Boundary Conditions Of Beam Research Articles

Validity of the Six-Flux model for photoreactors

In the literature for photocatalytic reaction modeling engineering, several simplified schemes applicable to ambient temperature radiative transfer for scattering media are available. A popular strategy is the Six-Flux method because it is simple and is not computer time demanding but its accuracy is not always explicit. In the present work we assess the accuracy of low order methods, including six flux case, by solving the radiative transfer equation in dimensionless form within a cubic enclosure, for i) a collimated beam and ii) a diffuse beam. Radiation enters the cube through a square window centered on one face and with an area one quarter the area of the face. The simulation considers fully absorbing walls, and isotropic scattering. The deviations of simplified models based on Discrete Ordinate or Finite Volume schemes, are compared to the mesh independent solution. The results indicate that for a collimated beam boundary condition the Six-Flux model and more refined models are moderate. The error of the total rate of energy absorbed (TREA) and that of the local volumetric rate of energy absorption (LVREA) with respect to a mesh independent solution are below 5% and 22% respectively. In contrast, it is found that for diffuse boundary conditions the Six-Flux model is very inaccurate since the corresponding errors are larger than 120%.

Staff Radiation Exposure Reduction

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the cross-section principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities

ABSTRACTThis article proposes a four-variable shear deformation refined beam theory for thermo-mechanical vibration characteristics of porous, functionally graded (FG) beams exposed to various kinds of thermal loadings by using an analytical method. Thermo-mechanical properties of functionally graded material (FGM) beams are supposed to vary through the thickness direction, and are estimated through the modified power-law rule in which the porosities with even and uneven types are approximated. The material properties of FGM beams are supposed to be temperature dependent. Porosities possibly occur inside FGMs during fabrication because of technical problems that lead to the creation of microvoids in these materials. The variation of pores along the thickness direction influences the mechanical properties. Thus, it is incumbent to predict the effect of porosities on the thermo-mechanical vibration behavior of FG beam in the present study. Four types of thermal loading, namely, uniform, linear, nonlinear, and sinusoidal temperature rises through the z-axis direction are discussed. The governing differential equations and boundary conditions of FG porous beams subjected to thermal loadings are formulated through Hamilton's principle, based on a four-variable refined theory that considers a constant transverse displacement and higher order variation of axial displacement through the depth of the beam without the need of any shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of porous FG beams subjected to various temperature fields. The impact of several specific parameters such as power-law exponent, porosity volume fraction, different porosity distribution, and thermal effect on the vibration of the porous FG beams is perused and discussed in detail. It is deduced that these parameters play a notable role on the thermo-dynamic behavior of porous FG beams. Presented numerical results can serve as benchmarks for the future analyses of FG beams with porosity phases.

Explicit formulation for natural frequencies of double-beam system with arbitrary boundary conditions

In this paper, free transverse vibration of two parallel beams connected through Winkler type elastic layer is investigated. Euler-Bernoulli beam hypothesis has been applied and it is assumed that boundary conditions of upper and lower beams are similar while arbitrary without any limitation even for non-ideal boundary conditions. Material properties and cross-section geometry of beams could be different from each other. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli-Fourier method. Explicit expressions are derived for the natural frequencies. In order to verify accuracy of results, the problem once again solved using modified Adomian decomposition method. Comparison between results indicates excellent accuracy of proposed formulation for any arbitrary boundary conditions. Derived explicit formulation is simplest method to determine natural frequencies of double-beam systems with high level of accuracy in comparison with other methods in literature.

Vibration and post-buckling of a functionally graded beam subjected to non-conservative forces

Vibration and post-buckling of beams made from functionally graded materials (FGM) subjected to uniformly and tangentially compressing follower forces are studied in this paper. Based on the accurately and geometrically nonlinear theory for extensible beams, the dynamic governing equations for FGM beams under non-conservative load are formulated. By using a shooting method to solve the non-linearly differential equations numerically, the responses of post-buckling and free vibration in the vicinity of post-buckling configuration are obtained, in which the hinged-fixed boundary conditions of beam are considered. Effects of material gradient parameter on the critical buckling, post-buckling and lower frequencies of the FGM beam are discussed in details.

Vibration analysis of pre-twisted functionally graded carbon nanotube reinforced composite beams in thermal environment

As a first endeavor, the free vibration behavior of the pre-twisted functionally graded carbon nanotube reinforced composite (FG-CNTRC) beams in thermal environment is studied. The governing equations are derived based on the higher-order shear deformation theory of beams by considering the temperature dependence of material properties and the initial thermal stresses. The free vibration eigenvalue equations are extracted by using the Chebyshev–Ritz method. In this regard, Chebyshev polynomials together with appropriate boundary functions are utilized as admissible functions of the Ritz method, which enables one to handle the problem with different sets of boundary conditions. The fast rate of convergence of the method is demonstrated numerically and its accuracy is verified by comparing the results in the limit cases with existing solutions in the literature. The effects of pre-twist angle together with carbon nanotubes (CNTs) distribution in thickness direction, the temperature dependence of material properties, the temperature rise, the geometrical shape parameters and boundary conditions on the frequency parameters are investigated. It is shown that the effects of the pre-twist angle on the natural frequencies depend on the beam boundary conditions and also the mode number.

Thermo-mechanical analysis of FG nanobeam with attached tip mass: an exact solution

Present disquisition proposes an analytical solution method for exploring the vibration characteristics of a cantilever functionally graded nanobeam with a concentrated mass exposed to thermal loading for the first time. Thermo-mechanical properties of FGM nanobeam are supposed to change through the thickness direction of beam based on the rule of power-law (P-FGM). The small-scale effect is taken into consideration based on nonlocal elasticity theory of Eringen. Linear temperature rise (LTR) through thickness direction is studied. Existence of centralized mass in the free end of nanobeam influences the mechanical and physical properties. Timoshenko beam theory is employed to derive the nonlocal governing equations and boundary conditions of FGM beam attached with a tip mass under temperature field via Hamilton’s principle. An exact solution procedure is exploited to achieve the non-dimensional frequency of FG nanobeam exposed to temperature field with a tip mass. A parametric study is led to assess the efficacy of temperature changes, tip mass, small scale, beam thickness, power-law exponent, slenderness and thermal loading on the natural frequencies of FG cantilever nanobeam with a point mass at the free end. It is concluded that these parameters play remarkable roles on the dynamic behavior of FG nanobeam subjected to LTR with a tip mass. The results for simpler states are confirmed with known data in the literature. Presented numerical results can serve as benchmarks for future thermo-mechanical analyses of FG nanobeam with tip mass.

Bending of thin-walled beams of open section with influence of shear—Part II: Application

This two-part contribution presents a novel theory of bending of thin-walled beams with influence of shear (TBTS). The theory is valid for general open thin-walled cross-sections. The loads are reduced to the cross-section principal pole, i.e. the cross-section shear center. In part I, the TBTS is established, in this part II, the theory is used to analyze various beam cross-section shapes, loads and boundary conditions. Beams with the extreme low ratios of beam length to beam cross-section contour dimensions are analyzed. The stress predictions of the TBTS, stresses as functions of the longitudinal coordinate as well as the cross-section curvilinear coordinate, are compared to those obtained by finite elements computations (FEM) as well as to exact solutions of the theory of elasticity. The results of the TBTS are given in closed analytic forms, i.e. parametric forms, suitable for general studies of thin-walled beam behavior under transvers bending loads, as well in the early design stage of thin-walled structures.

Simulation of stress wave attenuation in plain weave fabric composites during in-plane ballistic impact

An analytical method is presented for the simulation of longitudinal stress wave attenuation in typical plain weave fabric composites made of E-glass/epoxy and T300 carbon/epoxy under in-plane ballistic impact. The stress wave propagation/attenuation simulation studies are carried out for the case of in-plane impact of projectiles on composite beam targets. Geometrical modeling of plain weave fabric composites and an algorithm for stress wave propagation/attenuation are presented. Studies are carried out for both free edge and rigid boundary conditions for different beams. Based on the analytical simulation, for typical plain weave E-glass/epoxy composite beams studied, it is observed that the stress intensity is about 5% of the incident value when the longitudinal stress wave reaches a distance of 110mm from the point of in-plane impact at the representative section.

Bending and buckling of nonlocal strain gradient elastic beams

Featured by the two material length parameters in the nonlocal strain gradient theory, it is still unknown that what are the boundary conditions of nonlocal strain gradient beams, since the equations of motion and boundary conditions of these beam models appear in the same form as those of the classical ones. Based on the weighted residual approaches, this paper provides the boundary value problems of Euler–Bernoulli beams within the framework of the nonlocal strain gradient theory in conjunction with the von Kármán nonlinear geometric relation. The closed-form solutions for bending and buckling loads of nonlocal strain gradient beams are obtained. Numerical results show that the higher-order boundary conditions have no effect on the static bending deflection of beams for the cases studied. However, the higher-order boundary conditions and the material length parameters have a significant effect on the buckling loads. Finally, when the two material length parameters are the same, the buckling loads can not always reduce to the classical solutions, the findings of which violate our expectations. The results provided in this work are expected to be helpful for the applications of this theory to the analysis of engineering structures.

Effect of concrete carbonation on natural frequency of reinforced concrete beams

We carried out an experimental investigation to study the influence of concrete carbonation on the natural frequency of simply supported reinforced concrete beams. A total of 10 reinforced concrete test beams and 12 concrete-carbonation test specimens were subjected to different accelerated carbonation stages for 0, 7, 14, 21, and 28 days. Modal tests were performed on reinforced concrete test beams after the accelerated carbonation stages. In order to reduce the effect of other factors on the modal tests, constant temperature, relative humidity, and boundary conditions of the test beams were maintained in the experimental process. The experimental results show a trend of the natural frequencies of reinforced concrete test beams to decrease with the increase in concrete-carbonation depths. With statistical analyses of experimental data, this study established the relationship between concrete-carbonation depths and natural frequencies. Fitting lines for the drop in natural frequencies and carbonation depths are obtained for the first four modal frequencies. Based on the analysis of the physicochemical processes of concrete carbonation, the main reason behind the drop in natural frequencies is the increase in mass after concrete carbonation. The percentage composition of increase in mass after complete carbonation is obtained based on the analysis of the physicochemical process. This analysis demonstrates part of the reason for the drop in natural frequencies and proves that the experimental results are reliable and credible. This study provides further insight into the use of modal parameters to assess damage in concrete structures in structural health monitoring.

Nonlinear vibration analysis of FG-CNTRC sandwich Timoshenko beam based on modified couple stress theory subjected to longitudinal magnetic field using generalized differential quadrature method

In this paper, the magneto-mechanical nonlinear vibration behavior of rectangular functionally graded carbon nanotube reinforced composite (FG-CNTRC) sandwich Timoshenko beam based on modified couple stress theory (MCST) is investigated by the generalized differential quadrature method. The FG-CNTRC sandwich beam consists of two FG-CNTRC face sheets and homogenous core subjected to longitudinal magnetic field. In the FG-CNTRC face sheets, carbon nanotubes are disseminated in different patterns as uniform distribution, FG-X, FG-V, and FG-O. Based on von Kármán geometric nonlinearity, the nonlinear governing equations are derived by the Hamilton’s principle. Accuracy and convergence of this study are validated by comparing the numerical results with those found in literature. Various parameters effects are examined on the nonlinear 1st frequency of the sandwich beam. The results reveal that composition of the CNTRC face sheets and homogenous core in sandwich beam can be achieved remarkable stiffness in comparison to only CNTRC beam or only homogenous beam. For achieving the highest stiffness of FG-CNTRC sandwich beam, the value of thickness ratio is obtained about 0.34 and 0.27 in the presence and absence of Pasternak foundation, respectively. Moreover, the linear and nonlinear 1st frequencies increase with an increase in the magnetic field for all of CNT distribution types of sandwich beam and different boundary conditions whereas the frequency ratio decreases. Also the highest and lowest nonlinear 1st frequencies are corresponding to FG-A and FG-V distributions of CNT in face sheets, respectively.

Crack identification method for tapered cantilever pipe-type beam using natural frequencies

A method to detect location and size of a crack in tapered cantilever pipe-type beam using changes in natural frequencies is presented. The boundary conditions of cantilever beam are applied to a general solution for vibrating tapered beam. Then, an equivalent bending stiffness for cracked beam is used to obtain the natural frequencies. Numerical simulations are carried out to construct a set of training patterns of a neural network, and committee of neural networks is employed to identify the crack. Crack identifications are carried out for the 16 damage cases numerically and for the 3 damage cases experimentally. The identified crack locations and sizes agree reasonably well with the exact values.

COLLAPSE TEST ON RECTANGULAR STEEL BEAM WITH DIFFERENT SUPPORT CONDITION

The behavior of beams with different support condition was investigated through three-point bending test for the purpose of studying difference of maximum load and catenary action of a beam. Beams are made of SPSR400 which has 245MPa of nominal yield strength and 400MPa of nominal tensile strength. The parameters of the supports condition are simple support and rotationally semi-rigid connection with different horizontal reaction strength applied by anchors. The support boundary conditions of beams were classified into three types; (A) simply supported with no anchors, (B) embedded anchors with 50mm depth of φ10mm, and (C) embedded anchors with 80mm depth of φ10mm. Both ends of the beams were connected by fillet welded angles and supported on a rigid concrete wall through anchors. The test result shows how much the load carrying capacity is increased by catenary action after large deflection at the center of the beam. First peak loads from each types are the loads when first plastic hinge occurs at the mid-span. There were no significant differences among the first peak loads measured from three types. After that, tensile force at anchors due to catenary action increased the load carrying capacity by approximately 55%, which is called as the second peak load. However, second peak load happens when the anchors at a support fully resist a tensile force, therefore it doesn’t happen with type (A) and (B). In conclusion, support boundary conditions of a beam don’t have an effect on the first collapse load, but the second collapse load is increased as embedded length of an anchor becomes deeper.

Flexural natural vibration characteristics of composite beam considering shear deformation and interface slip

Based on Hamilton's principle, the flexural vibration differential equations and boundary conditions of the steel-concrete composite beam (SCCB) with comprehensive consideration of the influences of the shear deformation, interface slip and longitudinal inertia of motion were derived. The analytical natural frequencies of flexural vibration were compared with available results previously observed by the experiments, the results calculated by the FE model and the other similar beam theories available in the open literatures. The comparison results showed that, the calculation results of the analytical and Timoshenko models had a good agreement with the results of the experimental test and FE model. Finally, the influences of shear deformation and interface slip on the flexural natural frequencies of the SCCB were discussed. The shear deformation effect increases with the increase of the mode orders of flexural natural vibration, and the flexural natural frequencies of the higher mode orders ignoring the influence of shear deformations effect would be overestimated. The interface slip effect decrease with the increase of the mode orders of flexural natural vibration, and the influence of the interface slip effect on flexural natural frequencies of the low mode orders is significant. The influence of the degree of shear connection on shear deformation effect is insignificant, and the low order modes of flexural natural vibration are mainly composed of the rotational displacement of cross sections.

Analysis of smart nanobeams integrated with a flexoelectric nano actuator layer

This paper is concerned with the derivation of exact solutions for the static responses of simply supported nanobeams integrated with a flexoelectric layer acting as the distributed nano actuator. Considering both the direct and the converse flexoelectric effects, the governing equations and the associated boundary conditions of the overall beams are derived to obtain exact solutions for the displacements and the electric potential in the flexoelectric layer and the substrate beam. Due to the converse flexoelectric effect, the active flexoelectric layer significantly counteracts the deformations of the substrate beams caused by the applied mechanical load, resulting in the coupling of bending and stretching deformations in the substrate beams. For particular values of the length and thickness of the substrate beam and the applied voltage in the flexoelectric layer, the deflection of the substrate beam due to the converse flexoelectric effect increases with increasing thickness of the flexoelectric layer up to a certain value of the latter. Beyond this value, any further increase in the thickness of the flexoelectric layer causes a decrease in its ability to actuate the substrate beam. The electric potential varies linearly across the thickness of the flexoelectric layer. The benchmark results presented here may be useful for verifying further research and the present study suggests that the flexoelectric layer may be effectively exploited for advanced applications as a smart nano actuator.

Solving Large-Deflection Problem of Spatial Beam with Circular Cross-Section Using an Optimization-Based Runge–Kutta Method

Abstract Based on the Bernoulli–Euler beam theory, the nonlinear governing differential equations (GDEs) for a spatially deflected beam with circular cross-section are formulated, which are then reduced to first-order differential equations to be compatible with Runge–Kutta method. With the boundary conditions of a spatial beam, the governing equations are treated as an initial value problem (IVP) of ordinary differential equations. A Runge–Kutta method combined with an unconstrained optimization algorithm (RKUO) is presented to solve the IVP. The approach for determining the orientation of the cross-section plane at any position on the deflected beam is also provided. Finally, the comparison between the RKUO results and those achieved using nonlinear finite element (NFE) analysis and spatial pseudo-rigid-body model validate the accuracy and effectiveness of RKUO. The results also demonstrated the unique capabilities of RKUO to solve large spatial deflection problems that are outside the range of nonlinear finite element model.

Dynamical behavior of a double-beam system interconnected by a viscoelastic layer

As an idealized model of modern beam-type structures, the double-beam system has been specified in recent decades. While there have been various research efforts on the double-beam system, most of them are over-simplified, among which the viscoelastic damping mechanism of interlayer is often neglected. This paper presents a semi-analytical method to investigate the natural frequencies and mode shapes of a double-beam system interconnected by a viscoelastic layer. The two beams can be with different beam-masses, beam flexural rigidities and boundary conditions, as well as with and without the Winkler layer below lower beam, indicating that there is no restriction or assumptions on beams connected with the viscoelastic layer damping. The modal-expansion iterated method is further applied to determine the forced vibration responses in the double-beam system based on the natural frequencies and mode shapes obtained from the free-vibration analysis. A specific orthogonality condition for the double-beam system is derived, and then applied to decouple the differential equations of motion. Numerical examples are demonstrated and discussed in details to verify the efficiency of the proposed methodology, which can further help characterize the dynamic responses and design work for double-beam structures.

Modeling and analysis of nanobeams based on nonlocal-couple stress elasticity and surface energy theories

This paper aims to develop a new non-classical Bernoulli–Euler model, taking into account the effects of a set of size dependent factors which ignored by the classical continuum mechanics. Among those factors are the microstructure local rotation, long-range interactions between a particle and the other particles of the continuum and the surface energy effects. The model used the modified couple-stress theory to study the effect of the local rotational degree of freedom of a specific particle. Furthermore, the surface elasticity model developed by Gurtin and Murdoch has been used to determine the surface energy effects on the behavior of the particle. The effects of the local rotation and surface energy are investigated in the framework of nonlocal elasticity theory, which is employed to study the nonlocal and long-range interactions between the particles. In addition, Poisson׳s effect incorporated in the newly developed beam model. The equations of equilibrium and complete boundary conditions of the new beam are derived using the principle of virtual work.The developed model is validated, by comparing the obtained results with benchmark results. To illustrate the new model, analytical solutions for the static bending and critical buckling load are obtained. Numerical results reveal the significant effects of the nonlocal, microstructure, surface energy, length-to-height ratio and Poisson on the static bending and critical buckling load of nanobeams.

Variational formulations and extra boundary conditions within stress gradient elasticity theory with extensions to beam and plate models

• Extension to stress gradient elasticity of two variational principles of class elasticity. • Extra boundary conditions as congruence conditions at the boundary. • Extension to stress gradient elasticity of Timoshenko beam and Kirchoff plate. • Gradient-induced boundary conditions for beams and plates in stress gradient elasticity. • Comparison of stress gradient elasticity theories by Forest and Sab and by Polizzotto. The principle of minimum total potential energy and the primary principle of virtual power for stress gradient elasticity are presented as kinematic type constructs dual of analogous static type principles from the literature (Polizzotto, 2014; Polizzotto, 2015a). The extra gradient-induced boundary conditions are formulated as “boundary congruence conditions” on the microstructure’s deformation relative to the continuum, which ultimately require that the normal derivative of the stresses must vanish at the boundary surface. Two forms of the governing PDEs for the relevant boundary-value problem are presented and their computational aspects are discussed. The Timoshenko beam and the Kirchhoff–Love plate theories are extended to stress gradient elasticity under the assumption that stress gradient effects do not propagate transversally. It is shown that for beam models no extra gradient-induced boundary conditions are required, whereas for plate models such conditions must be enforced at the contour surface of the plate, where the normal derivative of the stress resultants are required to vanish. Appendix A is devoted to some basic aspects of the mechanics of the microstructure; Appendix B to a comparison between the present theory and an analogous theory from the literature (Forest and Sab, 2012).