Beam Of Rectangular Cross-section Research Articles

Evaluation of the errors of measuring residual stresses by the hole drilling method

A sample configuration with a specified distribution of residual stresses in depth was developed to elaborate modes for studying residual stresses using mechanical and physical methods. The sample formation technique is based on non-uniform plastic deformation of an aluminum beam of rectangular cross section according to the pure bending scheme. The control of the deformed state during loading was performed on the end surface of the sample by the field of normal deformations obtained using a digital image correlation system. The depth of the plastically deformed layer was 1.3 mm. The theoretical distribution of residual stresses obtained as a result of unloading of a plastically deformed sample was determined from the results of a numerical calculation of a finite element model, with allowance for the physical and mechanical characteristics, elastic-plastic hardening, as well as the deformation curve in true coordinates obtained as a result of uniaxial tension on elementary samples. A study of residual stresses inhomogeneous in depth was carried out by drilling holes in accordance with ASTM E837 on two opposite sides of the sample: the region of tension under loading and the area of compression, respectively. The control of the deformation response resulted from drilling was recorded using three-axis strain gauge rosettes. The results of the actual measurement of the longitudinal component of residual stresses were compared with their theoretical distribution obtained by a numerical method. The root-mean-square error in measuring residual stresses, relative to their theoretical distribution, for an aluminum alloy sample reaches 18.7 MPa. It should be noted that largest measurement errors were recorded at small depths, since they are characterized by small values of deformations comparable to the value of shot noise.

Load-Bearing Capacity of Reinforced Concrete and Fiber Concrete Cross-Beam Systems

The results of experimental studies and computer modeling of reinforced concrete and fiber concrete cross-beam systems are presented. The authors have made a special stand, the design of which allows to carry out researches of bearing capacity and deformability of such systems under the concentrated and distributed static loads. Samples made of reinforced concrete and steel fiber concrete and consisting of four mutually perpendicular beams of rectangular cross-section were tested. The specimens were reinforced in the lower zone with longitudinal reinforcement of 8 mm in diameter, two rods in each beam. Steel-fiber concrete specimens have an additional dispersed reinforcement of steel fiber with bent ends in an amount of 1% by volume of concrete. A procedure of finite-element modeling and calculation of cross-beam systems in the licensed program ANSYS 17.1 was developed, and a comparison of the results obtained experimentally and on the basis of computer modeling was performed. The tests showed that dispersed reinforcement of the cross-beam system with steel fiber leads to an increase in its bearing capacity by 1.23 times. During modeling and finite-element analysis in the ANSYS 17.1 program, the increase in bearing capacity was 1.18 times. At the same time, the load-carrying capacity of the reinforced concrete cross-beam system obtained by the finite-element method was 13% lower than in the experiment, and that of the fiber-reinforced concrete system was 15% lower. At the same time, the discrepancy in the approaches considering torsion and not taking it into account is about 5% for all the samples studied in the work.

Approximate determination of shear stresses for a 2D beam made of a non-Green elastic solid

Shear stresses for a two-dimensional (2D) beam are calculated modifying the classical method developed by Jouravski, for the case the linearized strain tensor is assumed to be a nonlinear function of the Cauchy stresses. Two problems are studied, namely, the case of a cantilever beam with a point load on its free edge (considering a rectangular and a circular cross-section) and the three-point flexural test for a beam of rectangular cross-section. Numerical results are obtained for the particular case of a bimodular constitutive model for rock, and the results for the shear stresses are compared with the predictions of the classical theory of strength of materials for such problems, assuming a linearized elastic body.

On the equivalent flexural and shear moduli of laminated beams: Definition and determination by bending tests

Equivalent flexural and shear moduli of laminated beams of rectangular cross-section made of orthotropic layers are defined by an analytical approach. Equivalent flexural and shear modulus are defined as those that would correspond to a homogenous beam. In order to check the suitability of the moduli defined, three-point bending tests varying the span are carried out in a sandwich material and in hybrid laminates. In the case of the sandwich material, it is shown that a three-point bending test could be considered a shear test from the stiffness point of view. In the case of hybrid materials, virtual tests are carried out by the Finite Element Method. The agreement between equivalent moduli obtained from the analytical approach and the bending tests has been checked, being quite good in all cases. Finally, the effect of dimension uncertainty in three-point bending tests is analyzed by a Monte Carlo simulation.

Electromechanical response of boron nitride nanosheet reinforced nanocomposite beam: A finite element study

The objective of the present study is to investigate the electromechanical response of a piezoelectric boron nitride nanosheet reinforced nanobeam accounting for the surface and the flexoelectric effects using finite element analysis. The finite element model was developed by using the size-based Euler–Bernoulli beam model, modified piezoelectricity theory, and Galerkin's weighted residual method. The boron nitride nanosheet-reinforced nanobeam was loaded with uniformly distributed load and point-loading conditions. Three common boundary conditions for beams such as clamped-free, simply supported, and clamped-clamped have been considered here. The electromechanical behavior of the boron nitride nanosheet reinforced nanobeam has been studied under the pure surface, pure flexoelectric, as well as combined surface and flexoelectric effects. It is observed that the integrated surface and flexoelectric effects are mainly responsible for enhancing the electromechanical performance of the nanobeam. For the thickness H = 20 nm, the maximum deflection of the nanobeam was reduced by ∼50% when both the flexoelectricity and surface effects are combined together for all the support conditions. Moreover, the circular cross-section beam becomes ∼30% stiffer than the rectangular cross-section beam under the integrated effect of surface and flexoelectricity under all loading conditions. Hence, the highly size-dependent surface and flexoelectricity must be explored in the accurate electromechanical behavior of the nanostructure. Furthermore, beam stiffness is highly influenced by the flexoelectric effect irrespective of the beam boundary conditions whereas the surface effect is largely reliant on the beam boundary conditions. This research work provides a methodology to design efficient boron nitride nanosheet reinforced nanostructures that may potentially be applied in the design and development of several nanoelectromechanical systems such as force and pressure-based nanosensors and actuators.

Experimental investigation on shear behaviours of textile reinforced concrete beams

An experimental study on eight concrete beams reinforced by combining GFRP bars for moment and glass textile sheets for shear resistances is conducted to examine the use of these non-corrosive reinforced materials in creating the structural concrete beams. Rectangular cross-section beams with two different aspect ratios, 1.1 and 1.7, are investigated. For each group, there are two control beams with and without GFRP bars for shear reinforcements. The other two beams in each group are reinforced with one and two glass textile grids for shear resistance. Fine-grained high-strength concrete with a similar grade is used for all tested beams. Three-point bending tests are employed to examine the shear performances of these specimens, including the load–deflection curves, cracking patterns, failure modes and strains in reinforcements. The experimental results indicate that the crushing failure is recorded in all tested short-span beams reinforced with shear reinforcements. Increasing the number of textile layers improves the shear capacities of the specimens. The shear strengths of the tested beams are significantly affected by the aspect ratio. Finally, the ratios of the effective experimental strength over the compressive strength of concrete are compared with the existing guidelines.

Thermoelastic damping in high frequency resonators using higher-order shear deformation theories

Thermoelastic damping (TED) in high frequency oscillators shifts their natural frequencies and attenuates the vibration amplitude. The amount of damping depends upon the temperature of the environment in which the device is operating. Here, we model the oscillator as a simply-supported beam of rectangular cross-section undergoing infinitesimal deformations. We consider three (the Levinson, the Timoshenko and a sinusoidal) through-the-thickness shear strain distributions, both translational and rotational inertia, and heat conduction to analytically delineate their effects on the shift in frequencies and the attenuation of their amplitudes for the lowest three vibration modes. The shear deformation distribution function multiplied by the Poisson ratio appears in the volumetric strain. The TED in the Timoshenko, the Levinson and a beam with sinusoidal variation of shear stresses is related to that in the Euler beam to recover the famous Lifshitz–Roukes formula. Various results are presented to qualitatively and quantitatively illustrate the dependence of the TED upon the lowest three frequencies and the resonator thickness. The analytical solutions presented here can be used as benchmark solutions for checking the numerical solutions.

Определение прогибов физически нелинейной балки на упругом основании

The article deals with the issues of determining the deflections of a beam lying on an elastic foundation with a constant coefficient of rigidity, while the beam is made of a material that has a non-linear relationship between stresses and deformations. The physical non-linearity of the beam material is taken into account by approximating the relationship between stresses and strains with a cubic parabola; such an approximation well describes the deformation curves of a non-linear elastic body with the same diagram of the work of the material in tension and compression. As an example, the deflections of a non-linear elastic beam of rectangular cross section, which lies on the Winkler base and carries a uniformly distributed load along the entire length, are considered for three cases of support fastenings at the edges: with two hinged supports, with two terminations, and with termination and hinged support. The method of successive approximations is used to obtain a solution to the non-linear equation for deflections, which depends on dimensionless parameters that take into account the influence of an elastic foundation, the physical non-linearity of the material, and a uniformly distributed load. The dependence of the change in the value of the maximum relative deflection on the coefficient of the bed and the distributed load, obtained by calculation, taking into account the influence of the physical non-linearity of the material for three cases of support fastenings, is given. The results of the calculations showed that the presence of additional bonds reduces the influence of the physical non-linearity of the beam material on its deflections.

SIMULATION OF TEMPERATURE EFFECTS IN MASSIVE BODIES USING THE MODIFIED METHOD OF LINES

This article presents the main ideas and possibilities of the modified straight line method for modeling temperature effects in massive bodies. The calculation model of a rectangular beam with a defined thermal state is adopted. The procedure for reducing the dimensionality of the original equations using the modified method of straight lines and the Bubnov-Galyorkin-Petrov method for boundary conditions is applied. The reduced equation of thermal conductivity and the reduced equation of heat balance are given. Conclusions are made regarding the addition of reduced second-order equations with components from the boundary conditions on the lateral surfaces, and finally reduced second-order equations in spatial variables are given.
 We are considering a beam of rectangular cross-section, the three dimensions of which are of the same order. The thermal state of such an object is three-dimensional and is therefore described systematically using three-dimensional heat conduction equations. When applying the modified method of straight lines to reduce the dimensionality of the original equations, it is convenient to use them in the form of a system of partial differential equations of the first order in spatial coordinates.
 According to the modified method of straight lines, two systems of straight lines parallel to the coordinate axes Oy and Oz are applied to the beam (a decrease in dimensionality in terms of spatial variables y and z is assumed). The choice of such straight lines is due to the fact that the reduced equations will depend on the variables x and t, that is, they will resemble the one-dimensional equations of the one-dimensional non-stationary problem of thermal conductivity, for which efficient and convenient numerical methods have been developed in the literature of mathematical physics at the moment, to which it is easy to adapt the reduced equation.

High-order torsion analysis of sandwich beams of rectangular cross-section

The structural response of soft-core sandwich beams of rectangular cross-section subjected to torsion is investigated. The paper uses the concepts of the extended high-order sandwich panel theory and generalizes the approach to account for the torsional response and its unique mechanisms. An analytical model that utilizes this generalized approach is formulated. The results of the model are presented and compared with the results of a reference model and a finite element analysis. Emphasis is placed on the torsional resistance mechanisms, their interaction, and the torsional-related localized effects. The results of the present model are in good agreement with the finite element analysis in terms of displacements and stresses, including the localized effects attributed to the soft-core. It is also shown that the present model can identify and quantify the localized effects while reference models available in the literature fail to do so. The results of the model contribute to the understanding of the torsional response of soft-core sandwich beams, including the transition between the warping and Saint-Venant torsional mechanisms and the localized effects associated with the soft-core and the application of boundary conditions to the different layers.

Tapered rainbow metabeam for wideband multimode acoustic blocking based on quadruple-mode resonators

We construct a rainbow metamaterial for multimode sound blocking over a broad range of sub-kHz frequencies in the form of a tapered rectangular cross section beam of machined cells based on elements that can, on average, simultaneously attenuate the majority of the possible elastic-wave polarizations. Using aluminum, we construct a five-cell structure containing sub-wavelength planar resonators with interconnected ribs, which couple to compressional, in-plane shear, flexural, and torsional vibrations. Backed up by numerical simulations, we verify that this tapered structure can, on average, strongly attenuate acoustic modes over the frequency range of the combined metamaterial bandgaps, that is over a frequency range representing ∼50% around ∼0.7 kHz. Applications include vibration isolation.

Vibrodiagnostics of the Presence of Surface Symmetrical Cracks in Beams of Rectangular and Round Cross-Sections by the Characteristics of Their Nonlinear Longitudinal Vibrations

Vibrodiagnostics of the Presence of Surface Symmetrical Cracks in Beams of Rectangular and Round Cross-Sections by the Characteristics of Their Nonlinear Longitudinal Vibrations

ВИЗНАЧЕННЯ ОПТИМАЛЬНОГО КРОКУ БАЛОК ДОЩАТОГО ПЕРЕКРИТТЯ

The article presents the results of optimization of the beams step of the building wooden floor according to the criterion of the minimum material consumption. The slab consists of parallel wooden beams of rectangular cross-section, on which the boardwalk is laid.

НЕЛИНЕЙНЫЙ РАСЧЕТ ЖЕЛЕЗОБЕТОННОЙ БАЛКИ НА УПРУГОМ ОСНОВАНИИ С ПОМОЩЬЮ ЗАВИСИМОСТИ «ЖЕСТКОСТЬ–КРИВИЗНА»

A reinforced concrete beam on an arbitrary elastic base under the action of an external load is considered. Its movements and the distribution of internal forces in its sections are determined. The calculation is performed by the variation-difference method using finite differences of increased accuracy. Initially, the beam is divided into identical rectangular sections and a stiffness matrix is constructed for the elastic base as the inverse of the malleability matrix. The functional of the total potential energy is compiled as the sum of the energy of the bending of the beam, the deformation of the elastic base and the work of the external load in the form of a quadratic function of the displacements of the centers of the sections on the beam. By differentiating the latter for each displacement, a system of linear algebraic equations is formed, the solution of which is the displacement of the centers of the sections on the beam. An iterative algorithm is organized, where at each iteration, according to the “stiffness-curvature” relation ship, the bending stiffness on each section of the beam is specified. Examples of calculation of rectangular cross-section beams on an elastic layer and a Winkler base are given.

Flexural performance and ductility of RC beams made using natural LWA

This paper evaluates experimentally the feasibility of utilizing local natural lightweight aggregates (LWA) for designing reinforced concrete (RC) beams. The aggregates investigated were of scoria origin, with an absorption capacity of 10%, and dry densities ranging between 860 and 1120 kg/m3. A series of full-scale 16 simply supported RC beams of rectangular cross section were designed using two concrete strengths, normal and high, and two tension steel ratios of the balanced reinforcement ratio, following ACI code requirements. For the lightweight concrete (LWC) mixes designed, the replacement level of normal weight aggregates (NWA) with coarse LWA was 100%, whereas the replacement level for fine LWA was 62.5%. Data presented include load-deflection, and moment-curvature graphs, cracking behavior, mode of failure and ductility indexes. Test results indicated that the flexural response of LWC beams was similar to that of normal weight concrete (NWC) beams. All the LWC beams tested failed in ductile flexural mode, exhibited relatively larger mid span deflections, higher curvatures, and wider cracks, compared to NWC beams of equal concrete strength and steel ratio. The most influential parameter on the ductility indexes of LWC beams was the steel ratio. Moreover, the scoria LWA investigated can be used successfully in structural design.

Optimizing the Load-Bearing Capacity of Cross Timber Beams Using Different Types of Timber

This article examines the operation of structures made of solid wooden beams and composite glued beams of two types, consisting of different types of wood. The object of the study was a beam structure, which is a homogeneous slab resting along a contour of constant stiffness under the action of an external load. The calculation of structures made of glued beams of rectangular cross-section has been performed. The calculation is based on the finite element method, analysis of the main methods and approaches for the optimal design of spatial structural structures/The main unknowns are the displacements and rotations of the nodes of the design scheme, based on this, the idealization of the structure is embodied in a form adapted to the use of this method: this system is presented in the form of a certain set of bodies of a standard type, which is attached to the nodes.

Design Aids for Beams of Rectangular Cross Section with Parabolic Haunches: Part 2

In this paper, the most used design aids for beams of rectangular section with parabolic haunches are presented. The beam is found subjected to concentrated load localized anywhere on the beam, and takes into account the bending and shear deformations to obtain the fixed-end moments factors, which is the main contribution of this research work. The first part of this work presents the design aids for the same type of beams to obtain the carry-over factors, the stiffness factors, and the fixed-end moments factors are obtained for a uniformly distributed load. The variables and the simplifications are the same as in the part 1. Also, a numerical example is shown using the design aids, and it is compared with the equations previously presented by the authors in a paper prior to this document, and the results are equal to a three-digit approximation. Therefore, design aids provide a great tool of help for structural engineers by the greatly time-savings.

Design Aids for Beams of Rectangular Cross Section with Parabolic Haunches: Part 1

In this paper, the most used design aids for beams of rectangular section with parabolic haunches are presented. The variables considered are: the heights at its two ends vary parabolically, the width of the beam is constant, and the height in the central part is constant. The beam is found subjected to a uniformly distributed load, which takes into account the bending and shear deformations to obtain the fixed-end moments factors, the carry-over factors and the stiffness factors, which is the main contribution of this research work. The simplifications are: the heights at its two ends are equal, as these are the most used by aesthetics in architectural constructions, and the total height of the beam in the central part is considered of 0.10 of the length of the beam according to the ACI (American Concrete Institute) code for deflection control. Also, a numerical example is shown using the design aids, and it is compared with the equations previously described by the authors in an article prior to this document, and the results are equal to a three-digit approximation. Therefore, design aids provide a great tool of help for structural engineers by the greatly time-savings.

Theoretical Analysis and Experimental Study on Normal Stress of Three Kinds of Beams Subjected to Pure Bending

The normal stresses of pure bending of a single rectangular cross-section beam, laminated beam and composite beam have been theoretically analyzed and calculated and experimentally tested by electrical measurement method in this paper. Through the analysis of the measured data, the correctness of theoretical calculations and experimental methods are verified, and the normal stress distribution laws of the three beams are revealed. The data shows that the normal stress of pure bending of the laminated beam is the largest, followed by that of the composite beam, and that of the single rectangular cross-section beam is the smallest. In addition, the reasons for the large errors of some experimental data are analyzed and discussed. By increasing the content and difficulty of the experiment, it is conducive to promote students to master more knowledge of material mechanics and exercise their practical skills. The result can provide a reference for the experimental teaching of material mechanics in other colleges and universities.

On pure bending in non-linear elasticity: A circular closed-form 2D solution for semi-linear orthotropic material

Large-deformation pure bending of a straight beam of uniform rectangular cross-section in plane strain is re-visited and generalised to an orthotropic linear-elastic material and the case of plane stress. The beam is bent into a completely shearless state by means of normal surface traction of a specific profile. For a linear-elastic material expressed in terms of Biot strain and stress tensors (orthotropic generalisation of the so-called semi-linear material) the solution still exists in closed form and is expressed in terms of the applied cross-sectional traction. As such, it provides a convenient benchmark reference for numerical methods in non-linear elasticity.In agreement with the isotropic case, through-the-thickness stresses develop in the form a hyperbolic cosine, but now with the magnitude actually depending on a measure of orthotropy in the material, while the longitudinal stresses change over the thickness in the form of a hyperbolic sine. In general, the unstrained axis is displaced from the centroidal axis.For an ideal material with infinite yield strength, the theoretical maximum of the magnitude of the longitudinal traction is limited to the amount of the geometric mean of the effective Young's moduli in the longitudinal and the through-the-thickness directions while, for a brittle material, bending strength reduces by approximately 715 of the square of the ratio between the magnitude of the applied traction and this mean value.