Beam Of Rectangular Cross-section Research Articles

Series solution of beams with variable cross-section

In structural engineering beams with non-constant cross-section or beams with variable cross-section represent a class of slender bodies, aim of practitioners’ interest due to the possibility of optimizing their geometry with respect to specific needs. Despite the advantages that engineers can obtain from their applications, non-trivial difficulties occurring in the non-prismatic beam modeling often lead to inaccurate predictions that vanish the gain of the optimization process. As a consequence, an effective non-prismatic beam modeling still represents a branch of the structural engineering of interest for the community, especially for advanced design applications in large spans elements.A straight beam of length l with variable inertia J(z) is provided in figure, subject to a generic live load condition q(z). The vertical displacement y(z) can be obtained from the solution of the differential equation of the elastic line, i.e., taking into consideration the inertia variability and neglecting, as first approximation, any shear contribution. Even if this solution is an approximate one, it is able to deal with the problem in its basic formulation.In this paper a solution for the problem stated is formulated using a series expansion of solutions, in a general load and cross section variability condition. Solution is thus obtained for a generic rectangular cross section beam with a variable height. Analytical solution is presented and evaluated using numerical evaluation of some cases of practical interest.

Simulation and Experimental Substantiation of Beam Deflection under Guided End Conditions

This paper studies the nonlinear deflection characteristics of a rectangular cross sectional beam with guided support conditions. In this study two different end conditions namely guided – guided and guided – simply supported have been examined. Beams made with two dissimilar materials for instance, aluminum and steel have been considered for this study. Different loading conditions namely point load and the amalgamation of uniformly distributed load and point load have been taken into account in this study. The nonlinear response of a beam under static loading condition is influenced by various parameters like sectional properties of the beam, material, loading and boundary conditions etc. A separate loading fixture was fabricated using steel to apply the load (Point load and uniformly distributed load). The loading fixture was validated by performing an experimental measurement of the deflection under various loading conditions on a simply supported beam. The corresponding theoretical displacement values were calculated using the findings in literature and compared with test results .Both the results were found matching with each other with an average variation of just 10%. Based on this validation lesson, the loading fixture was incorporated in the actual study. Displacement values from the nonlinear static analysis were predicted using Finite Element Method and correlation was made with the experimental values for the actual beam setup. Close correlation among the numerical and physical test results was achieved and the maximum error was 8%.

The Influence of the Height of Fiber Reinforcement Zone on the Crack Resistance of Rubcon Beams with Mixed Reinforcement

In order to research on the influence of the height of the fiber reinforcement zone on the normal sections crack resistance, we produced and tested rubcon beams of rectangular cross section with mixed reinforcement; our studies showed that the height of the fiber reinforcement zone has a similar effect on the normal sections crack resistance with the percentage of longitudinal reinforcement. It is important to note that the arrangement of fiber reinforcement at 3/4 of the section height greatly complicates the process of manufacturing the beam while not increasing the strength of the tensile zone compared to the beams with the location of the fiber over the entire height of the section.

Flexural Behavior of Lightweight Concrete Beams Reinforced with GFRP Bars and Effects of the Added Micro and Macro Fiber

This study evaluated the effect of macro steel fiber (SF), micro glass fiber (GF) and micro polypropylene fiber (PF) in lightweight aggregate concrete, (LWAC) beams reinforced with glass fiber reinforced polymer (GFRP) bars. Firstly, concrete mixtures with different volume fractions of GF, PF and SF were tested up to compressive strength, then determine the optimum fiber content GF, PF and SF added into LWAC mix by 0.3%, 0.8% and 0.25% by the volume of concrete, respectively. Meanwhile, eight rectangular cross-section beams with 100 mm (width) × 200 mm (depth) × 1500 mm (length) were tested by four-point bending beam test up to the ultimate load. The GFRP bars were used to reinforce all beams. The failure modes, load-deflection behavior, ductility, flexural capacity and energy absorption were compared in the test results. The experimental results shown added fibers into LWAC improved the flexural capacity, ductility and energy absorption also enhanced moment capacity by 10.07% to 110%. The results indicated that failure modes of GF and PF specimens were in good consistency with the ACI 440.1R-06 predicted failure modes, but for SF specimens, only concrete crushing failure modes accrued. At the end step, the correction factor ( ) obtained from calculated of the experimental results with the flexural capacity according to the ACI 440.1R-06 and ISIS design manual No. 3.

Assessment of Bending Reinforced Concrete Beams Crack Resistance

The problem of determining the stress state in a section of bending reinforced concrete beams with an initial and operational cracks has been solved analytically. To this end, a system of two nonlinear algebraic equations has been obtained from the equilibrium conditions of the part of the beam cut along the crack line. From this system, for a beam with an initial crack, the height of the compression zone and the nominal stress at the crack tip are determined, and for an operational crack, the height of the compression zone and the length of the crack are determined. The remaining parameters of the stress state are expressed in terms of these values. There is also determined the value of the external moment above which an increase in the initial length of the crack occurs.Determining the stress intensity factor (SIF) is based on the assumptions that the longitudinal forces are equal at the crack tip, with and without stress concentration. The size of the stress concentration zone is determined from the condition that the local stress is equal to the nominal stress. On this basis, the formula for determining the SIF is obtained. The paper analyzes the SIF dependence on the crack length and the bending moment.The method of calculation is valid for beams of arbitrary cross-section, but explicit dependences are given for beams of rectangular cross-section most frequently encountered in production.The results obtained allow estimating the bearing capacity of beams with a crack, as well as their crack resistance according to the force criterion of fracture mechanics.

Rectangular Cross Section Beams Calculation on the Stability of a Flat Bending Shape Taking into Account the Initial Imperfections

The article presents the derivation of the resolving equation for the calculation of lateral buckling of rectangular beams. When deriving the basic equation, the initial imperfections of the beam are taken into account, which are specified in the form of the eccentricity of the applied load, the initial deflection in the plane of least stiffness and the initial twist angle. The influence of initial imperfections on the process of beam stability loss is investigated.

Гармонический анализ упругого колосниково-карточного элемента динамически активной просеивающей поверхности вибрационного грохота

Поверхностные вибрирующие сита должны обеспечивать довольно сложное движение просеивания частиц сыпучего материала. Для обеспечения необходимых условий для работы просеивающих поверхностей предлагается использовать новую конструкцию динамической упругой поверхностно-активной просеивающей решетки самоочищающейся решетки - карты. Композитные упругие элементы решетки-карты без предварительной деформации, установленные в коробке вибрационного грохота. Консольные горизонтальные полки решетки элементов карты под воздействием виброустойчивых данных совершают сложные вибрационно-пространственные движения. Поскольку пласт URCC апертуры «активен» консольными горизонтальными полками, возникает необходимость в изучении их траекторий движения с учетом конструктивных параметров сита и вибрационной камеры вынужденных колебаний. Цель исследования: метод определения собственных частот и форм мод упругого (резинового) элемента, образующего тканевый ситовый вибратор, который достигнет наибольшей амплитуды колебаний при работе вибрационного грохота. Задача исследования: разработать модель решетчатого элемента карточного сита URCC для гармонического анализа методом конечных элементов. Поскольку элемент решетки имеет сложную форму, предлагается решение с использованием метода конечных элементов. Рассматривается решение задачи гармонического анализа движения составного элемента колосниково-карточного сита вибрационного грохота с помощью метода конечных элементов грохота. Для существующего элемента определено, что при диапазоне варьирования модуля упругости собственная частота горизонтальных колебаний изменилась в пределах 17…34 Гц, а вертикальных – 37…72 Гц.

Calculation of the Rectangular Cross-Section Beams On the Side Buckling Taking into Account Creep

The article presents the derivation of the resolving equation for calculating the flat form of bending stability of the beams, taking into account creep. When deriving the basic equation, the application of the load with eccentricity, as well as the initial curvature of the beam is taken into account. The solution of the test problem for a wooden cantilever beam is presented. The value of the long-term critical load was introduced and it was shown that the load acting on the beam should not exceed it. A new criterion for determining the critical time is proposed.

Perfect acoustic bandgap metabeam based on a quadruple-mode resonator array

Solid structures guide a multitude of elastic modes of different polarizations including both compression and shear, and the nature of the elastic constant tensor implies a much richer behavior than in optics. Here, we introduce a metamaterial in the form of a rectangular cross section beam of a single isotropic material that can simultaneously suppress all elastic-wave polarizations in the beam over a range of frequencies in the kHz range. This is experimentally achieved by machining replicas of a subwavelength unit cell in an aluminum metabeam based on a planar resonator with interconnected ribs, showing complex vibrational degrees of freedom that allow it to couple to compressional, in-plane shear, flexural and torsional vibrations, that is, all four existing mode types. The result is a lightweight structure that can forbid all possible acoustic modes over the metamaterial bandgap frequency range, an exotic behavior that opens up diverse applications in easily manufacturable vibration isolation structures and acoustic wave control.

Estudo analítico das teorias de viga de Bernoulli e Timoshenko para condições de contorno variadas

Neste artigo é analisado a diferença que ocorre nos deslocamentos utilizando as teorias de viga de Bernoulli e Timoshenko variando-se a relação entre o vão e a altura da seção transversal e alterando a condição de contorno do elemento estrutural. As análises efetuadas ficaram restritas a vigas de seção retangular, foi demonstrado que para vigas isostáticas o efeito da distorção devido ao cisalhamento (teoria de Timoshenko) tem pouca influência no aumento dos deslocamentos, em vigas continuas esse efeito passa a ter uma maior contribuição, enquanto que para viga biengastada esse efeito contribui de maneira significativa para o aumento dos deslocamentos na estrutura. Foram propostas relações limites de tamanho de vão e altura da seção transversal em função da condição de contorno para a consideração da distorção por cisalhamento.

Nonlocal strength criteria for the calculation of critical loads in bended structural elements, taking into account the nonlinearity of the material deformation

The destruction of artificial materials (plexiglas, ebonite) in non-uniform stress fields was experimentally investigated. Testing of one batch of samples is carried out using various methods: three-point and clean (four-point) bending of rectangular cross-section beams, Brazilian test (method of splitting a cylinder along a generator). The comparison of the strength characteristics obtained by the Brazilian test with the uniaxial tensile strength, as well as with the tensile strength during bending tests. Experiments on the fracture of beams on a three-point flexural test revealed higher values of ultimate tensile stress than those of a four-point flexural test. The processing of the experimental data obtained by the Brazilian test is carried out using various non-local strength criteria. A sequential analysis of the applicability of the criteria used in the determination of failure loads is carried out. Computer simulation was performed using the finite-element method for the destruction of a cylinder by splitting along a generator, three-point and four-point bending flexural test of a beam in a geometrically nonlinear formulation, taking into account the non-linearity of the material deformation diagram. The problem was solved in the updated Lagrangian formulation taking into account the physical and geometric nonlinearity based on the governing equations of the hypoelastic material. A comparative analysis of the numerical, analytical and experimental results is carried out. A qualitative correspondence was obtained between the experimental data and the results of numerical calculations.

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

This paper presents a mathematical model for rectangular beams with parabolic variation in the height (work object) under a uniformly distributed load taking into account the bending and shear deformations to obtain the fixed-end moments, carry-over factors and stiffness factors. The properties of the rectangular cross section of the beam vary along its axis “x”, i.e., the width “b” is constant and the height “h” varies along the beam, this variation is parabolic type. The compatibility equations and equilibrium are used to solve such problems, and the deformations anywhere of beam are found by means of the virtual work principle through exact integrations using the software “Derive” to obtain some results. The traditional model considers only bending deformations. Besides the effectiveness and accuracy of the developed models, a significant advantage is that fixed-end moments, carry-over factors and stiffness factors are calculated for any rectangular cross section of the beam using the mathematical equation presented in this paper, which is the main part of this research.

A closed-form model for nonlinear spatial deflections of rectangular beams in intermediate range

Modeling the nonlinear load-displacement relations for flexible beams has been a key objective in compliant mechanisms research. There have been several practically useful methods for modeling planar deflections, but less work has been done in modeling spatial deflections. This work proposes the load-displacement relations for rectangular beams by solving the nonlinear governing differential equations of the beams using the power series method and then simplifying the solution by Taylor series expansion and truncation. The solution is validated to be accurate by comparing with two commercial finite element software packages, ANSYS and Abaqus. This comparison shows that this approach is capable of capturing the relevant geometric nonlinearities in the intermediate deflection range defined as 10% of the beam length. The load-displacement relation offers a useful and parameterized tool for understanding the constraint (i.e. stiffness and motion) behavior of rectangular cross-section beams and generating compliant mechanism designs with nonlinear kinetostatic behaviors.

Effect of fibers and hybrid fibers on the shear strength of lightweight concrete beams reinforced with GFRP bars

The widespread use of concrete and concrete structures provide pursuant to the coming new requirements in technology and design of these structures. Combine different technologies to improve performance and increase the effectiveness of this type of structure is one of the paths of research and development. One of these ideas combine the use of lightweight concrete (LWC), micro and macro fibers and FRP bars. The main purpose of this experimental study is to investigate the effect of fibers and hybrid fibers on the shear strength of LWC beams reinforced with GFRP bars. Firstly, twelve concrete mixtures with different volume fractions of made with micro glass fiber (GF), micro polypropylene fiber (PF), and macro steel fiber (SF) were tested up to compressive strength to determine the optimum fiber content of GF, PF and SF added into LWC. Then, three LWC rectangular cross-section beams with 1500 mm length × 100 mm width × 200 mm depth were cast made with the optimum fiber content of GF, PF and SF and also four LWC beams made with hybrid fibers “GF + PF”, “GF + SF”, “PF + SF” and “GF + PF + SF” with the combinations of the optimum percentage which previously had been found for the fibers. Finally, one beam was made with pure LWC. All beams were reinforced with two GFRP bars having 8 mm diameter without shear reinforcement and tested up to failure under four-point bending. The experimental results showed that the fibers and hybrid fibers added into LWC improved the shear strength and increased ultimate load carrying capacity from 55% to 233% relative to the pure LWC beam. Meanwhile, a shear strength correction factor (α) for LWC reinforced with GFRP bars with and without added fibers determined, which was calculated from comparing the experimental results with ACI 440.1R-06, CSA-S806-02, ISIS Canada-01, and JSCE-97 formulas.

On the Application of the Simplified Crack Model in the Bending, Free Vibration and Buckling Analysis of Beams with Linear Variation of Widths

Adequate computational models are crucial for a reliable representation of the mechanical behaviour of structural elements and, therefore, numerous investigations are oriented towards the modelling of cracked structures. This paper studies the behaviour of transversely cracked beams of rectangular cross-sections with linearly-varying widths. The governing differential equations of bending are analytically solved for various beams with different boundary conditions. These simplified model’s solutions are further validated by the results from corresponding 3D finite models of the considered structures. Furthermore, strain and kinetic energy, as well as the work done by an external axial compressive force P are evaluated from the computed transverse displacement functions. These values allowed for estimations of the first eigenfrequency as well as the buckling load. These structural’s parameters were additionally evaluated by implementing dedicated polynomial functions for some cases considered.The results from the simplified model have exhibited very good agreement with the results from more detailed 3D FE models for all performed analyses. The simplified model thus yields an adequate, as well as accurate, approach for the modelling of cracked beams with a linear variation of width in engineering situations, where cracks have to be considered during analysis.
 The results from the simplified model have exhibited very good agreement with the results from more detailed 3D FE models for all performed analyses. The simplified model thus yields an adequate, as well as accurate, approach for the modelling of cracked beams with a linear variation of width in engineering situations, where cracks have to be considered during analysis.

Free out-of-plane vibration of cracked curved beams on elastic foundation by estimating the stress intensity factor

This paper is concerned with evaluating stress intensity factors (SIFs), for a cracked curved beam of rectangular cross section, applying an approach which allows us to estimate the strain energy release rate. The beam is located on an elastic foundation. The out-of-plane vibration of the beam is investigated. This approach requires an additional factor namely correction factor, on the basis of the energy release zone slope to approximate the SIFs. The initial curvature of the beam, however, adds some complication in using this factor. The second part of this study is investigating a numerical approach, namely differential quadrature element method (DQEM), to gain the natural frequencies of the cracked beam. This method is applied to show the application of the SIFs to calculate the compliance of the cracked section for modeling the crack. The other method which is used to obtain the natural frequencies is the finite element method (FEM). The results of these two methods are found to be in good agreement, which shows the precision of the stress intensity factors of the cracked beam.

МЕТОДИКА РАСЧЕТА ТРЕЩИНОСТОЙКОСТИ НОРМАЛЬНЫХ СЕЧЕНИЙ КАУТОНОВЫХ БАЛОК СО СМЕШАННЫМ АРМИРОВАНИЕМ

Rubcon-polymer concrete based on liquid rubber. Due to the high tensile strength of the rubcon, structures made of it are recommended as structures with increased requirements for crack resistance, also should be noted that the value of the ratio of compressive strength to tensile strength lies in the range of 7-8, while in the case of the addition of dispersed reinforcement, this ratio of strength indicators is ~6. For comparison, it should be noted that the ratio of compressive strength to tensile strength for concrete class B25 is almost 15. This indicates the relevance of the use of rubber polymer-concrete as a material for the manufacture of bending elements. The objects of the study are beams of rectangular cross section from the rubber polymer (Rubcon) size 60х120х1400mm. It that the addition of fibers from the steel cord improves the crack resistance of normal section of up to 1.5 times experimentally was established. As a result of the review of literature sources, it was found that the existing methods of calculating the normal sections of bending elements need to be adapted and obtain their own empirical dependences for the calculation of beams from rubber concrete. As a result of the calculation of crack resistance according to the proposed method, the maximum difference of values in the rubcon beams is 5.9%, and in fibrocauton beams, 4.1%.

Eigenfrequency and Euler's Critical Load Evaluation of Transversely Cracked Beams with a Linear Variation of Widths

For a truthful evaluation of the mechanical response of structures reliable and adequate computational models are essential. Consequently, various researches have been devoted to the mathematical representation of cracked structures. This paper studies the performance of the simplified crack model in estimations of fundamental eigenfrequency as well as elastic Euler's critical load for transversely cracked beams of rectangular cross-sections with linearly-varying widths. To obtain these solutions for different beams with diverse boundary conditions Rayleigh’s energy method which requires an assumed transverse displacement function can be applied. After the appropriate displacement function is being selected, kinetic and strain energy, as well as the work done by an external axial compressive force P are evaluated. From these values, the estimations of the fundamental eigenfrequency, as well as the critical load, are assessed. To obtain these preliminary estimates, static deflection functions were applied initially. These functions represent a wide group of suitable functions since they automatically satisfy the required kinematic boundary conditions. Afterwards, alternative functions constructed from a dedicated polynomial solution were applied. Since this mathematical form offers straightforward integration, the genuinely applied displacement functions were further upgraded, separately for eigenfrequency as well as for critical load estimation. All obtained simplified model’s solutions were afterwards compared to the results from equivalent and more detailed 3D finite models of the examined structures. The comparisons of the results demonstrated very fine agreements with the results from 3D FE models for all performed analyses. The considered simplified model thus clearly yields a suitable alternative in modelling of cracked beams with a linear variation of width in those situations, where cracks have to be considered within the analysis.

Numerical simulation of the transverse flow over spans of girder bridges

Introduction. The technique of numerical modeling of the transverse flow over span structures of bridges on the basis of the two-dimensional URANS (Unsteady Reynolds-averaged Navier-Stokes) approach used in the modern methods and software packages for computational fluid dynamics is verified. The work objective was debugging and experimental substantiation of this technique with the use of the database on the aerodynamic characteristics of the cross-sections of span structures of girder bridges of standard shapes pre-developed by the authors.Materials and Methods. A numerical simulation of the transverse flow of low-turbulent (smooth) and turbulent air flows around the bridge structures in a range of practically interesting attack angles is carried out. SST k − ω turbulence model was used as the closing one. The technique was preliminarily tested on the check problem for the flow of the rectangular crosssection beams. Calculations were carried out using the licensed ANSYS software.Research Results. The calculated dependences on the attack angle of the aerodynamic coefficients of forces (drag and lift) and the moment of the cross sections of the girder bridges of standard shapes are obtained. These data refer to the span structures at the construction phase (without deck and parapets, without parapets) and operation phase, under the conditions of model smooth and turbulent incoming flow. The latter allows us to outline the boundaries for more weighted estimates of the aerodynamic characteristics of thegirder bridges in a real wind current. The best agreement with the experimental data was obtained from the drag of the cross-section. The magnitude of the lifting force is more sensitive to the presence and extent of the separation regions, so its numerical determination is less accurate. The reproduction of the angle-of-attack effect on the aerodynamic moment of the cross-section is the most challenging for the majority of configurations.Discussion and Conclusions. Comparison of the calculated and experimental data indicates the applicability of the URANS approach to the operational prediction of the aerodynamic characteristics of the single-beam span structures. In the case of multi-beam span structures, where the aerodynamic interference between separate girders plays an important role, the URANS approach must apparently give way to more accurate eddy-resolving methods. The results obtained can be used in the aerodynamic analysis of structures and in practice of the relevant design organizations in the field of transport construction.

Natural frequency of bending vibration for stepped beam of different geometrical characters and materials

Based on the bending vibration theory of the beam with equal cross-sectional area, the free bending vibration equation of each subsection of stepped beam is given. The modal function of the stepped beam is assumed by the method of separating variable. First, substitute the modal function into free vibration equation so as to obtain the corresponding vibration shape function and then the modal solution with undetermined coefficients and natural frequency is developed. Next, substitute the modal solution into the boundary condition and the continuity condition to get the corresponding equations. Solving the equations by elimination method, the frequency equation of the step beam is obtained by the condition that the system of linear equations has non-zero solution. Taking the cantilever step beam and simplified supported step beam as examples, the first-order natural frequency of transverse bending vibration of stepped beams with different cross-sectional shapes, different materials, different areas, and different lengths for the two portions was obtained by solving the frequency equation with Mathcad software, and the corresponding curves of natural frequency for the circular and rectangular cross-section beam have been given.