The Mathematical Analysis of Bow Girders of Any Shape

1956 ◽  
Vol 23 (4) ◽  
pp. 522-526
Author(s):  
M. M. Abbassi
Keyword(s):  
Mathematical Analysis ◽  
Concentrated Loads ◽  
Middle Point ◽  
Circular Arc ◽  
Distributed Load ◽  
Approximate Formulas

Abstract By using parametric equations in which the parameter is the angle included between the tangent at any point on the bow girder and the tangent at the middle point, the analysis of bow girders of shapes other than the circular arc can be treated mathematically. Exact and approximate formulas are given for symmetrical bow girders of any shape carrying a distributed load or two equal concentrated loads placed symmetrically with respect to the middle point of the girder.

scholarly journals MECHANICAL CALCULATION OF FLEXIBLE WIRES WHEN THE CONCENTRATED FORCES ARE REPLACED BY A DISTRIBUTED LOAD

2018 ◽  
Vol 61 (2) ◽  
pp. 97-107 ◽  
Author(s):  
Yu. V. Bladyko
Keyword(s):  
Parametric Method ◽  
Climatic Conditions ◽  
Concentrated Loads ◽  
Mechanical Calculation ◽  
Overhead Lines ◽  
Distributed Load ◽  
Uniformly Distributed Load ◽  
Concentrated Forces ◽  
The Wire ◽  
The One

The objective of mechanical calculation of flexible wires of substations and overhead lines is to determine the sag and tension in different climatic conditions. A wire with a uniformly distributed load is considered as a homogeneous flexible thread having the form of a parabola. Concentrated loads from spacers, barrier balls, stubs, taps to electrical apparatus and other elements are replaced with the one distributed over the span. On behalf of a span without tension springs of insulators the action of concentrated loads on the wire is considered, an error is determined when replacing the concentrated forces with the one uniformly distributed along the span by the load. The sag for the equivalent wire is determined with the aid of the ratio of the increase of the sag, due to the presence of concentrated forces. An equation of state has been drawn up which makes it possible to determine the tension after changing the number of concentrated loads, e.g., after installing spacers, hanging the barrier balls, fixing the taps. The dependence of the maximum sag on the number of concentrated forces is given. The definition of the coefficient of concentrated forces as the ratio of the sum of the concentrated loads to the weight of the wire in the span is presented. A relationship between the load factors, the increase of the sag and the coefficient of concentrated forces is established. The formula has been deduced for determining the error in the replacement of concentrated forces by a uniformly distributed load along the span as a function of the number of concentrated forces and the coefficient of concentrated forces. A decrease in the error with an increase in the number of concentrated forces has been demonstrated. A more accurate calculation of mechanical tensions and sag is possible with the use of a vector-parametric method for calculating the flexible busbar of substations and air-line wires, where the design model of wires in the form of a flexible elastic thread is placed, taking into account the spatial disposition of all structural elements.

Camptograms for Beams in Compression

1947 ◽  
Vol 14 (3) ◽  
pp. A202-A208
Author(s):  
V. Rojansky ◽  
R. A. Beth
Keyword(s):  
Graphical Method ◽  
Flexural Rigidity ◽  
Bending Moment ◽  
Transverse Load ◽  
Circular Arc ◽  
Usual Assumption ◽  
Transverse Loads ◽  
Distributed Load ◽  
Circular Arcs ◽  
Stepwise Variation

Abstract Under the usual assumption that the square of the slope of the beam may be neglected compared to unity, the authors show that if the bending moment M is used as ordinate and a quantity proportional to dM/dx as abscissa, then the curve representing an axially compressed uniform beam carrying a uniformly distributed transverse load is a circular arc or a sequence of circular arcs. This result leads to a graphical method for evaluating bending moment. The procedure is illustrated by examples which include external torques, concentrated transverse loads, built-in ends, stepwise variation of distributed load, stepwise variation of flexural rigidity, and a protruding end. The diagrams, named “camptograms,” are simpler to draw and to interpret than the polar diagrams currently used for the same purpose. The construction of camptograms representing the slope and the deflection of the beam is outlined.

scholarly journals Accounting for Spacers in the Mechanical Calculation of Flexible Wires for Overhead Lines and Switchgears

2019 ◽  
Vol 62 (3) ◽  
pp. 219-231 ◽  
Author(s):  
Yu. V. Bladyko
Keyword(s):  
Concentrated Loads ◽  
Total Load ◽  
Mechanical Calculation ◽  
Overhead Lines ◽  
Distributed Load ◽  
Ice Loads ◽  
Concentrated Forces ◽  
The Difference

The mechanical calculation of flexible wires of overhead lines and switchgears, in which in-phase or phase-to-phase spacers are installed, is under consideration. Spacers are considered as concentrated loads acting on the split phase. The formulas for determining the sag are given for a different number of spacers as a function of their number and the coefficient of concentrated forces. This takes into account the difference in suspension heights, tension insulators strings, wind and ice loads. These formulas, being presented in a form that is convenient for consumers, can be used for computer execution of the mechanical calculation of flexible wires in different climatic regimes, both in the presence and in the absence of phase splitting. The errors of replacing the spacers with a distributed load are demonstrated. Formulas are proposed that give the smallest error when replacing spacers with a distributed load. The greater the value of the concentrated forces from the tap-off lines and loops, the greater the error in calculating the sag of the switchgears wires. Therefore, it is not possible to replace them with a distributed load obtained by simply dividing the total load by the length of the span in the presence of the tap-off lines and loops.

Density waves in disk galaxies

1967 ◽  
Vol 31 ◽  
pp. 313-317 ◽  
Author(s):  
C. C. Lin ◽  
F. H. Shu
Keyword(s):  
Mathematical Analysis ◽  
Spiral Structure ◽  
Stellar System ◽  
Collective Modes ◽  
Disk Galaxies ◽  
Spiral Pattern ◽  
Density Waves ◽  
The Galaxy ◽  
Detailed Mathematical Analysis

Density waves in the nature of those proposed by B. Lindblad are described by detailed mathematical analysis of collective modes in a disk-like stellar system. The treatment is centered around a hypothesis of quasi-stationary spiral structure. We examine (a) the mechanism for the maintenance of this spiral pattern, and (b) its consequences on the observable features of the galaxy.

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

Equilibrium Profile of Modern Belted Radial Ply Tires: Its Determination and Performance Benefits

2013 ◽  
Vol 41 (2) ◽  
pp. 127-151
Author(s):  
Rudolf F. Bauer
Keyword(s):  
Finite Element ◽  
Aspect Ratio ◽  
Finite Element Methods ◽  
Mathematical Analysis ◽  
Internal Stresses ◽  
Stress Concentrations ◽  
Equilibrium Profiles ◽  
Equilibrium Profile ◽  
And Performance ◽  
Beneficial Property

ABSTRACT The benefits of a tire's equilibrium profile have been suggested by several authors in the published literature, and mathematical procedures were developed that represented well the behavior of bias ply tires. However, for modern belted radial ply tires, and particularly those with a lower aspect ratio, the tire constructions are much more complicated and pose new problems for a mathematical analysis. Solutions to these problems are presented in this paper, and for a modern radial touring tire the equilibrium profile was calculated together with the mold profile to produce such tires. Some construction modifications were then applied to these tires to render their profiles “nonequilibrium.” Finite element methods were used to analyze for stress concentrations and deformations within all tires that did or did not conform to equilibrium profiles. Finally, tires were built and tested to verify the predictions of these analyses. From the analysis of internal stresses and deformations on inflation and loading and from the actual tire tests, the superior durability of tires with an equilibrium profile was established, and hence it is concluded that an equilibrium profile is a beneficial property of modern belted radial ply tires.

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