Electrical Resistance Network Research Articles

Creep deformation of statically indeterminate beams analysed by an electrical analogue

The creep bending of statically indeterminate beams is studied by an electrical resistance network. The moment-curvature relationship due to creep, which is non-linear and time-dependent, is expressed in graphical form. An elastic solution is first obtained and is modified to allow for the creep curvature. An iterative method has to be adopted and the technique of obtaining convergence is discussed. Particular problems which are considered include the propped cantilever and a two-span continuous built-in beam.

Predicted and experimental water table drawdown during tile drainage

Comparisons were made between experimental and predicted positions of the water table during drawdown by drains. Experimental positions were obtained from tank drainage data. The predicted water table positions were determined with a modified form of the Kirkham-Gaskell equation for drawdown. An electrical resistance network was used to obtain the components of the potential gradient which are required by this equation. Two modifications were made in the Kirkham-Gaskell equation: The capillary fringe replaced the water table as the upper boundary of the region in which flow occurred. Instead of using a constant drainable porosity, one was used which related drainable pore space to the water table depth. The agreement between experimental and predicted positions of the water table was good. When either of the aforementioned modifications was not made, significant discrepancies were noted. There was considerable deviation in both shape and mean position of the water table when a constant porosity was used in the theory. The proposed equation for drawdown offers a sound basis for studying drawdown by drains. An electrical resistance network as used in this study is convenient for eliminating the tedious calculations which are required by the equation. Because of extensive calculations, the theory is still not suited for design purposes. It will serve its greatest role in evaluating more practical drawdown equations.

Theoretical Aspects of Flow Above the Water Table in Tile Drainage of Shallow Homogeneous Soils

AbstractHypothetical conductivity‐tension relationships were employed in a study to determine the effect of the thickness of the capillary fringe and the relative conductivity above the capillary fringe on drainage flow. Solutions were obtained with an electrical resistance network using equivalent conductivities.A capillary fringe of 50 cm. increased the drainage coefficients in shallow soil by 24 to 57%, depending on the height of the water table. Smaller increases can be expected with decreasing thickness of the fringe and increasing depth of the impermeable layer. Agreement existed between the results of the network analyses and calculated drainage coefficients, including the effect of the capillary fringe with Hooghoudt's method.The effect of the flow above the capillary fringe on the drainage coefficient appeared to be small so that the flux across the top of the capillary fringe can be assumed to be uniform. Steady‐state drainage coefficients, therefore, can be used to calculate the rate of fall of the water table after cessation of rain when the region near the top of the capillary fringe becomes a source of almost uniform flow because of drainage of pore space.For a certain water‐table position midway between drains, the water table in the vicinity of the drains appeared to be higher with increasing thickness of the capillary fringe. Since hysteresis causes the fringe thickness to be greater for falling water tables than for rising water tables, lower water tables near the drains can be expected with the latter.

A Unifying Numerical Solution for Two‐Dimensional Steady Flow Problems in Porous Media With an Electrical Resistance Network

AbstractNumerical solutions of two‐dimensional steady flow systems that are partially or completely unsaturated involve satisfying the tension vs. conductivity relations of the media in addition to some approximative expression of the Laplace equation for nonhomogeneous conductivity. Where tensions and conductivities are initially unknown, assumed conductivity values are used in the first relaxation procedure after which the correct values are determined by a process of repeated relaxation analyses with conductivity to tension adjustments after each relaxation. Relatively rapid solutions can be obtained by use of an electrical resistance network for relaxing the system.A numerical procedure is developed for expressing the equivalent conductivity of the medium between any two adjacent points of the network in terms of the conductivities or tensions at the two points. This procedure is independent of the nature of the relation between tension and conductivity and utilizes experimentally determined values in the form of graphs or tables.The equivalent conductivity between adjacent points of the network differs from the arithmetic mean of the conductivities at the points, especially when the conductivity values are relatively far apart. The arithmetic mean appears to be an accurate measure of the equivalent conductivity only in event of horizontal flow with a linear relation between tension and conductivity.Examples of completed analyses include a solution of flow in tile‐drained soil with the saturated and unsaturated zone united into one system and a solution of unsaturated subirrigation flow from perforated underground tubing in the presence of a deep water table. The shape of the water table in the drainage example was immaterial to the solution of the problem, but it could be evaluated after the analysis as the atmospheric‐pressure contour. The streamlines above the water table deviated from the vertical direction causing the flow rate across the water table to be nonuniform.

Experiments with an Electrical Analogue for the Extension and Flexure of Flat Plates

SummaryDetails of the design, construction and use of an electrical resistance network for solving certain of the problems encountered in the flexure and extension of flat plates are given. Some typical experiments have been carried out with the aid of this network and the method of operation of the analogy is described. The results of these experiments are presented and they are compared with the corresponding theoretical values. The comparison is as good as could be expected in view of the limitations imposed by the very coarse mesh which was used; essentially the same results would have been obtained by the exercise of a relaxation computation, using the same size of net. No practical difficulties would be encountered in building a large network having a very fine mesh.

Electrical Resistance Networks for Beam and Column Problems

A similar formula in terms of the pitot tube reading shows that the overshoots indicated in Fig. 4 of reference 2 could be explained by a 1-5 per cent rise in static pressure pu in front of the pitot mouth. The preceding conjecture as well as the general up­ stream effect can be checked by taking simultaneous readings of pitot pressure at each y position and of static pressure from wall orifices under and slightly upstream of the pitot tip. If there is a demonstrable effect of upstream influence when a relatively large probe is outside of the layer, it would follow that similar or even larger effects could occur throughout the boundary layer and that some effect should be present for smaller probes. The presence of this effect and that due to the leading edge makes the near agreement with theory (which takes account of neither) in references 1, 2, and 3 the more remarkable. The question remains as to the non-dimensional parameters properly characterizing the above interference, with or without separation. Clearly, the probe height h has to be non-dimension alized with respect to some characteristic layer thickness in order to account for the observed effect of relative size. Just how this should be accomplished is not clear since it involves ques­ tions of pressure disturbance strength as function of y and the upstream viscous influence. The width of the probe must be significant since its increase means closer approach to the ob­ served two-dimensional critical pressure disturbance for sepa­ ration, P for instance, the upstream distance, s, through which a given (linearized) pressure disturbance at wall diminishes by a factor e1, measured with boundary layer thick­ ness as a unit, depends on the parameters approximately as fol­ lows :

THE ADJUSTMENT OF MISCLOSURES

The adjustment of misclosures in slope traverses by the method of least squares is effected by solving a system of simultaneous equations which may be written by inspection of the traverse diagram. These equations can be solved by measurements of currents in an analogous electrical resistance network. This paper is devoted mainly to the development of the electrical analogy.