other features of the solution are noted. Conclusions. It is shown that the induction method, previously applicable mainly to statics problems with one parameter (number of truss panels), is fully operational to the problems of the oscillations of system with two natural parameters. It should be noted that significant labor costs and a significant increase in the time symbolic transformations in such tasks

scholarly journals A New Approach for Solving the Undamped Helmholtz Oscillator for the Given Arbitrary Initial Conditions and Its Physical Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Alvaro H. Salas S ◽  
Jairo E. Castillo H ◽  
Darin J. Mosquera P
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Analytical Solution ◽  
Initial Conditions ◽  
Negative Ions ◽  
Nonlinear Problems ◽  
New Approach ◽  
Weierstrass Elliptic Function ◽  
Electronegative Plasma ◽  
The Given

In this paper, a new analytical solution to the undamped Helmholtz oscillator equation in terms of the Weierstrass elliptic function is reported. The solution is given for any arbitrary initial conditions. A comparison between our new solution and the numerical approximate solution using the Range Kutta approach is performed. We think that the methodology employed here may be useful in the study of several nonlinear problems described by a differential equation of the form z ″ = F z in the sense that z = z t . In this context, our solutions are applied to some physical applications such as the signal that can propagate in the LC series circuits. Also, these solutions were used to describe and investigate some oscillations in plasma physics such as oscillations in electronegative plasma with Maxwellian electrons and negative ions.

scholarly journals A Simple Free Energy for the Isotropic-Nematic Phase Transition of Rods

2016 ◽  
Vol 2016 ◽  
pp. 1-6 ◽  
Author(s):  
Remco Tuinier
Keyword(s):  
Free Energy ◽  
Nematic Phase ◽  
Analytical Form ◽  
Orientation Distribution ◽  
Scaled Particle Theory ◽  
Particle Theory ◽  
Aspect Ratios ◽  
Wide Range ◽  
Hard Rods ◽  
Simple Analytical Form

A free energy expression is proposed that describes the isotropic-nematic binodal concentrations of hard rods. A simple analytical form for this free energy was yet only available using a Gaussian trial function for the orientation distribution function (ODF), leading, however, to a significant deviation of the predicted binodals. The new free energy proposed here is based upon a rationalized correction to the orientational and packing entropies when using the Gaussian ODF. In combination with Parsons-Lee theory or scaled particle theory, it enables describing the isotropic-nematic phase coexistence concentrations of rods accurately using the simple Gaussian ODF for a wide range of aspect ratios.

scholarly journals The Use of Cubic Splines in the Numerical Solution of Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-16 ◽  
Author(s):  
W. K. Zahra ◽  
S. M. Elkholy
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Spline Function ◽  
Shooting Method ◽  
Polynomial Spline ◽  
Cubic Polynomial ◽  
Fractional Differential ◽  
Fractional Boundary Value Problems

Fractional calculus became a vital tool in describing many phenomena appeared in physics, chemistry as well as engineering fields. Analytical solution of many applications, where the fractional differential equations appear, cannot be established. Therefore, cubic polynomial spline-function-based method combined with shooting method is considered to find approximate solution for a class of fractional boundary value problems (FBVPs). Convergence analysis of the method is considered. Some illustrative examples are presented.

scholarly journals An Approximate Solution for Flow between Two Disks Rotating about Distinct Axes at Different Speeds

2007 ◽  
Vol 2007 ◽  
pp. 1-16 ◽  
Author(s):  
H. Volkan Ersoy
Keyword(s):  
Reynolds Number ◽  
Approximate Solution ◽  
Angular Velocity ◽  
Analytical Solution ◽  
Viscous Fluid ◽  
Approximate Analytical Solution ◽  
Velocity Difference ◽  
Small Angular Velocity

The flow of a linearly viscous fluid between two disks rotating about two distinct vertical axes is studied. An approximate analytical solution is obtained by taking into account the case of rotation with a small angular velocity difference. It is shown how the velocity components depend on the position, the Reynolds number, the eccentricity, the ratio of angular speeds of the disks, and the parameters satisfying the conditionsu=0andν=0in midplane.

The Shearing of a Rectangular Block Between Rough Plates

1953 ◽  
Vol 20 (2) ◽  
pp. 270-272
Author(s):  
J. W. Craggs
Keyword(s):  
Approximate Solution ◽  
Analytical Solution ◽  
Shear Stresses ◽  
Rectangular Block ◽  
Uniform Shear ◽  
Published Paper ◽  
General Method

Abstract In a recently published paper, W. T. Read gives an approximate solution of a problem in elasticity which occurs in the testing of materials. This is the problem of determining the stresses in a rectangular block when a pair of opposite faces are mutually displaced in the direction of an edge. The conditions are not those of uniform shear, for that would imply shear stresses on the free faces of the block. In the present paper an analytical solution of the same problem is obtained and the results are compared for a particular case. The method used is a particular case of the general method described by G. Pickett.

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