Two-layer Beam Research Articles

Functionally Graded Concrete: Structural Design and Effect on Sustainability Parameters

Researchers have become interested in functionally graded concrete (FGC) in recent years due to its potential to enhance the desired performance. Functional gradation can be carried out in a continuous or stepped/layered manner. Most studies have been conducted on two layered FGC beams by substituting the richer mixes in either tension or compression zone. Previous studies have incorporated and presented the structural design approach of such two-layered beams. Moreover, the layer in the compression zone was assumed to bear entire compressive stresses. Previous studies exhibited savings in cement by up to 37% using a layer of the concrete class defined by exposure conditions around reinforcement and the concrete of minimal structural class. However, using structural concrete of minimal class in the remaining segments may result in a higher reinforcement requirement, reducing the benefits of savings in embodied CO2 offered by reduced cement consumption. This paper examines how designing the beam as a Functionally Graded Reinforced Concrete (FGRC) beam following Indian Standard 456:2016 affects the cost and embodied CO2 based on cement and steel consumption through the durability approach of design, wherein the substitution of the layer is considered in the tension zone. The study revealed that a 10% and 16% reduction in cost and embodied CO2 could be accomplished.

A Two-Layer Beam under Conditions of Creep and Damage Growth with Three-Point Loading

A Two-Layer Beam under Conditions of Creep and Damage Growth with Three-Point Loading

Optimum depth of a lower concrete grade at the tension zone in a two-layer reinforced concrete beam

This paper presents the results of an investigation into the optimum depth of the lower concrete grade (LCG) at the tension region in a two-layer reinforced concrete beam. A total of nine (9) simply supported two-layer RC beams (1100 x100 x150 mm) were studied. Two 8 mm and two 6 mm diameter rods were used as reinforcement at the bottom and top of each two-layer beam, respectively. The beam samples were grouped into two: the first group comprises two-layer RC beams produced with 1:2:4 as the higher grade and 1:3:6 as the lower grade; the second group comprises two-layer RC beams cast with 1:2:4 as the higher grade and 1:4:8 as the lower grade. The depth of LCG adopted for each group is 25 mm to 100 mm at a step of 25 mm out of the total beam depth of 150 mm. The beams were subjected to two-point static loading to evaluate the load resistance and deflection. The results show that the greater the depth of the layer under compression, the stiffer the beam. The two-layer RC beam has an equal loading carrying capacity as the beam made entirely of higher grade. The depth of the layer of RC beam under tension in two-layer beams should be kept between 40 and 50% of the overall beam depth, which would be desirable structurally.

A linear finite element for timber–concrete layered beams with interlayer slip

In this paper, a beam theory for linear two-layer beams is presented in the framework of the Finite Element (FE) method. In particular, the beam theory aims to represent the static and dynamic response of timber–concrete layered structures, in which a concrete slab is supported by timber girders. Since the timber layer has a rather low shear stiffness, a shear-flexible formulation of the displacement field is adopted. In general, it cannot be assumed that the layers of such members are perfectly bonded. Therefore, the relative interlayer displacement in the longitudinal direction is considered, which is determined by a linear slip modulus in a smeared approach. Based on these assumptions, the stiffness matrix, mass matrix and load vector of a finite beam element are derived. Modal analyses on a single-span beam with clamped, free and hinged supports are performed and compared to the outcomes of 3D FE simulations to verify the applicability of the proposed beam theory over a wide frequency range. Static analyses on two structural systems illustrate the accuracy of the beam deflection and the interlayer slip, as well as the realistic assessment of normal and shear stresses in the two layers.

Fracture initiation sequence of double-layer beam layers under three-point loading

The article deals with brittle failure of a two-layer beam under three-point loading conditions depending on the parameters under consideration — different proportions of thicknesses, Young’s moduli and strengths of both layers. Based on the equations of force and moment equilibrium, the dependences of the position of the beam neutral axis and its curvature are derived and the regions of parameters are determined, at which the failure begins earlier in the layer to which the load is applied than in the outer opposite layer to the load.

Influence of partial shear connection on the behaviour of U-shaped steel concrete beams with L-shaped shear connectors

This paper presents a study on the influence of partial shear connection on the behaviour of the U-shaped steel concrete beams (USCB) with L-shaped shear connectors. An experimental campaign of three full-scale experimental tests on the USCB with a degree of shear connection ranging from 0.4 to 1 has been conducted under sagging bending moment. The test results showed high ductility of the USCB with both full and partial shear connections. The failure mode of the USCB was governed by the plastic buckling of the upper flanges of the U-shaped steel profile at the location of the shear connectors. The plastic moment resistance of the USCB with partial shear connection predicted by the full plastic analysis approach and by the simplified approach in Eurocode 4 agreed well with the experimental results. Furthermore, the flexural stiffness of the USCB with partial connections predicted by American and Australian standards fit well with the experimental results. A numerical model based on the two-layer beam element formulation taking into account the interlayer slips with continuous connection in a co-rotational framework was also adapted to determine the behaviour of the USCB. This model was validated against experimental tests as well as against the analytical approaches. It was finally used to carry out a parametrical study in order to investigate the effect of selected parameters such as the degree of shear connection, the concrete strength and the steel strength of U-shaped profile.

Experimental case study for practical implementation of real pre-stressed two layer reinforced concrete beams

The present study summarizes a long-term 14 - year research program, focused on investigation of reinforced concrete (RC) two - layer beams (TLB). In the frame of this program the TLB idea was proposed in 2007. The idea was based on using high strength steel fibered concrete (SFHSC) in the compressed zone of RC bending elements and normal strength concrete (NSC) in the tensile one. Further development of the above-mentioned idea included testing SFHSC at material level, experimental investigation of non – pre - stressed simple supported 3 m and continuous 4 m (2 × 2 m) two-span TLBs and testing pre-stressed simple supported 3 m TLBs (PTLBs) that represented a model of long - span prestressed concrete beams. The present paper presents a case study, aimed at investigating real 8 m long PTLB behavior at four-point bending and its comparison with pre-stressed single – layer beam with the same span. Like in the previous research stages, interaction of concrete layers in a PTLB was examined. No de-bonding between the layers was observed that demonstrates proper interaction between the layers, like in the 3 m long PTLBs. Results, obtained in the present study, enable to recommend PTLBs for practical implementation as effective real pre-stressed concrete elements for new structures and also for repairing existing ones.

Performance of Glued Two-Layer Beams in Cantilever Structures under Abrasive Wear

Performance of Glued Two-Layer Beams in Cantilever Structures under Abrasive Wear

Formation of multiple aerial LED signs in multiple lanes formed with AIRR by use of two beam splitters

We propose multiple aerial imaging system by use of two beam splitters in the optical system of aerial imaging by retro-reflection (AIRR). The AIRR optical system consists of a light source, a beam splitter, and a retro-reflector. Use of two-layered beam splitters enables us to show multiple aerial signs to drivers in multiple driving lanes. The purpose of this paper is to confirm feasibility that an aerial display by use of AIRR for a novel traffic information provision. First, we explain an optical configuration to show aerial images to multiple road lanes, a driving lane and an overtaking lane with AIRR by use of two beam splitters. Next, we have developed a prototype optical system by use of two large beam splitters and a large high-brightness LED sign that is designed to be used for actual road displays. The light source was an LED display with a height of 450 mm and a width of 390 mm, which shows a character. We observed aerial images by naked eyes as experiments. And we discuss the issues for practical application that the experiments have revealed.

Evaluating the interfacial shear strength and bonding properties of recycled aggregate concrete for two-layer beams application

Evaluating the interfacial shear strength and bonding properties of recycled aggregate concrete for two-layer beams application

Free and forced small flexural vibrations of slightly curved slender composite beams with interlayer slip

This paper presents a beam theory for analyzing the dynamic bending response of slender slightly curved composite beams whose layers are flexibly connected and therefore subject to interlayer slip. The equations of motion and boundary conditions are derived using Hamilton’s principle, assuming separately for each layer the applicability of Euler–Bernoulli theory and a linear elastic relationship between the interlayer slip and the shear traction. For the problem of a three-layer slightly curved single-span beam with symmetric layer arrangement and soft-hinged bearings, analytical expressions for the natural frequencies and the eigenfunctions are derived. For the arbitrarily supported two-layer beam, on the other hand, a numerical solution scheme of the combined initial boundary value problem is presented. Several examples show how important it is to consider even very small deviations from the straight beam axis in the prediction of the dynamic response for slender beams with interlayer slip, in particular when all supports are immovable. The comparison of the beam solutions with the results of much more expensive FE analyses based on plane stress elasticity proves the accuracy of the presented theory.

Quantifying chaotic dynamics of nanobeams with clearance

We investigate chaotic dynamics and contact interaction of geometrically nonlinear (according to the Kármán model) beam nanostructures with a small clearance. Contact interaction is described by the Kantor’s model. The constructed mathematical models are based on the first-order (Euler–Bernoulli) and the second-order (Timoshenko) kinematic approximations with the help of the modified couple stress theory. The fundamental governing dynamical equations of two-layer beam nanostructures are yielded by Hamilton’s principle.The paper substantiates the validity of the obtained solutions for the problem of contact interaction of two nanobeams, subject to different kinematic hypotheses. When solving problems by numerical methods, errors accumulate, which are very often mistaken for chaotic oscillations. In this paper, we propose a methodology for identifying “true” chaos for mechanical beam size-dependent structures and obtaining reliable results.The governing PDEs are reduced to ODEs by finite difference method (FDM) of the second-order and the Bubnov–Galerkin method in higher approximations. The obtained ODEs are solved by the Runge–Kutta and Newmark methods. Results of convergence versus the number of partition points along the spatial coordinate in the finite difference method, the number of series members in the Bubnov–Galerkin​ method and time steps are investigated. The revealed chaotic/hyper-chaotic vibrations are studied by nonlinear dynamics methods, wavelet analysis, and the spectrum analysis of the Lyapunov exponents. The numerical methods are validated by estimating the temporal and spatial convergence, while the reliability of the obtained solution is validated by the Lyapunov exponents obtained through the analysis of the results from four different qualitative methods. The numerical solution is obtained by several methods at each simulation step. The influence of control parameters (the transverse harmonic load and the size-dependent parameter) on the two-layer beam vibrations is reported.

Two-dimensional elasticity model for composite beams with deformable shear connectors: Analytical and numerical solutions

A novel and efficient exact two-dimensional (2D) model is developed for predicting the flexural response of two-layered composite beams made of different materials based on the elasticity theory. The beam is considered as a specific case of plane stress problem laying in the longitudinal vertical plane passing through the centroid of the beam section. The proposed two-dimensional model considers the orthotropic properties of materials used in composite beams. The partial shear interaction between two material layers in the form of shear slip at their interface is also considered in this study. The deformable shear connectors between these layers are modeled using distributed shear springs along the beam length. The present 2D model's accuracy is validated by comparing its results with published results as well as numerical results generated by a thorough 2D finite element model using ABAQUS. Furthermore, the stretching effect over the direction of beam depth on the flexural response of two-layered composite beams is investigated by comparing the results produced by the one-dimensional (1D) HBT model and the proposed 2D elasticity model.

Shear Behavior of Two-Layer Beams Made of Normal and Lightweight Concrete Layers

This study investigates the shear strength behavior of two-layer reinforced concrete beams consisting of two different types of concrete. One of the layers made of lightweight concrete (LWC) and the other was normal weight concrete (NWC). A total of 16 shear deficient reinforced concrete beams were fabricated and cast with NWC, LWC, and two-layer beam of both material with different configuration. All the beams were tested under four-point loading after 28 days. The variables of the experimental program include the ratio of thickness of the lightweight concrete layer to the overall depth of beam (hLW/h), and concrete compressive strength. Experimental results which include load-deflection response curves along with failure modes for NWC, LWC and two-layer beams. The results showed that all beams failed in a similar mode, due to diagonal tension shear crack. Based on the experimental results it can be also concluded that the shear load is governed by compressive strength of lower layer of the concrete when the shear span to overall depth (a/h) of the beams is 2.75 or more. While for the a/h 2.375 and 2.00 the two-layer beam has a significant reduction in the shear capacity compared to the NWC beams and increasing compared to LWC beam. The ratio of experimental shear stress divided by the root square of concrete compressive strength (vexp⁄√(f_c^' )), which demonstrates the diagonally cracked concrete's ability to transfer strain and shear was maintained for all configurations greater than 0.17, which is the minimal value recommended by ACI318-19.

Softening-Spring Phenomenon in Large Amplitude Vibration of Two-Layer Bi-Material Beams

The purpose of this study is to analyze the nonlinear vibration behavior of two-layer bi-material beams by using the Euler beam theory and Timoshenko beam theory. It is assumed that material properties satisfy a power law distribution along the thickness direction. The differential quadrature (DQ) method in conjunction with an iterative algorithm is employed to solve the nonlinear governing equations for two-layer bi-material beams. Numerical results are presented to show the influence of Young’s modulus ratio and thickness ratio on nonlinear frequency ratio–amplitude curves of two-layer bi-material beams. It is observed that the nonlinear frequency ratio–amplitude curves are unsymmetrical with the hinged end because of bending-extension coupling effect in two-layer beams. It is also observed that the hinged–hinged two-layer beams may occur the softening-spring nonlinearity when the amplitude of oscillation is small. This softening-spring nonlinearity is dependent on the thickness ratio, Young’s modulus ratio, vibration amplitude, layer number and boundary condition. The greater the difference of material properties between the two layers is, the more obvious the softening-spring nonlinearity becomes.

Influence of the nonlinear behaviour of ballast on the dynamics of simply supported railway bridges

In this work, the vertical motion of a simply supported railway bridge which is subjected to the circulation of high speed trains was studied. A system consisting of two-layer beam was considered to model the dynamics of the bridge structure. The upper beam represents the rails with the sleepers and the lower beam the bridge deck. These two beams are coupled through distributed nonlinear springs that model the ballast action. The characteristics of these elements were identified from experimental measurements performed on real rail track. Considering the circulation of high speed train at given velocity, the influence of the nonlinear stiffness of the ballasted track on the response of the bridge system was analyzed. This was achieved by using the Galerkin method and the Runge-Kutta scheme to solve numerically the nonlinear partial differential equations governing the motion of the two beams. It was found that the nonlinear behaviour of the ballast affects notably the dynamics of the bridge, especially when the ballast stiffness is low. The proposed modelling enables to get more understanding regarding the vertical dynamics of ballasted track bridge in high speed line.

Mathematical and Numerical Calculation of the Interlayer Slip of a Two-Layer Glued Beam

This study analyses the spatial variation of the deflection, rotation, and slip at the interface of two-layer bamboo scrimber-concrete composite beam simply supported under uniform transverse load on its entire length. The Timoshenko model is considered, and the equations are solved using the analytical methods and direct numerical simulation based on finite differences schemes and a Fortran code. The results obtained show that it is a good agreement between the results from the mathematical calculation and those obtained from the numerical simulation.

Flexural behavior of two-layer beams made with normal and lightweight concrete layers

In this paper, twelve concrete beams with two different layers of concrete were evaluated as a simply supported beam under four-points loading. The beams assembled of two different types of concrete layers, one of which was normal-weight concrete (NWC) and the other was lightweight aggregate concrete (LWAC). The investigated parameters were the thickness of the lightweight concrete to the overall depth of beams (hLW/h), and the compressive strength of normal and lightweight concrete. Due to the weak lightweight aggregates used, lightweight aggregate concrete exhibits more brittleness and lower stiffness. Therefore, the viability of compensating for this degradation and providing a layer of normal concrete seems to be very interesting in such beams. The behavior of beams was evaluated based on cracking, failure mode, flexural strength, maximum deflection, stiffness, and toughness. The results showed slight variations on the majority of the above-mentioned performance aspects of two-layer beams compared to fully normal concrete beams. While there were great enhancements compared to fully LWAC beams. The variants were mainly attributable to the efficacy of using LWAC in providing lower stiffness and lower tensile strength. The experimental results have been compared to predicted values using the ACI 318-19, with some modifications for the equations to be matched with two-layer beams, the comparison was in terms of the deflection due to service load, moment capacity, and cracking moment.

A two-layer beam model with interlayer slip based on two-dimensional elasticity

The objective of this paper is the computation of the Airy stress function of a two-layer composite beam with interlayer slip. For the calculation of the upper and lower layer stress functions an iterative solution procedure from Boley-Tolins is adapted, which was originally introduced for isotropic single layer beams under plane stress assumptions. Considering the continuity equations for the interface stress, for the vertical deflection and a linear law between the interlayer slip and the shear stress, Boley-Tolins’ method may be extended to derive an ordinary differential equation for the slip. The order of the ordinary differential equation depends on the number of iterations and the desired accuracy, ranging from an elementary, second order ordinary differential equation for the slip to a more refined, fourth order ordinary differential equation for the slip. Finally the derived slip models are compared by considering a thick simply-supported two-layer beam subjected to a sinusoidal transverse load. The results from the iterative approaches are compared to two-dimensional results under plane stress assumptions for two limit cases: a perfectly bonded beam and a composite beam without bonding. It is found that the refined higher order slip model almost coincides with the elasticity solution, even for very thick composite beams, whereas the elementary slip model only yields accurate results for a thin beam.

ENVIRONMENTAL BENEFIT OF TWO-LAYER STEEL FIBERED HIGH-PERFORMANCE CONCRETE BEAMS

ABSTRACT This study evaluated Life-Cycle Assessment (LCA) of two different designs of high-performance concrete beam: (1) a single-layer beam (SLB) that consisted of steel fibered high-strength concrete in both the compression and tensile zones and (2) a two-layer beam (TLB) that consisted of steel fibered high-strength concrete and normal-strength concrete in the compression and tensile zones, respectively. The SLB and steel fibered high-strength concrete layer of the TLB were of the same concrete class C70/85. LCAs of the SLB and TLB were conducted using the ReCiPe2016 midpoint and endpoint-single-score methods. The difference between the two endpoint-single-score results was evaluated using a two-stage nested analysis of variance. The ReCiPe2016 midpoint results showed that replacing the SLB with the TLB reduces the environmental impact of global warming potential, terrestrial ecotoxicity, water consumption, and scarcity of fossil resources by 15%, 17%, 11%, and 17%, respectively. The ReCiPe2016 endpoint-single-score results showed that the environmental damage from the TLB compared to the SLB was statistically reduced (p = 0.0256). Therefore, considering two different designs of steel fibered high-strength concrete beams, the TLB design was found environmentally preferable to SLB design on both, midpoint and endpoint-single-score evaluations.