Beam Of Rectangular Cross-section Research Articles

Edgewise Bending Vibration Analysis of a Rotating Sandwich Beam with Magnetorheological Elastomer Core

The vibration of a rotating sandwich beam with magnetorheological elastomer (MRE) as a core between two elastic layers is theoretically analyzed in this paper. This study is focused on the bending vibration along the edgewise direction of a sandwich beam of rectangular cross-section, which, to the best of our knowledge, has not been addressed yet. The classical Euler–Bernoulli beam theory is used to model the dynamic behavior of the elastic layers. In the modeling, the effect of the MRE layer is considered by incorporating its shear strains and the inertia due to shear deformation and bending motion. The governing equations of motion of the rotating sandwich beam are derived by using the Ritz method and the Lagrange’s equations. The effects of the applied magnetic field, core layer thickness, rotational speed, setting angle and hub radius on the natural frequencies and the corresponding loss factors are investigated parametrically. The results show the significant effect of the magnetic field intensity and the MRE layer thickness on the modal characteristics of the MRE sandwich beam.

A modeling method for vibration analysis of cracked laminated composite beam of uniform rectangular cross-section with arbitrary boundary condition

This paper establishes an analysis model to study the vibration behavior of a cracked laminated composite beam with uniform rectangular cross-section based on the Jacobi-Ritz method and the first-order shear deformation theory (FSDT). The boundary conditions of both ends of the cracked laminated beam are modeled as the elastic spring and the beam is divided into two parts by the crack section. The continuous conditions at the connecting face are modeled by the inverse of the flexibility coefficients of the fracture mechanics theory. Ignoring the influence of boundary conditions, displacements admissible functions of cracked laminated beam can be set up as Jacobi orthogonal polynomials. The accuracy and robustness of the present method are evidenced through comparison with previous literature and the results achieved by the finite element method (FEM). Numerical examples are given for free vibration analysis of cracked laminated composite beams with various boundary conditions, which may be provided as reference data for future study.

Scaling laws for static displacement of linearly elastic cracked beam by energy method

Experimental studies on the problem of crack identification can be simplified and enhanced through the concept of structural similarity since the behaviors of a full-scale structure can be investigated using a reduced-scale model. The structural similarity of a rectangular cross-section beam with a through-thickness edge crack and elastically supported was investigated. The beam was subjected to concentrated forces perpendicular to its longitudinal axis and concentrated bending moments. Complete similarity conditions and scaling laws for static deflection and slope of the cracked beam were derived using the principle of conservation of energy and the strain energy release rate approach within the framework of linear elastic fracture mechanics. Accuracy of the derived complete similarity conditions and scaling laws were experimentally and numerically verified. The concept of incorporating the similarity theory to problems of structural health monitoring was explained.

A single variable shear deformable nonlocal theory for transversely loaded micro- and nano-scale rectangular beams

In this paper, a simple single variable shear deformable nonlocal theory for bending of micro- and nano-scale rectangular beams is presented. To incorporate small size effects, the theory uses Eringen’s nonlocal differential constitutive relations. The theory has only one fourth-order governing differential equation involving a single unknown variable. The governing equation and the expressions for the bending moment and shear force of the present theory are strikingly similar to those of nonlocal Euler-Bernoulli Beam Theory (EBT) formulated based on Eringen’s nonlocal elasticity theory. The theory assumes that the axial and lateral displacements have bending and shear components such that the bending components do not contribute towards shear force, and the shear components do not contribute towards bending moment. Also, the chosen displacement functions of the theory give rise to a realistic parabolic transverse shear stress distribution across the beam cross-section. Efficacy of the proposed theory is demonstrated through bending of simply supported, cantilever and clamped-clamped micro- and nano-scale beams of rectangular cross-section. The numerical results obtained by using the present theory are compared with those predicted by other nonlocal first-order and higher-order shear deformation beam theories. The results obtained are quite accurate.

Vibration of deploying rectangular cross-sectional beam made of functionally graded materials

Transverse vibration and stability of deploying rectangular cross-sectional cantilever beam made of functionally graded material are investigated. The functionally graded material beam is assumed to be constructed with ceramics and metal phases, and the corresponding equivalent parameters of functionally graded material are found to continuously vary across the cross-sectional height with a simple power law. The differential equations of motion of deploying functionally graded material cantilever beam are derived by Hamilton’s principle. Based on the assumed modal method, the beam deflection function is expanded into a series, in which each term is expressed to admissible function multiplied by generalized coordinate. The eigenfunctions of cantilever beam in which the length of the beam is time-dependent are chosen as admissible functions. Galerkin method is employed to discretize the differential equation to a set of time-coordinate-dependent ordinary differential equations, and then the eigenvalue problem depending on time is obtained. The deployment motion of beam is considered as a constant speed in this study, and some numerical results, which are variations of tip deflection response and complex frequencies, are obtained. Finally, the effects of gradient index of functionally graded material, deploying speed, initial length and protruded length, the cross-sectional height on dynamical characteristics, and divergence instability of the deploying functionally graded material beam are discussed.

Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the cross-section (i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.

Effect of stress concentration in a beam of rectangular cross section in the region of attachment of the longitudinal efforts

Relevance. To ensure the safe operation of buildings and structures, it is necessary to more accurately determine the stress-strain state (SSS) of structural elements, to identify areas of stress concentration. The distribution of stresses in the region of the fastening bars in three-dimensional formulation is relatively little studied. In these areas, there may be significant stress concentrations that contribute to the occurrence and development of cracks and splits, which are a harbinger of destruction. The development of modern methods of calculation, software systems and the growth of computing capabilities allow refining the design scheme: to move from one-dimensional to two-dimensional calculation scheme, from two-dimensional to three-dimensional calculation scheme. All this makes it possible to more accurately assess the SSS of structural elements and structures, to identify areas of stress concentration, as well as to investigate the effect of the Poisson's ratio on the stress concentration. Methods of research. It is noted that cracks and breaks in the edges under the influence of longitudinal loads occur in the rods (racks) of square cross-section. Three-dimensional elements based on the spline version of the finite element method and the LIRA computational complex are used to estimate the stress-strain state. The spline finite element method, thanks to the synthesis of the idea of parametrization and the finite element method (FEM) with cubic approximation of all three required variables within each element, allows obtaining consistent three-dimensional finite elements. On the basis of the mentioned methods and complexes, numerical studies of the stress concentration in the bars of square and rectangular cross-sections fixed at one end and perceiving the tensile forces at the other end are performed. Conclusions. It is found that in the angular points of the cross section in the area of fastening of straight bars, perceiving axial tensile forces, there are stress concentrations. Away from the mounting area of the bar, the voltages are aligned. By increasing the Poisson's ratio, the stress concentration increases faster than at low values. The transition from a onedimensional design model to a two-dimensional one, and even more so to a three-dimensional model allows to determine the stress concentration, both in plan and in thickness. Information about the concentration of stresses in elements of structures will allow designers to more accurately design structures and facilities, and the operators to promptly identify the defective region.

Численное и экспериментальное исследование чистого изгиба балок из титанового сплава АБВТ-20 в условиях ползучести с учетом различных свойств на растяжение и сжатие

Рассматривается решение задачи чистого изгиба балки прямоугольного сечения в режиме ползучести с учетом различных свойств ползучести на растяжение и сжатие. Построен алгоритм и разработано программное обеспечение для математического моделирования процесса перераспределения напряжений по высоте балки с учетом накопления повреждений. Моделирование процессов ползучести разупрочняющегося материала происходит на основе уравнений кинетической теории ползучести и повреждаемости. Численное решение задачи проводится на основе метода Рунге - Кутты - Мерсона. Проведено сравнение результатов моделирования с экспериментальными данными чистого изгиба балок прямоугольного сечения из титанового сплава АБВТ-20 при действии знакопеременного изгибающего момента в условиях продолжительного воздействия температуры ($750^\circ$C), которое показало удовлетворительное соответствие результатов расчета экспериментальным данным.

A critical study of stresses of FG Timoshenko Beams using B-spline Collocation

A functionally graded material beam of rectangular cross-section is studied for bending and shear stresses under uniformly distributed mechanical load. The gradation is assumed to be along the height of the beam and follows power law. The governing equations are derived using principle of virtual work in the framework of Timoshenko beams which is solved numerically using sixth order B-spline collocation technique with Greville abscissa as collocation points. The variation in the position of neutral axis, variations in bending (axial) and shear stresses at different positions along a cross-section corresponding to different power law index values have been reported. It has been observed that shift in neutral axis governs the stress distribution along the cross-section.

Tsallis entropy modeling of Pressure Stimulated Currents when cement-based materials are subjected to abrupt repetitive bending loadings

Pressure Stimulated Currents (PSCs) are emitted while brittle materials of high electrical resistivity are subjected to mechanical loading. The PSCs are related to the generation of cracks and the consequent evolution of cracks’ network in the bulk of the specimen. In the experimental protocol described here cement-mortar beams of rectangular cross-section were subjected to Three-Point Bending (3PB) repetitive loading/unloading loops. The qualitative and quantitative characteristics of the recorded PSCs match corresponding recordings previously published. More specifically, during the load increase a spike-like PSC emission was recorded followed by a relaxation of the PSC, after reaching its final value. The relaxation process of the PSC was then studied in terms of non-extensive statistical physics (NESP) based on Tsallis entropy equation. The behavior of the Tsallis parameter was studied in relaxation PSCs in order to investigate its potential use as an index for monitoring the crack evolution process with a potential use in non-destructive laboratory testing of cement-based specimens of unknown internal damage level. This analysis of the experimental data indicated that the value of the entropic index q exhibits a characteristic decrease attaining the critical value of 1.25 while reaching the ultimate strength of the specimen, and thus could be used as a forerunner of the expected failure.

Iterative techniques for analyzing nonlinear vibrating dynamical systems in the frequency domain

The nonlinear response of prototypical structures experiencing harmonic excitation is studied using novel techniques called iterative harmonic analysis (IHA) and iterative modal analysis (IMA). First, a simple damped oscillator with a cubic hardening stiffness nonlinearity is studied, and IHA is used in this single-degree-of-freedom system. In this first section, a high-order harmonic balance is applied, and IHA is employed in order to find the amplitude coefficients for different harmonics and their codependence. Additionally, a set of nested sums are identified that describe the harmonic coupling explicitly. Secondly, a pinned–pinned nonlinear beam of rectangular cross-section is studied, and IMA is applied to find the amplitude coefficients for different modes and their codependence. The nonlinearity is introduced through the membrane effect, where axial strain due to transverse deflection becomes a significant contributor to the system behavior. Typical frequency-domain methods cannot be easily applied to these systems as the solutions of the differential equations lead to intricate coupling between coefficients of the solution, and no analytical expression exists for those coefficients. Based upon these examples, other nonlinear systems may also be considered in future work using either a modal-based or finite element model. Finally, the advantages of the new method (reduced computational cost) as well as the limitations (effectively the same as those of an nth-order harmonic balance) are emphasized.

STUDIES OF DEFLECTIONS OF STATICALLY DETERMINED REINFORCED CONCRETE BEAMS BASED ON A NONLINEAR DEFORMATION MODEL

In this paper theoretical studies of statically determinated bending reinforced concrete beams of rectangular cross-section on deformation indexes under the eff ect of a short-time uniformly distributed load are viewed. These theoretical studies are based on the main points of the nonlinear deformation model that takes into account the nonlinear work of concrete and fi tments taking into account discrete cracking. The results of calculating the deformation of beams by the method of SP 63.13330.2012 and by authors’ method are proposed as well as the results of a numerical experiment with the identifi cation of the stress-strain state of statically determinated beams in the form of a fi nite element model in the program complex «Lira CAD-2017R3» using a linear and non-linear sett ing of characteristics of concrete and reinforcement. Based on the results of theoretical studies, the calculation results by the methodology of the current standard with a numerical experiment, as well as with the authors’ calculation methodology are compared. All calculations and loading schemes in this technique are given taking into account the possibility of further experimental studies.

Lateral torsional buckling of anisotropic laminated thin-walled simply supported beams subjected to mid-span concentrated load

In this paper, a generalized semi-analytical approach for lateral-torsional buckling of simply supported anisotropic, thin-walled, rectangular cross-section beams under concentrated load at mid-span/mid-height was developed using the classical laminated plate theory as a basis for the constitutive equations. A closed form buckling expression was derived in terms of the lateral, torsional and coupling stiffness coefficients of the overall composite. These coefficients were obtained through dimensional reduction by static condensation of the general 6x6 constitutive matrix mapped into an effective 2 × 2 coupled weak axis bending-twisting relationship. The resulting two coupled stability differential equations are manipulated to yield a single governing differential equation in terms of the twisting angle. This differential equation with variable coefficients, subjected to applicable boundary conditions, was solved numerically using Mathematica. The resulting solution was found to correlate with the effective lateral-flexure, torsional and coupling stiffness coefficients to yield a general semi-analytical solution. An analytical formula was possible to extract, which was verified against finite element buckling solutions using ABAQUS for a wide range of lamination orientations showing excellent accuracy. The stability of the beam under different geometric and material parameters, like length/height ratio, layer thickness, and ply orientation, was investigated.

Stress-strain state of reinforced bimodulus beam on an elastic foundation

The paper provides the calculation theory of an arbitrary supported and arbitrary loaded reinforced beam filled with bimodulus material. The formulas determining normal stresses, bending moments, shear forces, rotation angles and a deflection of a rectangular crosssection beam reinforced with any number of bars aligned parallel to the beam axis have been obtained. The numerical study has been carried out to investigate an influence of a modulus of subgrade reaction on values of maximum normal stresses, maximum bending moments and a maximum deflection of a hinged supported beam loaded with a point force or uniform distributed load. The estimation is based on the method of initial parameters for a beam on elastic foundation and the Bubnov-Galerkin method. Values of maximum deflections, maximum bending moments and maximum stresses obtained by these methods coincide. The numerical studies show that taking into consideration the bimodulus of material leads to the necessity to calculate the strength analysis of both tensile stresses and compressive stresses.

Lateral-torsional buckling of simply supported anisotropic steel-FRP rectangular beams under pure bending condition

Abstract In this paper, a generalized analytical approach for lateral-torsional buckling of simply supported anisotropic hybrid (steel-FRP), thin-walled, rectangular cross-section beams under pure bending condition was developed using the classical laminated plate theory as a basis for the constitutive equations. Buckling of such type of hybrid members has not been addressed in the literature. The hybrid beam, in this study, consists of a number of layers of anisotropic fiber reinforced polymer (FRP) and a layer of isotropic steel sheet. The isotropic steel sheet is used in two configurations, (i) in the mid-depth of the beam sandwiched between the different FRP layers and (ii) on the side face of the beam. A closed form buckling expression is derived in terms of the lateral, torsional and coupling stiffness coefficients of the overall composite. These coefficients are obtained through dimensional reduction by static condensation of the 6 × 6 constitutive matrix mapped into a 2 × 2 coupled weak axis bending-twisting relationship. The stability of the beam under different geometric and material parameters, like length/height ratio, ply orientation, and layer thickness, was investigated. The analytical formula is verified against finite element buckling solutions using ABAQUS for different lamination orientations showing excellent accuracy.

Modeling of 1D acoustic device composed of a planar beam and a fluid gap with discontinuity in thickness

Precise modeling of 1D acoustic devices (passive or active, miniaturized or not) containing a planar beam loaded by a fluid gap and cavities is of interest in a variety of applications (transducers, acoustic filters, metamaterials, etc.). An analytical approach presented herein enables to describe the vibration of the planar elastically supported rigid beam of rectangular cross-section surrounded by very thin slits and loaded by the fluid gap, which is divided in three parts of different thicknesses (the central part being the thinner one), the thermoviscous losses originating in the fluid being taken into account. Comparing to the case of the fluid gap of uniform thickness, such a geometry provides more parameters which can be adjusted in order to achieve the required behavior (resonant or damped, etc.). The analytically calculated beam displacement is presented and compared to the numerical solution provided by finite element method (a reference against which the analytical results are tested).

Elastic Stability of Uniform and Hollow Columns

The stability of elastic beams and columns is investigated. The beams are subjected to the axial pressure at the elastically supported end whereas the other end of the beam is clamped.The attention is confined to the case of the beam of rectangular cross-section of piece wise constant thickness. The both, the uniform and the hollow-sectional beams are considered. It is assumed that the beams are weakened by stable cracks.The influence of crack parameters on the critical buckling load is assessed making use of the method of massless rotational spring.

STRENGTH AND CRACK RESI STANCE OF REINFORCED CONCRETE BEAM STRUCTURES UNDER THE ACTION OF LOW-CYCLE LOADS OF CONSTANT SIGN AND CYCLIC ALTERNATING LOADINGS OF HIGH LEVELS

Purpose. To investigate and to analyze the behavior of the areas near supports of reinforced concrete beam elements under the action of low-cycle loads of constant sign and repeated alternating loadings taking into account the influence of constructive factors and factors of external action for the future using of obtained results for improvement of calculated engineering technique. Methodology. Three series of experiments were performed to research this problem. Test specimens are reinforced concrete beams of rectangular cross section with sizes 200´100 mm, with length 1975 mm, that were reinforced with two flat welded framework with symmetrically located top and bottom longitudinal reinforcement 2Ø14А500С and transverse reinforcement 2Ø3, 4, 5ВрI with relative shear span = 1, 2, 3, that were made from concrete С 16/20, С 30/35, С 40/50, and that were tested by monotonic load, low-cycle alternating loading (= ±0,50; ±0,65; ±0,80) and repeated load of constant sign (= 0…0,50; 0…0,65; 0…0,85). Findings. This material considers and analyses obtained and processed experimental data, that discover the influence of low-cycle loads of constant sign and repeated alternating loadings of high levels on the main proofs of crack resistance and strength of inclined and vertical sections of areas near supports of reinforced concrete structures tacking into account the change of selected by planning of an experiment constructive factors and factors of external action. Originality. In this paper the systematized experimental data are given, that reflect new features of reinforced concrete beam elements work under the action of mentioned loads. Practical value. The research results are being introduced to the practice of design and calculations of areas near supports of reinforced concrete elements that are subjected to the action of low-cycle loads, and also the results can be used in educational process of technical academies.

On the analysis of pure bending of rigid-plastic beams in strain-gradient plasticity

Abstract The complete stress field, including the microstress, the moment-stress, and the line forces are derived for the pure bending of a rigid-plastic beam of rectangular cross-section in the model of strain-gradient plasticity. The workless spherical parts of the microstress and the moment-stress tensors are incorporated in the analysis. Their determination is shown to be of importance for the fulfilment of the higher-order traction boundary conditions, the physical interpretation of line forces, and their contributions to bending moments. Three equivalent methods are used to derive the moment-curvature relationship for any of the gradient-enhanced effective plastic strain measures from the considered broad class of these measures. Specific results are given for the selected choice of the stress-strain relationship describing the uniaxial tension test. Closed-form analytical expressions are obtained in the case of linear hardening, and in some cases of nonlinear hardening. The analysis of the plane-strain bending of thin foils is also presented. In this case there are two sets of line forces along the edges of the beam. The relationships between the applied bending moment and the curvature, and between the lateral bending moment and the curvature are derived and discussed. The lateral bending moment along the lateral sides of the beam, needed to keep the plane-strain mode of deformation, is one-half of the applied bending moment.

Distortions and Residual Stresses at Layer-by-Layer Additive Manufacturing by Fusion

Thermal shrinkage of the added material can distort the manufactured part and generate residual stresses. Experiments are carried out on growing the beams of rectangular cross section. The beams bend with formation of a concave top surface. The distortion is characterized by the curvature radius. The curvature radius significantly increases with the beam height, however, its variation with the layer thickness is within the experimental uncertainty. The proposed mathematical model assumes sequential addition of thermally expanded elastic layers. It explains the experiments and indicates the existence of finite limits for the stress and the deformation fields and the curvature radius at small layer thickness. The proposed model can be applied to predict residual stresses and deformations arising in complicated parts.