Uniform Rectangular Cross-section Research Articles

Hydrodynamic derivation of a variable parameter Muskingum method: 2. Verification

Abstract The hydrodynamic derivation of a variable parameter Muskingum method and its solution procedure for estimating a routed hydrograph were presented in Part I of this series (Perumal, 1994a). In this paper, the limitations of the method, the criterion for its applicability and its accuracy are discussed based on the assumptions used. The method is verified by routing a given hypothetical inflow hydrograph through uniform rectangular cross-section channels and comparing the results with the corresponding numerical solutions of the St. Venant equations. The stage hydrographs as computed by the method are also compared with the corresponding St. Venant solutions. It is demonstrated that the method closely reproduces the St. Venant solutions for the discharge and stage hydrographs subject to the compliance of the assumptions of the method by the routing process.

Dynamic stability of radially stiffened annular plates

The dynamic stability of radially stiffened annular plates with both edges subjected to in-plane forces is studied. For this purpose a sector and a beam finite element is used. The method is applied to annular plates with several stiffeners of uniform rectangular cross-section located at equal angular intervals. The instability regions are determined for a wide range of exciting frequencies with different boundary conditions using Bolotin's method by varying the number and size of the stiffeners. Also the effect of static forces are taken into account.

A finite-element model for the analysis of thick, Anisotropic, Bimodular, Fibrous-composite plates

A finite-element model is developed to analyze the behavior of a shear-deformable, bimodular, anisotropic plate with a uniform rectangular cross-section. The effect of shear-deformation is taken into account by assuming a linear variation for normal strains through the thickness of the plate. Results for transverse deflection and position of neutral surface are presented for a simply-supported plate under uniformly distributed load. The bending behavior is studied for different angles of fiber orientation, plate aspect ratios, and bimodularity ratios.

Steady Wave‐Drift of Modeled Ice Floes

Systematic laboratory measurements are performed to determine the steady drift of isolated, tabular ice-floe models in regular waves. A relatively simple expression for the steady wave-induced drift of two-dimensional floes of uniform thickness and density in deep water is developed. This empirical wave-drift formula indicates that the drift velocity depends primarily on the ratio of wave period to ice-floe roll period, wave steepness, and length of the floe. The highest drift velocity is produced by waves with period somewhat greater than the ice-floe roll period, with fundamentally different drift behavior evident on either side of this maximum (long-wave and short-wave drift response). The dependence on wave height is found not to be linear but increases with wavelength (from approximately \IH\N¹\u/² to \IH\N³\u/². A simplified equation for the roll period of a prismatic body of uniform density and rectangular cross section is developed and found to be of practical value. Floe curvature (in the vertical plane) is found to affect drift rates substantially.

Vibration and stability of stiffened annular plates

The vibration and stability of radially stiffened annular plates subjected to an in‐plane force uniformly distributed at the edges are analyzed by the energy method. For this purpose, the transverse deflection of a stiffened annular plate is written in a series of the products of the deflection function of a sectorial beam with a small angle and the trigonometric function of angular coordinate. The deflection function of the sectorial beam is approximately expressed by a quintic spline function satisfying the boundary conditions at the edges, and the frequency equation of the plate is derived by the Ritz method. The method is applied to annular plates with several stiffeners of uniform rectangular cross section located at equal angular intervals; the natural frequencies and the critical buckling loads are calculated numerically and the effects of stiffeners on them are studied.

The fluid flow around a manoeuvring tanker in restricted water

Using a three-dimensional finite-element analysis and potential flow theory, the fluid flow fields around a tanker after a sway or yaw disturbance are illustrated. The tanker is positioned either on the centerline or near a wall in a canal of uniform rectangular cross-section and the flows existing in two water-depth conditions are compared.

The optimization of modal sound attenuation in ducts, in the absence of mean flow

The optimization of modal, axial sound attenuation rates, in the sense defined by Cremer, is examined in detail for zero mean flow in the duct. Cremer's result, for the lowest order mode pair in the absence of flow, is generalized so that the optimum impedance and attenuation rate can now be obtained for any mode pair in ducts of uniform rectangular, circular or annular cross-section. One qualification to Cremer's result emerges in the derivation of the Green's function : at the optimum condition the mode pair degenerates into a single mode with the expected exponential attenuation rate but this is offset by an amplification rate which is directly proportional to the distance from the source. In spite of this effect the axial decay rate of the Green's function can reach a maximum for a wall impedance close to Cremer's optimum value. Computed results for the attenuation compare favourably with available experimental values. The computation method produces slightly more accurate values of the optimum wall impedance for maximum attenuation of the “least damped mode” than those previously obtained by Cremer and by Morse. The theoretical approach developed is suitable for extension to attenuation (and attenuation optimization) studies in ducts carrying mean flow.

Load transfer from a web to two parallel sheets

This paper deals with the elastostatic transfer of a tensile load from an infinite strip-shaped web of uniform rectangular cross section to two identical parallel sheets of infinite extent. The web is perpendicular to the sheets and is continuously bonded to the latter along finite portions of its “top” and “bottom” edges. The sheets are treated within the two-dimensional theory of generalized planes stress. On the other hand, two alternative models for the web are considered: (a) the web likewise is regarded as a two-dimensional continuum within the conventional theory of generalized plane stress and (b) the web is idealized as a one-dimensional elastic continuum. In either case, an integral equation with a Cauchy-type singular kernel is obtained for the density of the bond force between the web and the sheets. This equation, in turn, is reduced to a Fredholm integral equation of the second kind. Numerical results for the axial force in the web and the bond force are presented for several combinations of the governing material and geometrical parameters.

Effect of Slenderness Ratio on the Natural Frequencies of Pre-Twisted Cantilever Beams of Uniform Rectangular Cross-Section

This paper is concerned with the vibrational characteristics of pre-twisted cantilever beams of uniform rectangular cross-section allowing for shear deformation and rotary inertia. A method of solution of the differential equations of motion allowing for shear deformation and rotary inertia is presented which is an extension of the method introduced by Dawson (1)§ for the solution of the differential equations of motion of pre-twisted beams neglecting shear and rotary inertia effects. The natural frequencies for the first five modes of vibration are obtained for beams of various breadth to depth ratios and lengths ranging from 3 to 20 in and pre-twist angle in the range 0–90°. The results are compared with those obtained by an alternative method (2), where available, and also to experimental results.

Strength of Near-Perfect Single Crystals of Cadmium

The ultimate strength of thin single crystals of cadmium has been measured. The crystals were initially completely free from dislocations and other imperfections, in so far as this could be established by observations with the electron microscope. They were grown from the vapor in an atmosphere of argon and had surfaces parallel to {0001} and {011̄0} planes,[2̄110] axes, and uniform rectangular cross sections. The crystals were strained in the electron microscope with a pneumatic tensile device, and the maximum elastic strain which they would sustain was determined directly from measurements of diffraction patterns. The near-perfect crystals failed at measured elastic strains between 3.8% and 4.5% along the[2̄110] direction. The corresponding normal stress was also measured directly with the tensile device. When the elastic strain along the[2̄110] axis of tension is 4%, the resolved shear stress on the glide systems which could be activated varies between G/11 and G/15, where G is the relevant shear modulus. This is within the range of estimated theoretical maximum strengths of crystals for deformation by a homogeneous shear process.

Anchorage zone stresses in axially post-tensioned members of uniform rectangular section

Synopsis Anchorage zone stresses in axially loaded post-tensioned members of uniform rectangular cross-section are determined by using a finite element procedure for the elastic analysis of three-dimensional solids. Particular attention is paid to the distributions of bursting stresses for a wide range of concentration ratios between the dimensions of the anchor plate and the cross-section of the member. The results are compared with those predicted by conventional methods.

Hydraulic Flow in Coupled Parallel Channels of Varying Cross-Sectional Area

The problem considered is that of flow in the two parallel channels formed by the introduction of a long thin plate into a duct of uniform rectangular cross section. The plate is slightly inclined and does not, in general, span the channel so that transverse flow may occur. The flow is described by applying the one-dimensional approximation. Boundary conditions were established by regarding the stagnation streamline as defining two separate channels upstream of the plate leading edge. The solution was then obtained numerically with an IBM 650 computer. Computed hydraulic parameters were in good agreement with those obtained experimentally.

Coupled Bending Vibrations of Pretwisted Cantilever Beams

The author describes a method of determining the vibrations of cantilever beams of non-uniform pretwist and cross-section. The known condition for orthogonality of normal modes for a beam in simple bending in one transverse plane is extended to that of coupled bending-bending modes of vibration in two transverse planes. This condition of orthogonality is then used in the Stodola process to determine the frequencies and modes of vibration for cantilever beams of uniform pretwist and cross-section. In this method the modes of vibration may be coupled or uncoupled. The resonant frequencies of the beam are determined in ascending order of magnitude and the modal patterns deduced without requiring any preliminary knowledge of the vibrational characteristics of the beam. It is considered that this is a considerable advantage on the alternative methods of solution. An analysis of experimental and theoretical results is made for three cantilever beams of uniform rectangular cross-section and angles of pretwist 0, 90·9 and 877° respectively. The frequencies and modal patterns of the first five modes of vibration were obtained experimentally and show good agreement with those calculated from the theory. The computed results for the first five frequencies are then given for a series of cantilever type beams in coupled bending-bending vibrations. The cantilevers considered had angles of pretwist from zero to 180° and were of uniform rectangular cross-section having breadth to depth ratios ranging from unity to 16. The calculated frequencies have been compared with known experimental results and show good agreement. It has also been shown that some of the larger errors may be anticipated for certain critical frequencies.

Vibratory power convertor systems

A brief description of the underlying principles involved in vibratory power convertor systems is given, and then formulae are derived for use in system analysis and design. These consist of expressions which enable the compliance forces and effective masses of two different arrangements of conventional systems to be determined under all conditions throughout the cycle of vibration, and equations which govern the performance of the systems. Included are equations giving the deflection at which vibrating contacts open, and their optimum position along the reed has been determined. The characteristics of the systems are examined in a general analysis, and more specific details are discussed in a particular design.In a number of instances, simplification of the formulae has been undertaken and these are presented graphically. They apply as well to non-conventional systems such as single-contact transformer-vibrators or multi-contact vibrator mechanisms. The formulae are not directly limited by power considerations or by a range in dimensions for normal degrees of vibration, but they are confined in their use to systems embodying reeds of uniform rectangular cross-section. Information regarding reeds of variable cross-section is given in an Appendix.

Discussion: “Analysis of Buckling Column Spring With Pivoted Ends and Uniform Rectangular Cross Section” (Blake, Alexander, 1961, ASME J. Eng. Ind., 83, pp. 61–66)

Discussion: “Analysis of Buckling Column Spring With Pivoted Ends and Uniform Rectangular Cross Section” (Blake, Alexander, 1961, ASME J. Eng. Ind., 83, pp. 61–66)

Closure to “Discussion of ‘Analysis of Buckling Column Spring With Pivoted Ends and Uniform Rectangular Cross Section’” (1961, ASME J. Eng. Ind., 83, pp. 66–67)

Closure to “Discussion of ‘Analysis of Buckling Column Spring With Pivoted Ends and Uniform Rectangular Cross Section’” (1961, ASME J. Eng. Ind., 83, pp. 66–67)

Analysis of Buckling Column Spring With Pivoted Ends and Uniform Rectangular Cross Section

Design formulas and working charts are derived for predicting load-deflection characteristics and maximum bending stresses in initially straight buckling column springs, of uniform cross section, considered to be pin jointed at the supports. Load-deflection analysis is based on the study of a slender bar, compressed beyond critical buckling, made by Lagrange. Stresses are calculated using the elementary strength of materials theory. The predicted load-deflection curves for typical spring proportions are compared with the experimental results.

Vibrations of Pre-Twisted Cantilever Blading

This paper reports a study of the vibrations of pre-wisted cantilever blading. The blading is pre-twisted linearly about the centroid of its cross-section up to an angle of π/2 rad and is considered to be mounted encastré at the root. Accurate prediction of the frequencies of vibration of such blading is of considerable practical importance. Two theoretical approaches to the problem are discussed. The first one, involving a direct solution of the equations of motion, is not made use of, although the characteristic equations are derived in an appendix for reference. The second approach is based on the application of Rayleigh's principle. For blades of uniform rectangular cross-section, frequency equations are derived for the fundamental mode of lateral motion and all the modes of torsional motion. Similar types of equation are shown to apply to aerofoil cross- section blading since direct coupling effects between bending and torsion, being extremely small, can be neglected. Vibration tests were conducted on sets of blades of both uniform rectangular and uniform aerofoil cross-section pre-twisted between 0 and π/2 rad, the objects being to determine the frequencies and corresponding nodal patterns contained within a specified frequency band. The effect of pre-twist on the measured frequencies of the lateral, torsional, and, for the aerofoil cross-section, coupled modes of vibration is shown, with certain exceptions, to be considerable. For the modes of motion where a theoretical solution is provided, reasonable agreement is shown to exist between the calculated and corresponding measured frequencies. Hence, the use of Rayleigh's method is justified, giving, as it does, relatively simple expressions for the natural frequencies of vibration. The torsional frequencies of pre-twisted blades are dependent on the torsional stiffness values. Hence, a theoretical study of torsional stiffness variation with pre-twist is presented, the results obtained being confirmed experimentally by a series of simple torsion tests.

Deflection of a Thick Ring in Diametral Compression by Test and by Strength-of-Materials Theory

Abstract Experimental and theoretical deflection studies are briefly described for several steel rings, of uniform rectangular cross section, compressed by two forces along a diameter and having D/d ratios ranging from 1.3 to 1.9. The calculations are based on the principle of Castigliano and expressions for strain energy due to bending, normal, and shear stresses. Discrepancies between the theoretical estimate and the tests are shown.

A Photoelastic Study of Maximum Tensile Stresses in Simply Supported Short Beams Under Central Transverse Impact

Abstract Simply supported short Castolite beams of uniform rectangular cross section were subjected to central transverse impact by a heavy mass. Photoelastic streak photographs were taken of the transverse section of symmetry for a wide range of spans, widths, and impact velocities at exposures of less than 1 microsec. The maximum tensile stresses were determined. Comparison with the elementary theory for long beams shows that while this theory is satisfactory for long beams, it does not agree with the results from short beams. An approximate theory for short beams under central impact is developed which gives satisfactory agreement. The duration of impact also was determined and the appearance of isotropic points is discussed.