Narrow Rectangular Cross Section Research Articles

Lateral buckling of the compressed edge of a beam under finite bending

This paper investigates the critical condition whereby the compressed edge of a beam subjected to large bending exhibits a sudden lateral heeling. This instability phenomenon occurs through a mechanism different from that usually studied in linear theory and known as flexural–torsional buckling. An experimental test device was specifically designed and built to perform pure bending tests on soft materials. Thus, the experimental campaign provides not only the moment-curvature behavior of beams of narrow rectangular cross section, but also information regarding the post-critical lateral buckling behavior. To study the local bifurcation phenomenon, an analytical model is proposed in which a field of small transversal displacements, typical of the linear stability of thin plates, is superimposed on the large vertical displacement field of an inflexed beam in the nonlinear elasticity theory. Furthermore, numerous numerical simulations through nonlinear FE analysis have been performed. Finally, the results provided by the different methods applied were compared and discussed.

Flow behavior of nano-silica solutions in cement cracks/fractures

One of the most prevalent causes of failure in oil and gas wells is the formation of cracks in the wellbore cement. The cracks can lead to loss of zonal isolation and fluid leakage to the surface. Compared to cement slurry which is difficult to squeeze into the cracks due to its high-solid content, nanoparticle (NP) gels have gained significant attraction for sealing fractures/cracks in oil and gas wells. It is highly desirable to know the flow behavior of NP solutions in the cracks. This study investigated the flow of nano-silica solutions through narrow rectangular cross-sections simulating sealant flow in cracks of oil well cement sheath. Pressure gradient was tested at 4 flow rates for various aspect ratios of the rectangular cross-sections. For a given cross-section of conduit, pressure drop increased with flow rate. For a given conduit height, the pressure drop decreased with conduit width due to increased flow area. Pressure drop increased as the conduit height reduced. These characteristics of nan-particle solution flow are like those found for conventional fluids. The pressure gradient of nano-silica solution flow in a conduit of a given rectangular cross-section is nearly proportional to flow velocity at tested flow rates, indicating laminar flow conditions in the tested flow conditions. The proportionality factor (rate of increase) increases as the aspect ratio of the cross-section decreases (slot becomes narrower). This is because a system with a lower aspect ratio has four boundaries that generate a 2-dimensional velocity profile, while a system with a higher aspect ratio has essentially two boundaries that generate a 1-dimensional velocity profile. More boundaries mean more frictional resistance to fluid flow. Hydraulics models for conventional fluid flow through conduits of rectangular cross-sections under-predict the pressure gradient of nano-silica solution. The discrepancy/difference is over 90% in pressure gradient prediction in general. The hydraulics model presented by Guo et al. (2022) with aspect ratio is better than the classical hydraulics model with hydraulic diameter. However, both models fail to predict the pressure gradient along the conduits with slot-like cross sections. The excessive pressure drop is attributed to the interactions between the nano-particles and conduit walls. Further investigations are required to gain a thorough understanding of nano-silica flow.

A Geometrically Exact Beam Finite Element for Non-Prismatic Strip Beams: The 2D Case

In this paper, a new finite element for 2D non-prismatic (curved and tapered) elastic beams with narrow rectangular cross-section is proposed. The element is geometrically exact, hence capable of handling large displacements and finite rotations, and allows for cross-section arbitrary warping through the inclusion of additional degrees of freedom (DOFs). These additional DOFs make it possible to capture the non-standard shear stress distributions that are developed in tapered members. All expressions required to implement the proposed finite element are derived and written in a simple vector/matrix form. To assess the accuracy and demonstrate the capabilities of the proposed element, several numerical examples are presented. For comparison purposes, results obtained with refined meshes of standard 2D finite elements are also shown. It is concluded that the proposed element leads to very accurate results with a small computational cost.

Effect of Discrete Ribs on Heat Transfer and Friction Inside Narrow Rectangular Cross Section Cooling Passage of Gas Turbine Blade

The performance of a gas turbine during the service life can be enhanced by cooling the turbine blades efficiently. The objective of this study is to achieve high thermohydraulic performance (THP) inside a cooling passage of a turbine blade having aspect ratio (AR) 1:5 by using discrete W and V-shaped ribs. Hydraulic diameter (Dh) of the cooling passage is 50 mm. Ribs are positioned facing downstream with angle-of-attack (α) of 30° and 45° for discrete W-ribs and discerte V-ribs respectively. The rib profiles with rib height to hydraulic diameter ratio (e/Dh) or blockage ratio 0.06 and pitch (P) 36 mm are tested for Reynolds number (Re) range 30000-75000. Analysis reveals that, area averaged Nusselt numbers of the rib profiles are comparable, with maximum difference of 6% at Re 30000, which is within the limits of uncertainty. Variation of local heat transfer coefficients along the stream exhibited a saw tooth profile, with discrete W-ribs exhibiting higher variations. Along spanwise direction, discrete V-ribs showed larger variations. Maximum variation in local heat transfer coefficients is estimated to be 25%. For experimented Re range, friction loss for discrete W-ribs is higher than discrete-V ribs. Rib profiles exhibited superior heat transfer capabilities. The best Nu/Nuo achieved for discrete Vribs is 3.4 and discrete W-ribs is 3.6. In view of superior heat transfer capabilities, ribs can be deployed in cooling passages near the leading edge, where the temperatures are very high. The best THPo achieved is 3.2 for discrete V-ribs and 3 for discrete W-ribs at Re 30000. The ribs can also enhance the power-toweight ratio as they can produce high thermohydraulic performances for low blockage ratios.

Elastic bending wave on the edge of a semi-infinite plate reinforced by a strip plate

Elastic waves localised near the edge of a semi-infinite plate reinforced by a strip plate are considered within the framework of the 2D classical theory for plate bending. The boundary value problem for the strip plate is subject to asymptotic analysis, assuming that a typical wavelength is much greater than the strip thickness. As a result, effective conditions along the interface corresponding to a plate reinforced by a beam with a narrow rectangular cross-section are established. They support an approximate dispersion relation perturbing that for a homogeneous plate with a free edge. The accuracy of the approximate dispersion relation is tested by comparison with the numerical data obtained from the ‘exact’ matrix relation for a composite plate. The effect of the problem parameters on the localisation rate is studied.

Flame Propagation in a Narrow Closed Channel: Effects of Aspect Ratios, Blockage Ratio, and Mixture Reactivity on Flame Speed and Pressure Dynamics

ABSTRACT A set of experiments is conducted to study flame acceleration in a narrow rectangular cross-section channel in order to examine the effects of mixture reactivity, channel height, and blockage ratio (BR). The channel had an ignition chamber at one end and a dump chamber at the opposite end. The initial conditions are mixtures of hydrogen-air at 1 bar and 293 K. Five different compositions of 10, 20, 30, 40, and 50% hydrogen volume fraction in air are studied. The rectangular chamber is 500 mm long and 40 mm in width, with an adjustable height from 2 to 10 mm. To study the relation between channel height and BR, combustion behaviors with three different B.R. (BR of 0.1, 0.5, and 0.9), 5 different obstacle sizes, and different channel heights (from 2–10 mm) are presented. Pressure waves are monitored with seven fast response pressure transducers placed along the chamber, and the flame propagation is observed with a high-speed Schlieren imaging system. There was no detonation observed during this series of experiments. The results show that the flame propagation is affected by both the channel dimensions and the obstacles, with a maximum flame velocity achieved for partially obstructed channels. The evolution of the flame’s propagation and the pressure dynamics through the various obstructed and unobstructed channels, including the dynamics contributed by the ignition and dump chamber, are discussed.

Side busking of the cantilever beam with narrow rectangular cross section

The problem of lateral buckling of a cantilever strip with a constant narrow cross section loaded with a concentrated force at the end of the span is considered. In the study of lateral buckling of beam energy method was used. For the case of load application in the center of gravity, the problem is reduced to a generalized secular equation. The relationship between the magnitude of the critical force and the position of the point of application of the load. A comparison of the results obtained by the authors with an analytical solution using infinite series and a numerical iterative method is shown.

Flow rate–pressure drop relation for deformable shallow microfluidic channels

Laminar flow in devices fabricated from soft materials causes deformation of the passage geometry, which affects the flow rate–pressure drop relation. For a given pressure drop, in channels with narrow rectangular cross-section, the flow rate varies as the cube of the channel height, so deformation can produce significant quantitative effects, including nonlinear dependence on the pressure drop (Gervaiset al.,Lab on a Chip, vol. 6, 2006, pp. 500–507). Gervaiset al. proposed a successful model of the deformation-induced change in the flow rate by heuristically coupling a Hookean elastic response with the lubrication approximation for Stokes flow. However, their model contains a fitting parameter that must be found for each channel shape by performing an experiment. We present a perturbation approach for the flow rate–pressure drop relation in a shallow deformable microchannel using the theory of isotropic quasi-static plate bending and the Stokes equations under a lubrication approximation (specifically, the ratio of the channel’s height to its width and of the channel’s height to its length are both assumed small). Our result contains no free parameters and confirms Gervaiset al.’s observation that the flow rate is a quartic polynomial of the pressure drop. The derived flow rate–pressure drop relation compares favourably with experimental measurements.

Lateral Torsional Buckling of Steel Beams under Transverse Impact Loading

This study employs experiments and numerical simulation to analyze the dynamic response of steel beams under huge‐mass impact. Results show that lateral torsional buckling (LTB) occurs for a narrow rectangular cross‐section steel beam under transverse impact. The experiments were simulated using LS‐DYNA. The numerical simulation is in good agreement with experimental results, thus indicating that the LTB phenomenon is the real tendency of steel beams under impact. Meanwhile, the study shows that LS‐DYNA can readily predict the LTB of steel beams. A numerical simulation on the dynamic response of H‐shaped cross‐section steel beams under huge‐mass impact is conducted to determine the LTB behavior. The phenomenon of dynamic LTB is illustrated by displacement, strain, and deformation of H‐shaped steel beams. Thereafter, a parametric study is conducted to investigate the effects of initial impact velocity and momentum on LTB. The LTB of H‐shaped cross‐section steel beams under transverse impact is primarily dependent on the level of impact kinetic energy, whereas impact momentum has a minor effect on LTB mode.

High-speed flames and DDT in very rough-walled channels

An experimental investigation of flame acceleration in a narrow rectangular cross-section channel with various lower boundary conditions is presented. The boundary conditions investigated include a smooth surface and a rough surface with and without porosity. Tests were performed with a mixture of nitrogen diluted stoichiometric methane–oxygen, i.e., CH4+2(O2+2/3N2), at initial pressures in the range of 9–15kPa. The explosion front velocity was measured via ionization probes and pressure transducers and the explosion wave front structure was visualized by high-speed shadowgraph video. For tests performed with a porous rough surface the flame achieves a velocity that approaches the speed of sound of the combustion products. In the case of the rough surface without porosity, a globally quasi-steady high-speed flame is produced with a velocity between the combustion products speed of sound and the Chapman–Jouguet detonation velocity. It is proposed that flame acceleration beyond the speed of sound of the combustion products is possible due to enhanced burning that is driven by chemical–acoustic interactions. Such interactions occur as the turbulent flame brush is traversed by transverse pressure waves produced by the passage of the explosion front shock wave over the rough surface. It is also observed that under certain conditions a smooth transition from a diffusion driven flame to a shock ignition detonation is possible.

Treatment surfaces with atomic oxygen excited in dielectric barrier discharge plasma of O2 admixed to N2

This paper describes the increase in surface energy of substrates by their treatment with gas composition generated in plasmas of DBD (Dielectric Barrier Discharge) in O2 admixed with N2. Operating gas dissociation and excitation was occurred in plasmas developed in two types of reactors of capacitively-coupled dielectric barrier configurations: coaxial cylindrical, and flat rectangular. The coaxial cylindrical type comprised an inner cylindrical electrode encapsulated in a ceramic sheath installed coaxially inside a cylindrical ceramic (quartz) tube passing through an annular outer electrode. Components of the flat rectangular type were a flat ceramic tube of a narrow rectangular cross section supplied with two flat electrodes mounted against one another outside of the long parallel walls of this tube. The operating gas, mixture of N2 and O2, was flowing in a completely insulated discharge gap formed between insulated electrodes of the devices with an average velocity of gas inlet of about 7 to 9 m/s. Dielectric barrier discharge plasma was excited in the operating gaps with a bipolar pulse voltage of about 6 kV for 2 ms at 50 kHz repetition rate applied to the electrodes of the coaxial device, and of about 14 kV for 7 ms at 30 kHz repetition rate for the flat linear device. A lifetime of excited to the 2s22p4(1S0) state in DBD plasma and streaming to the surfaces with a gas flow atomic oxygen, responsible presumably for treating surfaces, exceeded 10 ms in certain cases, that simplified its separation from DBD plasma and delivery to substrates. As it was found in particular, surfaces of glass and some of polymers revealed significant enhancement in wettability after treatment.

Shape optimization of beam due to lateral buckling problem

Based on a non-linear mathematical model of lateral buckling of a slender beam with a narrow rectangular cross section, the variational formulation of the two-parametric optimization problem is given in the dimensionless form. An optimal shape is obtained by solving the variational problem using the Rayleigh–Ritz method with the orthogonal system of trigonometric functions. By a partial solution of the Euler–Lagrange differential equation of the variational problem, a proof is given that in the case of the optimal shape, a maximal reference stress according to the total strain energy theory is constant along the beam. An example of extrapolation of the two-parametric optimization problem solution is represented.

Ethanol-Promoted Fabrication of Tungsten Oxide Nanobelts with Defined Crystal Orientation

A controlled method for the production of tungsten oxide nanobelts through metal oxidation in presence of ethanol is proposed. At the optimal synthesis conditions, up to 20 μm long vertically aligned ribbon-like structures with a narrow rectangular cross section can be obtained in a tuned manner with preferential formation of WO2 in the presence of graphitic like carbon. Bulk and local-scale studies suggest that carbon diffusion to the surface of the material leads to the formation of mainly WO3 nanobelts from simple annealing treatments at 450 °C. This represents one alternative method to the common tungsten oxidation in air, opening the possibility to use C-containing compounds with negligible formation of carbide traces. An in-depth characterization of these materials has been performed, and the possible growth mechanisms are here discussed.

Uniform Flow of Modified Bingham Fluids in Narrow Cross Sections

A model of modified Bingham fluid is used to investigate uniform mudflows in narrow cross sections of the kind typical of hillslope incisions. The problem is tackled both numerically and experimentally. Experiments were performed using carboxymethylcellulose, a shear thinning fluid whose rheological behavior can be interpreted by a modified Bingham fluid model. For narrow rectangular, trapezoidal, and triangular cross sections, the capability to flow has been expressed by a general relationship that defines a dimensionless coefficient (f) as a function of two dimensionless parameters: the Bingham number and an aspect ratio of the channel. We also derive plots for the critical Bingham number for incipient flow of modified Bingham fluids as a function of the aspect ratio of the channel. The present calculations and the experimental results demonstrate that the narrow character of the channel as well as its shape strongly affect the flow conductance and the critical value of the Bingham number.

Analytical solution to the bending of a nonlinearly hardening wide curved bar

An analytical solution to the partially plastic deformation of a nonlinearly hardening wide curved bar is derived. The bar considered has a narrow rectangular cross-section and is under pure bending. The analytical treatment is based on Tresca’s yield criterion, its associated flow rule and a Swift-type nonlinear hardening law. Taking numerical limits, the complete solution is verified in comparison to the linear hardening solution available in the literature.

A concise analytical treatment of elastic‐plastic bending of a strain hardening curved beam

AbstractPlane stress and plane strain analytical solutions to partially plastic deformation of a curved beam are presented. The beam has a narrow rectangular cross section and it is subjected to couples at its end sections prevailing pure bending conditions. The analysis is based on Tresca's yield criterion, its associated flow rule and linear strain hardening material behavior. The solutions are verified in comparison to the ones available in the literature. It is shown that plane stress and plane strain solutions agree well in the elastic and in the elastic‐plastic deformation stages. It is also observed that the changes in the dimensions of the beam as it deforms are negligibly small.

Dynamic stability of a thin-walled beam subjected to axial loads and end moments

The dynamic stability problem of thin-walled beams subjected to combined action of axial loads and end moments is studied. Each force and moment consists of a constant part and a time-dependent stochastic function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability and uniform stochastic stability conditions are obtained as the function of stochastic process variance, damping coefficient, and geometric and physical parameters of the beam. The almost sure stability regions for I-cross section and narrow rectangular cross section are shown in the plane of variances of stochastic parts axial force and end moment. Uniform stochastic stability regions are shown in intensity of stochastic loadings and constant parts of axial loads and end moments.

Dynamic lateral torsional post-buckling of a beam-mass system: Experiments

The phenomenon of static lateral torsional buckling of a beam with a narrow rectangular cross-section is well known. Specifically, a beam is clamped at one of its ends and is subjected to a shear force at its other end which causes deformation in the principal plane with stiffest resistance to bending. Above a critical value of load, a bifurcation occurs and the beam twists and experiences out-of-plane deformation which tends to transfer bending to the plane of weakest resistance. Here, attention is focused on an experimental study of dynamic lateral torsional buckling. In the experiment, a beam is attached to the shaft of a motor at one of its ends and a relatively large mass is attached to its other end. Rotation of the motor causes deflection of the beam in its principal plane of stiffest bending resistance. By increasing the excitation frequency and/or amplitude of oscillation of the motor's shaft, the shear force applied by the mass on the beam's end exceeds a critical value which causes dynamic lateral torsional buckling of the beam. Special techniques have been developed to produce and measure this phenomenon and the data has been presented in a form that can be used for future validation of analytical or numerical models.

Elastic Buckling Analysis of Die Fins during Extrusion

In this study, elastic buckling criteria for die fins during extrusion of complex-shaped parts are analytically derived. In this analytical model, the fins of the extrusion die are regarded as cantilever beams with a narrow rectangular cross section. The load intensity acting on each fin is assumed to be distributed uniformly, which is quite similar to the actual distribution. The elastic critical load intensity of buckling is successfully obtained by solving simultaneously the moment equilibrium differential equations. A commercial finite element package “DEFORM 3D" is also employed to simulate the flow pattern of the billet inside the die in extrusion of a CPU heat sink. The stress distribution pattern on the fins of the extrusion die is discussed.

Almost sure stability of the thin-walled beam subjected to end moments

The dynamic stability problem of the thin-walled beams subjected to end moments is studied. Each moment consists of constant part and time-dependent stochastic non-white function. Closed form analytical solutions are obtained for simply supported boundary conditions. By using the direct Liapunov method almost sure asymptotic stability condition is obtained as function of stochastic process variance, damping coefficient, geometric and physical parameters of the beam. The stability regions for I-cross section and narrow rectangular cross section are shown in variance - damping coefficient plane when stochastic part of moment is Gaussian zero-mean process with variance ?2 and harmonic process with amplitude A.