Two-layer Beam Research Articles

Hybrid analytical-numerical modelling of ground vibrations from moving loads in a tunnel embedded in the saturated soil

The dynamic response of a floating-slab track and underground railway tunnel embedded in saturated soils under moving loads are investigated by a hybrid analytical-numerical method. The tunnel is modelled as thin cylindrical shell surrounded by saturated soil of infinite radial extent. A two-layered beam structure is used to model the floating-slab track. In the continuous floating-slab track, the slab is an infinitely long Euler–Bernoulli beam while in the discontinuous one, the slabs are modelled as a periodic structure consisting of beams of finite length. The coupled equations of the tunnel-soil system are solved in the frequency-wavenumber domain with the aid of Fourier transform and Fourier decomposition. The track structure and tunnel-soil model are coupled by force and displacement continuity conditions at the tunnel invert. Numerical results for the track, tunnel and soil responses are presented for a moving point and train load. It is found the critical velocity of a track-tunnel-soil system approaches the shear wave speed of the soils. The natural frequencies of the track have significant influence on the soil displacement and pore pressure. Furthermore, an increase of the soil permeability leads to a decrease of the displacement as well as the pore pressure in the free field.

Closed-form solutions for modelling the rotational stiffness of continuous and discontinuous compliant interfaces in two-layer Timoshenko beams

An analytical model for a two-layer Timoshenko beam with a compliant interface is presented. A family of closed-form solutions is given for several cases of contact conditions at the interface that can include or exclude interlayer slip and relative rotations of the layers’ cross sections (distortion), with particular attention devoted to the latter. Each kinematic field at the interface is related to a corresponding traction by means of a linear-elastic law. The novelty of the proposed model is the rotational stiffness at the interface, completely separate from the tangential stiffness, that can control the amount of interlayer distortion at the interface. The derived closed-form solutions allow an exact stiffness matrix for any case of admissible boundary conditions to be obtained, as well as for continuity conditions in the case where multiple elements with closed-form solutions are mutually connected. The application of the closed-form solutions for continuous interface (adhesive joints) or for discontinuous interface (shear connectors or adhesive defects) is presented. Accuracy and applicability of the closed-form solutions have been assessed in two representative numerical examples allowing to conclude that the rotational stiffness at the interface can strongly affect the behaviour of a two-layer beam.

Post-fire flexural behavior of functionally graded fiber-reinforced concrete containing rubber

The optimal distribution of steel fibers over different layers of concrete can be considered as an appropriate method in improving the structural performance and reducing the cost of fiber-reinforced concrete members. In addition, the use of waste tire rubber in concrete mixes, as one of the practical ways to address environmental problems, is highly significant. Thus, this study aimed to evaluate the flexural behavior of functionally graded steel fiber-reinforced concrete containing recycled tire crumb rubber, as a volume replacement of sand, after exposure to elevated temperatures. Little information is available in the literature regarding this subject. To achieve this goal, a set of 54 one-, two-, and three-layer concrete beam specimens with different fiber volume fractions (0, 0.25, 0.5, 1, and 1.25%), but the same overall fiber content, and different volume percentages of the waste tire rubber (0, 5, and 10%) were exposed to different temperatures (23, 300, and 600oC). Afterward, the parameters affecting the post-heating flexural performance of concrete, including flexural strength and stiffness, toughness, fracture energy, and load-deflection diagrams, along with the compressive strength and weight loss of concrete specimens, were evaluated. The results indicated that the flexural strength and stiffness of the three-layer concrete beams respectively increased by 10 and 7%, compared to the one-layer beam specimens with the same fiber content. However, the flexural performance of the two-layer beams was reduced relative to those with one layer and equal fiber content. Besides, the flexural strength, toughness, fracture energy, and stiffness were reduced by approximately 10% when a 10% of natural sand was replaced with tire rubber in the threelayer specimens compared to the corresponding beams without crumb rubber. Although the flexural properties of concrete specimens increased with increasing the temperature up to 300oC, these properties degraded significantly with elevating the temperature up to 600oC, leading to a sharp increase in the deflection at peak load.

Dynamic Analysis of Two-layered Elastic Cantilever Beam under the Action of Concentrated Load

Dynamic Analysis of Two-layered Elastic Cantilever Beam under the Action of Concentrated Load

COMPARATIVE STRUCTURAL STUDY OF INNOVATIVE STRENGTHENING PROPOSALS OF TIMBER BEAMS IN HISTORICAL BUILDINGS IN CAIRO USING TWO-LAYERED COMPOSITE SYSTEM

Traditional restoration and rehabilitation work of flat timber ceilings at historical Islamic buildings in Cairo generally require the dismantling of most of their timber parts (i.e. timber planks, secondary beams, etc.) to expose the timber main beams and conduct the work. Hence, these parts are liable to damage and lose during the dismantling that requires time, money and effort to achieve. Hence, the author conducted this study to overcome the dismantling work and propose effective strengthening techniques for the timber beams utilizing two-layered composite system. This paper reported the limited application of the layered system in the construction of the historical timber beams in Cairo. It studied a number of innovative strengthening proposals of the timber beams from their upper-side merely while leaving the exposed bottom surface intact, apart from its indispensable shoring. Hence, a small experimental and a wide comparative analytical investigation of the strengthening proposals utilizing two-layered composite system are conducted. Several 2D Finite Element numerical models using well-known software were conducted. The modelling techniques of various interfacial conditions of the two-layered beams are established and evaluation of the strengthening proposals is conducted. Finally, conclusions and recommendations are derived.

The analysis of a free two-layer composite beamon the elastic foundation

Abstract
 Introduction. This article offers a methodology for the analysis of a two-layer composite beam on an elastic foundation that represents a one-parameter Winkler model. The behaviour of two-layer beams is described by A.R. Rzhanitsyn in his theory of composite rods that have rigid transverse and elastic-yielding longitudinal connections between layers. The theory of composite rods allows to study the stress-strain state (SSS) of multilayer foundation beams, having a layer featuring low thermal conductivity, and perforated beams.
 Materials and methods. However, analytical solutions to these problems involve certain difficulties; therefore, they are often inapplicable. We propose to apply a numerical method, a method of successive approximations (MSA), developed by professor R.F. Gabbasov at the Department of structural and theoretical mechanics of the Moscow State University of Civil Engineering. MSA has proven to be an effective and highly accurate method designated for analyzing static/dynamic loads, applied to beams, slabs and shells, and for making stability calculations. The difference-based variation of the MSA method has a number of advantages over difference equations of the “classical” finite difference method (FDM). The proposed methodology allows to take into account various types of boundary conditions without involving contour points. Сoncentrated forces, concentrated moments, and piecewise distributed loads can be taken into account as loading types.
 The article describes a problem solving algorithm. The system of initial differential equations is solved using difference analogs. Typical difference equations for regular and boundary points are provided.
 Results. The analysis of a composite free-lying beam on an elastic foundation illustrates the proposed approach. The qua-lity of the analysis results is controlled by performing a numerical study of the solution convergence using several nested meshes.
 Conclusions. The proposed method can be used in the engineering practice of design organizations and the educational process of higher educational institutions training civil engineering specialists.

The contact layer stiffness influence assessment on the stress-strain state of a multilayer beam

The article is devoted to the stress-strain state of multilayer beams’ study of the. The beam model is represented by the main and connecting layers. For the main layers, the classical Euler-Bernoulli hypotheses are used. The contact layer model is used to describe the connecting layers. As a result, a system of resolving equations for solving numerous problems is obtained. As an example, the analytical solution to the problem of bending a two-layer beam is given. The resulting solution describes the stress-strain state of the beam in the range from the ideal contact of the main layers to the complete absence of the contact. It is shown that the used model of the contact layer allows satisfying all the boundary conditions, including the fact that the tangent stresses at corner points are equal to zero. The influence of the contact layer’s stiffness on the model’s stress-strain state is analyzed.

Experimental investigation and comparison of prestressed single‐layer and repaired two‐layer reinforced concrete beams

AbstractThis study continues a series of theoretical and experimental investigation of two‐layer beams (TLBs), performed by the authors. Such beams consist of normal strength concrete (NSC) layer in the tensile zone and steel fibered high strength concrete (SFHSC) in the compressed one. The present research is focused on comparison of prestressed single layer beam and three repaired prestressed two‐layer beams. The research includes experimental investigation of full‐scale simple supported beams under four‐point bending. A TLB represents possibility of strengthening by addition of high strength steel fibered concrete layer. Therefore, nonrepaired beam section height is equal to that of the NSC layer in repaired ones. All prestressed specimens with equal section height were produced at real industrial conditions. After that, a SFHSC layer was added in laboratory conditions to obtain three identical TLBs (one single‐layer beam was used as a control specimen). The main aim of the research is to study the effect of repairing, using a SFHSC layer on load‐bearing capacity. All specimens were tested from initial loading up to failure. The study was focused on cracking and deflections in TLBs, compared to single‐layer one. It was found that initial cracks' development in the investigated beams starts at same mid span deflection, but the corresponding load in the repaired beams is about 50% higher, compared to the control specimen. Additionally, proper interaction between the layers without de‐bonding was verified. The load bearing capacity in the control specimen is about 77% lower, compared to the repaired ones. Results, obtained in the present study, enable to recommend using SFHSC layer for strengthening prestressed beams. This method is a promising way for repairing existing elements and creating attractive structural systems.

Durability increase of mining dump trucks through the use of protective coatings of the body

The share of trucking in the total cost of open-cast mining is 50% or more. Therefore, reducing transportation costs is one of the tasks of increasing the technical and economic indicators of an open development method, which is an urgent task. Performance dump truck is estimated either by the volume of transported rock mass (in cubic meter or tons), or the value of cargo turnover (ton-kilometers). However, these indicators do not take into account the percentage of adhering rock that is transported uselessly and reduces productivity. The paper proposes methods for protecting the body of a mining dump truck from clay rocks adhering. Detail the method of protection with the help of vibration using metal structure is considered. A mathematical model of the stress-strain state of a two-layer beam is proposed. The dependence of the stress in the body on the stiffness of the elastic supports and impact energy when using a rubber lining is shown.

A SIMPLE MIXED FINITE ELEMENT MODEL FOR COMPOSITE BEAMS WITH PARTIAL INTERACTION

A simple and effective mixed finite element model is proposed for studying the behavior of beams with composite section, with particular reference to two-layer beams with partial interaction, namely, steel-concrete, timber-concrete, and two-layer laminated glass beams. The model is based on a mixed variational formulation, which as independent fields assumes the displacements of each layer and the shear stresses transmitted by the connections between the layers. The effectiveness of the model, which automatically satisfies the Newmark kinematic hypothesis for a composite beam, is evaluated by performing a set of numerical tests and comparing results with existing analytical and numerical solutions; furthermore, the convergence of the model to the well-known layered and monolithic beam behavior is shown. Numerical tests are performed in the linear elastic field, but the possible model extensions to the field of material and geometric nonlinearity are highlighted, together with a further refinement of the numerical model.

Experimental investigation of prestressed two layer reinforced concrete beams

AbstractThis study continues a series of investigation of two‐layer beams (TLB), performed by the authors. This series includes developing an idea of a TLB, testing steel fibered high strength concrete (SFHSC) as material, testing simple supported and continuous two‐span TLBs. The present research is focused on testing full‐scale prestressed simple supported TLBs (PTLBs). The study is aimed at investigating PTLBs behavior at four‐point bending and its comparison with that of nonprestressed ones. Like in the previous research stages, interaction of concrete layers in a PTLB was studied to prove the efficiency of such beams for real long‐span reinforced concrete beams. No de‐bonding between the SFHSC and normal strength concrete layers was observed up to the ultimate limit state of the tested beams that demonstrates proper interaction between the layers. Results, obtained in the present study, enable to recommend PTLB for further investigation of full‐scale reinforced concrete elements with longer spans (9–12 m) and their practical application as effective and economical bending elements.

Bending of two-layer beams under uniformly distributed load – Analytical and numerical FEM studies

The subject of the paper are two-layer beams with various mechanical properties, thicknesses and widths of the layers. The original nonlinear hypothesis-theory of planar cross section is developed. Based on the principle of stationary potential energy three differential equations of equilibrium are obtained. The system of the equations is analytically solved and the deflections and normal and shear stresses of example beams are calculated. The analytical calculation results are compared with numerical solutions obtained with FEM (SolidWorks). These results are presented in Tables and Figures.

A novel linear matrix method to analyze adhesive joints

A novel linear matrix method to analyze balanced and unbalanced adhesive joints is presented in this paper. The stiffness matrix, loading vector and transfer functions for the analysis of adhesive joints such as stiffened plate, single-strap and single-lap joints are derived in a classic manner. A matrix formulation is derived for a two-layer beam composed of two elements (adherends) and an adhesive. The proposed method takes into account the following effects: a) longitudinal and transversal relative displacements along the interface; b) the interaction between the normal and shear stresses; c) the coupling between the axial, bending and shear deformations in the adherends; and d) the transverse load applied on the two adherends. The expressions developed for the load vector are general for any type, or combination, of transverse load that fits a second-order polynomial curve, including uniformly distributed transverse load, trapezoidal and parabolic loads. The transfer functions necessary to determine the adhesive stresses, axial and shear forces, bending moments, deflections and rotations along the members are also presented in detail. Two comprehensive examples are presented to show the effectiveness and validity of the proposed method and corresponding equations.

An Investigation of Damping Mechanism in Two-Layered Beam with General Sectional Shape

Objectives The paper aims to develop a theory model of clamped layered structures for parameter study of a mechanical damper. The model can be applied to many cantilever friction cases such as general shape cantilever beams, different material cantilever beams, different cross-section beams, non-uniform thickness beams, etc.

Energy Dissipation during Vibrations of Heterogeneous Composite Structures: 2. Method of Solution

This paper describes the method of numerical solution of decaying vibration equations for heterogeneous composite structures. The system of algebraic equations is generated by applying the Ritz method with Legendre polynomials as coordinate functions. First, real solutions are found. To find complex natural frequencies of the system, the obtained real natural frequencies are taken as initial values, and then, by means of the third-order iteration method, complex natural frequencies are calculated. The paper discusses the convergence of numerical solution of the differential equations describing the motion of layered heterogeneous structures, obtained for an unsupported rectangular two-layered plate. The bearing layer of the plate is made of unidirectional CRP, its elastic and dissipation properties within the investigated band of frequencies and temperatures are independent of vibration frequency. The bearing layer has one of its outer surfaces covered with a layer of “stiff” isotropic viscoelastic polymer characterized by a temperature-frequency relationship for the real part of complex Young’s modulus and loss factor. Validation of the mathematical model and numerical solution performed through comparison of calculation results for natural frequencies and loss factor versus test data (for two composition variants of a two-layered unsupported beam) has shown good correlation.

On the stressed state of the joint upon bending beam with an elastic coating

Joint deformation of two elastic layers is considered, one of them being a coating applied to the surface of another thicker layer. It is assumed that slippage is absent in the plane of layer contact. External load is normal to the coating surface and unchanging in width. In the absence of lateral loading, the junction can be considered a two-layer beam and described by the equations of the plane elasticity using the asymptotic method. The required functions, normal and tangential stresses, and also the components of the displacement of an arbitrary point of the beam and coating are expanded in power series of the small parameter (accepted as a half thickness of the corresponding layer). In contrast to the known asymptotic expansions, an alternative asymptotic method is proposed in which all the required functions are asymptotically equivalent, i.e., expanded in asymptotic series of the same structure. When all positive powers of the small parameter are present in the series, the asymptotic algorithm leads to the appearance of two independent recurrent systems of linear equations, which greatly simplifies their solution. In each approach for one elastic layer the algorithm generates five indefinite functions of the longitudinal coordinate. Due to a rapid convergence of the asymptotic series, the first asymptotic approximation is used to construct the mathematical model of the deformation of a beam-to-coating coupling. Ten indefinite functions of the coordinate x provide fulfilling of eight conditions of continuity of the stress-strain state in the linkage depth, two functions remaining indefinite. Equilibrium equations (a system of two linear differential equations of the sixth order) are derived for them proceeding from the principle of minimum potential energy of deformation and corresponding boundary conditions (that can be realized both in static and kinematic variants). Numerical results are obtained in static approximation for two types of surface loading — steady and sinusoidally varying — along the length of the coating. An emphasis is made on the necessity of allowing for tangential stresses when studying the stress state of the coating as they can be rather significant upon bending.

On the contact interaction of a two-layer beam structure with clearance described by kinematic models of the first, second and third order approximation

Abstract In this article, the contact interaction of two geometrically non-linear beams with a small clearance is studied by employing various approximations of the kinematic beam models. Our investigation concerns two-case studies, i.e. when the upper beam is governed by the second order Timoshenko model (i) or by third order Pelekh-Sheremetev-Reddy-Levinson model (ii), whereas in both cases the lower beam is described by the kinematic (Bernoulli-Euler) model of the first approximation. The upper beam is subjected to the transversal, uniformly distributed harmonic load, whereas the beam interaction follows the classical Kantor’s model. The problem is highly non-linear due to occurrence of the geometric von Karman non-linearity and the contact interaction between beams (structural non-linearity). The governing PDEs are reduced to ODEs by the method of finite differences (FDM) of the second order. The obtained ODEs are solved by a few Runge-Kutta type methods of different orders. Results of convergence versus the number of partition points along the spatial co-ordinate and time steps are investigated. New non-linear phenomena of the studied structural package are detected, illustrated and discussed with emphasis on the “true” chaotic vibrations. In particular, three qualitatively different algorithms for computation of largest Lyapunov exponents are employed, and the differences between geometrically linear versus non-linear problems are reported.

The edge effects in layered beams

The article presents the resolving system of equations for solving the problem of the stress-strain state of multilayer beams. The system of resolving equations allows us to solve a wide range of problems, such as shear, bending, normal separation for any number of layers. In the proposed model the interaction between layers is accomplished by the contact layer. The contact layer is an anisotropic medium which can be considered as a "brush" of short elastic rods. The use of contact layer allows avoiding problems such as the endless shear stresses at the boundary of layers near the beam end. Another feature of the proposed model is the strict satisfaction of boundary conditions. Due to the complexity of resolving system of equations we consider a two-layer beam as an example. In this case it is possible to obtain an analytical solution and to produce its analysis. The edge effects arising in a two-layer beam are considered in detail. The article shows that the distribution of tangential stresses in the zone of the edge effect is a complex nonlinear function. At the moment the adhesive interaction is characterized by a single value – the average strength. This value is the ratio of the breaking load to the bonding area. Since the nature of stress distribution is a nonlinear function, the average strength is not true. The obtained solution makes it possible to establish a relationship between the average and true strength of the adhesive interaction, and to define the parameters of the contact layer based on experimental data. The influence of mechanical and geometric characteristics of the model on stress-strain state of the beam is analyzed.

Plates and beams of asymmetric structure in thickness

The theory of plates and beams of asymmetric structure in thickness is constructed. It is shown that, in general, the complete problem for this kind of a structure is not divided into a plane problem and a bending problem. Here stress-strain state of plates and beams of asymmetric structure in thickness is analyzed by the mathematical method without using any simplifying assumptions. Simple theories were obtained for practical applications by an asymptotic method. As examples, the dynamic problems for a two-layer beam and a plate with a gradient of properties in thickness were calculated.

Dynamics of Laminated Timoshenko Beams

This paper is concerned with long-time dynamics of laminated beams modeled from the well established Timoshenko system. Of particular interest is a model of two-layered beam proposed by Hansen and Spies which describes the slip effect produced by a thin adhesive layer uniting the structure. In a more general setting, involving a nonlinear foundation, we establish the existence of smooth finite dimensional global attractors for the corresponding solution semigroup.