Bars Of Rectangular Cross Section Research Articles

Curvature Analysis in a Bi-layered Non-linear Elastic Bar under Uniform Heating

The present theoretical paper is concerned with analysis of the curvature in a bi-layered bar of rectangular cross-section. The bar is subjected to uniform heating. The two layers of the bar have different thickness and material properties. Besides, the layers exhibit non-linear elastic mechanical behaviour that is treated by applying a power-law stress-strain constitutive relationship. The bi-layered bar curvature is studied analytically. For this purpose, the mechanical response of the bar to uniform heating is investigated. The strains in the two layers due to heating are analyzed. An equation is compiled by using the fact that the prolongations of the bar layers are equal. Two equations are worked out by considering the equilibrium of the layers. The analysis of the curvature presented in this paper is checked-up by comparing with a known solution for the curvature in a bi-layered linear-elastic bar under uniform heating. The uniform heating induced curvature in a bi-layered bar whose layers are continuously inhomogeneous in longitudinal direction is also studied.

Оптимизация сжатых деревянных стоек переменного сечения по критерию максимума критической нагрузки

The article presents a numerical and analytical solution to the problem of optimizing compressed rods according to the criterion of the maximum critical load with a possible loss of stability in two planes. We consider a wooden bar of rectangular cross section, the height of which varies according to a linear law, and the width is constant. The calculation is performed in an elastic formulation. A constraint on the constancy of the rod mass is introduced. Variable parameters are the ratio of the minimum section height to the maximum, as well as the ratio of the section width to the maximum height. Numerical search for the optimal solution is performed in the MATLAB environment using the method of sequential quadratic programming. It has been established that the use of rods with a cross-sectional height that varies linearly along the length is effective only if the rod is fixed in at least one of the planes according to the “pinched-free end” scheme. It has been established that the increase in the critical load when using rods with a linearly varying cross-sectional height can be up to 22%.

Calculation and experimental procedure for determining the modulus of elasticity of porous coatings on a substrate during bending. Part 1. Theoretical foundations

The purpose of this work is to develop a design and experimental technique for determining the modulus of elasticity of one of the layers of a two-layer bar of rectangular cross section during bending, provided that the modulus of elasticity of the other layer is known. In this part of the work, using the transition formulas of relatively parallel axes, a mathematical expression has been obtained to determine the bending stiffness of a two-layer rectangular section, which does not include distances from the centers of weight of sections of layers to the pivot center of weight of the entire section, which, in turn, depend on the elastic modules of layers. The presence of such an expression made it possible to form an algebraic equation in which the modulus of elasticity of one of the layers can act as an unknown. As a result, the mathematical connection of the unknown modulus of elasticity with the known modulus of elasticity of the other layer, geometric dimensions of layers and bending stiffness of the entire section was established, which should be determined experimentally.

Saint Venant’s torsion of homogeneous and composite bars by the finite volume method

We propose a new finite-volume based approach to the solution of Saint Venant’s torsion problems of homogeneous and composite prismatic bars subjected to pure torsion. The semi-inverse approach is employed to construct an approximate displacement field characteristic of pure torsion within the subvolumes of the beam’s discretized cross section such that the governing differential equation for the warping function is locally satisfied in each subvolume in an integral sense. Orthotropic subvolumes are admitted in the formulation to enable analysis of bars comprised of heterogeneous microstructures, including composite materials with orthotropic shear moduli. Continuity of both displacements and tractions across subvolumes’ interfaces is satisfied in a surface-average sense, together with traction-free lateral boundary conditions. Closed-form expressions for the stiffness matrix elements that relate surface-averaged displacements to tractions are provided for ease of programming. The convergence and accuracy of the proposed method are assessed and verified upon comparison with the exact elasticity solutions for prismatic homogeneous and composite bars of rectangular cross section and the corresponding finite-difference solution for the homogeneous bar. The utility of the developed solution methodology is first illustrated by analyzing torsion of homogeneous T and box beams, focusing on the singular behavior at the re-entrant corners and the concomitant membrane analogy’s applicability. Then, composite cross sections reinforced by angle braces as well as continuous fiber-reinforced wraps with graded shear moduli are investigated with the aim of mitigating the re-entrant corner shear stress singularities, not easily analyzed using the finite-element or finite-difference approaches.

Geometric Design of Rectangular Cross-Sectional Springs<sup></sup>

In order to study the influence of the cross-sectional shape on the stiffness of a spring, a finite element analysis software DEFORM is used to simulate and analyze the torsion of rectangular cross-section bars. The material of the bar is TS1800 SAE9254 and the cross-section of aspect ratio (w / h) is 1.5. From literature it is known that when the rectangular section bar is twisted, the shear stress at the four corners is zero, so elliptical corners can decrease the volume and increase the stiffness with the same volume. Five levels for the long side of the elliptical corner are set as 1 to 5 mm, and 3 levels are set for the short side. Torsion of the rectangular section bars under 15 kinds of geometric designs are simulated to find the preferred cross-sectional shape design by evaluating the cross-sectional area, load, and the maximum shear stress. The objective of the design is obtaining a uniform stress distribution with a larger spring stiffness and lighter weight. The optimal cross section of the bars is established as the spring geometry, and the pre-loading processing of the spring is simulated. The required load and the maximum shear stress data are obtained. The effects of load, cross-sectional area and maximum shear stress on the springs performance are investigated.

Research on Shear Horizontal Waves in Bars of Rectangular Cross-Section

Research on Shear Horizontal Waves in Bars of Rectangular Cross-Section

О рациональном распределении пористости при кручении бруса прямоугольного поперечного сечения

О рациональном распределении пористости при кручении бруса прямоугольного поперечного сечения

Fracture toughness testing of nanocrystalline alumina and fused quartz using chevron-notched microbeams

We show that chevron-notched samples offer an attractive approach to the measurement of fracture toughness in micron-scale samples of brittle materials and use the method to characterize quartz and nanocrystalline alumina. Focused ion beam milling is used to carve bend bars of rectangular cross-section a few micrometres wide and containing a notch with a triangular ligament. Load-controlled testing is conducted using a nanoindentation apparatus. If the notch is appropriately machined, cracks nucleate and propagate in a stable fashion before becoming unstable. Sample dimensions are measured using a scanning electron microscope, and are used as input in finite element simulations of the bars’ elastic deformation for various crack lengths. The calculated compliance calibration curve and the measured peak load then give the local fracture toughness of the material. Advantages of the method include a low sensitivity to environmental subcritical crack growth, and the fact that it measures toughness at the tip of a sharp crack situated in material unaffected by ion-milling. The approach is demonstrated on two materials, namely, monolithic fused quartz and nanocrystalline alumina Nextel™ 610 fibres; results for the latter give the intrinsic grain boundary toughness of alumina, free of grain bridging effects.

Additional Results to the Structural Analysis of Eccentrically Compressed Cross-Section Made of no Tension Material

Abstract The mechanical behaviour of eccentrically compressed stocky bars of rectangular cross section, having no-tension material is well known. The aim of the structural analysis is to determine the distribution of the stresses arisen from the effect of an eccentrically compressive force, i.e. to describe the shape of the stress block, and to calculate the compressive resistance of the bar. Some important details of the phenomenon could not be found in the technical literature. In our paper we study these details. We evaluate the analytical functions of the curves, bounding areas lain around the kern or the point of gravity, on which areas the eccentric force acts a pentagonal compressed zone arisen. Furthermore, we determine the resistance of the stocky bar in the cases of triangular and rectangular compressive zone.

Normal waves in elastic bars of rectangular cross section

This paper addresses a theoretical study of guided normal waves in elastic isotropic bars of rectangular cross-section by an analytical superposition method. Dispersion properties of propagating and evanescent modes for four families are analyzed in detail at various geometric and physical parameters of the bar. A comparison of the obtained results with the well-known properties for waves in infinite plates and circular cylinders is provided. The complicated structure of dispersion spectra is explained. High-frequency limiting values for phase and group velocities of normal waves are established for the first time. Calculated data agree well with the available experimental results.

Effects of the forming processes and Y 2O 3 content on ODS-Eurofer mechanical properties

This work investigates the effects of manufacturing processes on the behaviour of ODS alloys based on the Eurofer composition (Fe–9%CrWVTa). Materials were produced by mechanical alloying with two fractions of Y 2O 3 strengthening particles (0.3% and 0.5% in weight). The ODS powders were consolidated by hot isostatic pressing (HIP) and hot extrusion (HE). In case of HE, two distinct extrusion dies geometry were used to obtain circular or rectangular cross-sections bars. Cold rolling (CR) trials were also applied to rectangular cross-section bars to obtain 1 mm thick plates with a high cumulated cold work level ( ε = 80%). Microstructure, impact and tensile properties of selected samples are compared to assess the effects of the consolidation process (HIP or HE), the Y 2O 3 content and the shape of the extrusion die.

Fission of a longitudinal strain solitary wave in a delaminated bar

The aim of the paper is to show that splitting of a waveguide leads to fission of bulk solitons in solids. We study the dynamics of a longitudinal bulk solitary wave in a delaminated, symmetric layered elastic bar. First, we consider a two-layered bar and assume that there is a perfect interface when x<0 and complete debonding (splitting) when x>0 , where the axis Ox is directed along the bar. We derive the so-called doubly dispersive equation (DDE) for a long nonlinear longitudinal bulk wave propagating in an elastic bar of rectangular cross section. We formulate the problem for a delaminated two-layered bar in terms of the DDE with piecewise constant coefficients, subject to continuity of longitudinal displacement and normal stress across the "jump" at x=0 . We find the weakly nonlinear solution to the problem and consider the case of an incident solitary wave. The solution describes both the reflected and transmitted waves in the far field, as well as the diffraction in the near field (in the vicinity of the jump). We generalize the solution to the case of a symmetric n -layered bar. We show that delamination can lead to the fission of an incident solitary wave, and obtain explicit formulas for the number, amplitudes, velocities, and positions of the secondary solitary waves propagating in each layer of the split waveguide. We establish that generally there is a higher-order reflected wave even when the leading order reflected wave is absent.

Stress state of a composite shell with a sizable opening

A mechanomathematical model for bending of packages of transversely isotropic bars of rectangular cross section is proposed. Adhesion, slippage, and separation zones between the bars are considered. The resolving equations for deflections and tangential displacements are supplemented with a system of linear differential equations for determining the normal and tangential contact stresses, and boundary conditions are formulated. A scheme for analytical solution of two contact problems—a package under the action of a distributed load and a round stamp—is considered. For these packages, a transition is performed from the initial system of differential equations for determining the contact stresses, where the unknown functions are interrelated by recurrent relationships, to one linear differential equation of fourth order and then to a system of linear algebraic equations. This transition allows us to integrate the initial system and get expressions for the contact stresses.

ねじり試験による直交三次元織物ＳｉＣ繊維／ＳｉＣ基複合材料のせん断弾性率特性

The present study examines the in-plane and out-of-plane shear moduli of an orthogonal 3-D woven SiC fiber/SiC matrix composite. A prismatic composite bar of rectangular cross section was subjected by a small torsional moment, and the torsional rigidity was measured by an optical method. Conventional Lekhnitskii's equations for orthotropic materials subjected by torsion were applied to estimate the shear moduli from the torsional rigidities obtained from the specimens with different cross sections. The estimated in-plane shear modulus agreed with the modulus measured by ±45°off-axis tensile testing. The effect of 0°tensile loading on the shear stiffness degradation was also investigated by the repeated torsion tests after step-wise tensile loading. As a result, it was revealed that both in-plane and out-of-plane shear moduli were reduced by more than 50% of the initial values due to 0°tensile load. Finite element analyses (FEA) based on the homogenization method were conducted in order to evaluate the shear moduli estimated by the torsion tests. By optimizing the void fraction between fiber bundles, the numerical results by FEA were in good agreement to the experimental results.

Account of the cross-sectional deplanation of a composite bar in determining the critical force

A method for determining the critical force for an axially compressed composite bar with account of deplanation of its cross sections is considered. It is proposed to consider the cross-sectional deplanation, which decreases the critical force, by lowering the basic flexural stiffness of the bar by the value resulting from this effect. General relationships for calculating this value for composite bars are obtained. Numerical results are presented, which show the decrease in the critical force as a function of some parameters for transversely isotropic bars of rectangular cross section.

Shear wave focusing for three-dimensional sonoelastography

A new vibration scheme is shown to provide localized vibration fields for three-dimensional sonoelastography imaging. The theoretical vibration distributions of double strip loads vibrating normally to the surface of a semi-infinite elastic space are calculated. A localization or focusing of shear waves inbetween the double-strip loads is predicted. Experimentally, two parallel rigid rectangular cross-section bars are mounted on an electromagnetic shaker. Driven by the signal source, the bars vibrate against the surface of a tissue-mimicking phantom. The double-bar source is also used to propagate shear wave into an ex vivo prostate phantom with a 6 mm "tumor" in it. A combination of high frequencies (400-600 Hz) is used to drive the double-bar applicator. In the phantom experiments, a shear wave focal zone with higher vibration amplitude and uniformity predicted by the theory was confirmed. The position of the focal zone is controllable when adjusting the separation of the bars as the theory shows. When this vibration scheme was used in a prostate phantom experiment, high-resolution tumor images with clear boundaries are obtained. The parallel bar is an ideal applicator to create more uniform vibration within a controllable localized volume. The field has uniformity especially in the direction along the bars.

A numerical solution of torsional stress wave propagation in layered orthotropic bar of rectangular cross-section

A numerical scheme based on the method of numerical integration along bicharacteristics is developed for the numerical analysis of torsional wave propagation in layered orthotropic media. The dynamic behavior of a layered orthotropic bar of rectangular cross-section due to impact torque is studied. Moreover, the stability of the present method is investigated.

Torsional stress wave propagation in orthotropic bar of rectangular cross-section

In this paper, a torsional wave propagation problem of anisotropic medium is analyzed by using the method of numerical integration along bicharacteristics. The dynamic behavior of an orthotropic bar of rectangular cross-section due to an impact torque on the end surface of the bar is studied and the stability of the present method is investigated.

On Dynamics of Bar of Rectangular Cross Section

The propagation of nonstationary waves in semi-infinite elastic rectangular bars is studied. It is assumed that two opposite lateral surfaces of the body are free of forces, while the two others are subjects to cross conditions. By introducing three new potential functions, the author succeeded in getting closed-form solutions in Laplace and Fourier transform parameters. Inversion of the transform solutions, carried out by an original method of inversion, is suggested herein.

Anisotropy of torsional rigidity of sheet polymer composite materials

Wide application of polymer composite materials (PCM) in modern technology calls for detailed evaluation of their stress-strain properties in a broad temperature range. To obtain such information, we use the dynamic mechanical analysis and with the help of a reverse torsion pendulum measure the dynamic torsional rigidity of PCM bars of rectangular cross section in the temperature range up to 600 K. It is found that the temperature dependences of the dynamic rigidity of the calculated values of dynamic shear moduli are governed by the percentage and properties of the binder and fibers, the layout of fibers, the phase interaction along interfaces, etc. The principles of dynamic mechanical spectrometry are used to substantiate and analyze the parameters of anisotropy by which the behavior of a composite can be described in the temperature range including the transition of the binder from the glassy into a highly elastic state. For this purpose, the values of dynamic rigidity are measured under low-amplitude vibrations of the PCM specimens with a fiber orientation angle from 0 to 90°. It is shown that for unidirectional composites the dependence between the dynamic rigidity and the fiber orientation angle is of extreme character. The value and position of the peak depend on the type of the binder and fibers and change with temperature. It is found that the anisotropy degree of PCM is dictated by the molecular mobility and significantly changes in the temperature range of transition of the binder and reinforcement from the glassy into a highly elastic state (in the case of SVM fibers). The possibility of evaluating the anisotropy of composites with other reinforcement schemes, in particular, of orthogonally reinforced PCMs, is shown.