Dynamic analysis of elastic plates reinforced with beams of doubly-symmetrical cross section

In this paper a solution to the dynamic problem of elastic plates reinforced with beams of doubly – symmetrical cross section is presented. The adopted model takes into account the resulting inplane forces and deformations of the plate as well as the axial forces and deformations of the beam, due to combined response of the system. The analysis consists in isolating the beams from the plate by sections parallel to the lower outer surface of the plate. The method of analysis is based on the capability to establish a flexibility matrix with respect to a set of nodal mass points using the Analog Equation Method (AEM) for the static plate problem. A lumped mass matrix is constructed from the tributary mass areas to the nodal mass points. Both free and forced transverse vibrations are considered and numerical examples with great practical interest are presented. The discrepancy in the obtained eigenfrequencies using the presented analysis, which approximates better the actual response of the plate-beams system, and the corresponding ones ignoring the inplane forces and deformations justify the analysis based on the proposed model.     

Flexural response predictions of reinforced concrete beams strengthened with prestressed CFRP plates

Existing experimental studies showed that the reinforced concrete (RC) beams strengthened with prestressed carbon fiber-reinforced polymer (CFRP) plates had three possible flexural failure modes (including the compression failure, tension failure and debonding failure) according to the CFRP reinforcement ratio. Theoretical formulas based on the compatibility of strains and equilibrium of forces were presented to predict the nominal flexural strength of strengthened beams under the three failure modes, respectively, and a limitation on the tensile strain level developed in the prestressed CFRP plate was proposed as the debonding failure occurred. In addition, the calculation methods for cracking moment, crack width and deflection of strengthened beams were provided with taking into account the contribution of prestressed CFRP plates. Experimental studies on five RC beams strengthened with prestressed CFRP plates and a nonlinear finite element parametric analysis were carried out to verify the proposed theoretical formulas. The available test results conducted by other researchers were also compared with the predicted values.     

An extended BEM formulation for plates reinforced by rectangular beams

This article presents a BEM formulation developed particularly for analysis of plates reinforced by rectangular beams. This is an extended version of a previous paper that only took into account bending effects. The problem is now re-formulated to consider bending and membrane force effects. The effects of the reinforcements are taken into account by using a simplified scheme that requires application of an initial stress field to locally correct the bending and stretching stiffness of the reinforcement regions. The domain integrals due to the presence of the reinforcements are then transformed to the reinforcement/plate interface. To reduce the number of degrees of freedom related to the presence of the reinforcement, the proposed model was simplified to consider only bending and stretching rigidities in the direction of the beams. The complete model can be recovered by applying all six internal force correctors, corresponding to six degrees of freedom per node. Examples are presented to confirm the accuracy of the formulation and to illustrate the level of simplification introduced by this strong reduction in the number of degrees of freedom.     

Shear strengthening of reinforced concrete beams using different configurations of externally bonded carbon fibre reinforced plates

This paper deals with research undertaken at Oxford Brookes University into shear strengthening of reinforced concrete beams using externally bonded carbon fibre reinforced plastic plates (CFRP). Thirty-eight reinforced concrete beams of 1.8 m length were constructed and tested and are described in this paper. The relative performance of a group of sixteen beams with the same steel reinforcement but with different amounts of shear strengthening is discussed. All the beams were designed to fail in shear using a spreadsheet program. The spreadsheets were designed to ensure that the beams' flexural capacity exceeded the shear capacity after strengthening. The variables were: main reinforcement ratio, spacing between links in the shear span and different configurations of CFRP plates on shear spans. The concrete had an average compressive strength of 61.76 N/mm². The majority of the beams tested showed a significant improvement in shear strength by the addition of CFRP plates, with increases of between 19–122% over the control beams.     

Analysis of plates stiffened by parallel beams

In this paper a general solution for the analysis of plates stiffened by parallel beams subjected to an arbitrary loading is presented. According to the proposed model, the stiffening beams are isolated from the plate by sections in the lower outer surface of the plate, taking into account the arising tractions in all directions at the fictitious interfaces. The aforementioned integrated tractions result in the loading of the beams as well as the additional loading of the plate. Their distribution is established by applying continuity conditions in all directions at the interfaces. The analysis of both the plate and the beams is accomplished on their deformed shape taking into account second‐order effects. Six boundary value problems with respect to the plate transverse deflection, to the plate inplane displacement components, to the beam transverse deflections, to the beam axial deformation and to the beam non‐uniform angle of twist are formulated and solved using the analog equation method (AEM), a boundary element method (BEM)‐based method employing a boundary integral equation approach. The solution of the aforementioned plate and beam problems, which are non‐linearly coupled, is achieved using iterative numerical methods. The adopted model describes better the actual response of the plate beams system and permits the evaluation of the shear forces at the interfaces in both directions, the knowledge of which is very important in the design of prefabricated ribbed plates. The evaluated lateral deflections of the plate–beams system are found to exhibit considerable discrepancy from those of other models, which neglect inplane and axial forces and deformations. Copyright © 2006 John Wiley & Sons, Ltd.     

A theoretical study on shear strengthening of reinforced concrete beams using composite plates

The present paper deals with shear strengthening of reinforced concrete beams by means of thin fiber-reinforced composite plates. First, the theoretical model is presented. It is then applied to the shear strengthening of a T-beam. This beam is also used to investigate the effects of some design parameters on the ultimate shear strength.     

The strengthening and deformation behaviour of reinforced concrete beams upgraded using prestressed composite plates

Externally bonded steel plates have been used world-wide for over twenty years to strengthen concrete members, but the disadvantages include the transportation and installation of heavy plates and steel corrosion. Polymeric composites avoid these disadvantages and provide an equally effective method of strengthening. The advantages of composites are exploited further by prestressing them before bonding to the concrete. This paper is concerned with the response of such prestressed members to applied load. After strengthening with externally bonded carbon fibre reinforced polymer (CFRP) plates, a number of beams of 1.0 m and 4.5 m lengths were tested in four point bending. The plates were bonded without prestress and with prestress levels ranging from 25% to 50% of the plate strength. The ultimate capacities of the non-prestressed beams were significantly higher than those of the unplated members and plate prestress brought about yet further strengthening. Prestressed plates are utilised more efficiently than non-prestressed plates, since a given plate strain is associated with a lower structural deformation in a prestressed member. The internal steel rebars yielded at a higher proportion of the ultimate capacity in the prestressed beams. Prestressing lowers the position of the neutral axis so more of the concrete section is loaded in compression, making more efficient use of the concrete.

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Résumé Le renforcement des éléments en béton par placage de plaques d’acier est pratiqué partout dans le monde depuis plus de vingt ans; cependant cette méthode présente les inconvénients du transport et de l’installation des plaques lourdes d’acier, ainsi que de la corrosion de l’acier. Des matériaux composites-polymère évitent ces inconvénients tout en fournissant une méthode aussi efficace de renforcement. Les avantages présentés par les composites peuvent être davantage améliorés par leur précontrainte avant leur placage au béton. Ce article traite de la réponse de tels éléments précontraints sous mise en change. Après renforcement par placage de plaques en polymère renforcé de fibres de carbone, des poutres ayant une longueur de 1,0 m et de 4,5 m ont été soumises à des essais de flexion-4 points. Les plaques étaient appliquées sans précontrainte et avec des niveaux de précontrainte allant de 25 à 50% de leur résistance. Les capacités ultimes des poutres non-précontraintes se sont montrées significativement plus hautes que celles des éléments non-renforcés par des plaques, et la précontrainte des plaques a apporté une résistance encore meilleure. Les plaques précontraintes sont plus efficaces que les plaques non-précontraintes, la contrainte sur une plaque donnée étant associée avec une déformation structurelle moindre dans l’élément précontraint. Dans les élément précontraints, les armatures métalliques internes ont cédé à une plus haute proportion de la capacité ultime. La précontrainte diminue la position de l’axe neutre et une plus grande partie de la section en béton est donc chargée en compression, résultant en une utilisation plus efficace du béton.

     

Analysis of pultruded glass reinforced plastic beams with semi-rigid end connections

The influence coefficient method of analysis has been used to derive closed-form expressions for the mid-span deflection and end rotations of shear deformable uniform section beams with semi-rigid end connections. The formulae have been recast into performance indices which define the reduction in mid-span deflection, the increase in load carrying capacity and the increase in span relative to an otherwise identical simply supported beam for two practical load distributions: (1) a point load at mid-span and (2) a uniform load over the entire span. Expressions are also presented for the required rotation capacity of the semi-rigid end connections. Initial rotational stiffness data, derived from full-scale tests on web and web and flange cleat connections between two sizes of pultruded glass reinforced plastic (GRP) WF-section are used in the formulae to evaluate the performance indices for the practical range of span-to-depth ratios for load case (1). The values obtained quantify the benefits to be derived from exploiting semi-rigid end connection stiffness in the design of pultruded GRP beams with the current, very limited, range of section sizes.     

A BEM formulation for linear bending analysis of plates reinforced by beams considering different materials

In this work, a boundary element method (BEM) formulation to perform linear bending analysis of building floor structures where slabs and beams can be defined with different materials is presented. The proposed formulation is based on Kirchhoff's hypothesis, the building floor being modelled by a zoned plate, where the beams are treated as thin sub-regions with larger rigidities. This composed structure is treated as a single body, the equilibrium and compatibility conditions being automatically taken into account. In the final integral equation, the tractions are eliminated along the interfaces, therefore reducing the number of degrees of freedom. The displacements are approximated along the beam cross-section, leading to a model where the values remain defined on the beam skeleton line instead of their boundaries. The accuracy of the proposed model is shown by comparing the numerical results with a well-known finite element code.     

Analysis of fatigue crack growth in reinforced concrete beams

Mechanical behavior of reinforced concrete members is influenced by the action of unknown crack bridging reactions of rebars. Under cyclic loading, due to progressive growth of cracks, this bridging action contributes to the overall strength, stiffness and hysteretic behavior of the member. In this work, fatigue behavior of reinforced concrete beams are studied using a crack propagation law, developed using dimensional analysis for plain concrete with the effect of reinforcement being simulated through constraint exerted on the crack opening. The parameters considered in the model are fracture toughness, crack length, loading ratio and structural size. A numerical procedure is followed to compute fatigue life of RC beams and the dissipated energy in the steel reinforcement due to the shake down phenomenon under cyclic loading. Through a sensitivity study, it is concluded that the structural size is the most sensitive parameter in the fatigue crack propagation phenomenon. Furthermore, the residual moment carrying capacity of an RC member is determined as a function of crack extension by including the bond-slip mechanism.     

Shakedown with softening in reinforced concrete beams

Critical softening parameters for first-formed highes at any location in a two-span beam are derived, and the relationship between positive load increments and softening parameter less than critical for midspan or interior support highes are also determined. The effect of non-critical softening on the well known static collapse and shakedown loads for an elastic-plastic beam is found. It is shown that the combined effect of softening and residual moments (such as caused by differential settlements) on the static collapse and shakedown loads may be dramatic.     

PLGA bone plates reinforced with crosslinked PPF

In this study a matrix of poly(propylene fumarate) (PPF) was crosslinked with N-vinylpyrrolidone (NVP), 2-hydroxyethylmethacrylate (HEMA), or a mixture of NVP and ethyleneglycol dimethacrylate (EGDMA) in the presence of poly(lactide-co-glycolide) (PLGA) to reinforce and preserve the form of PLGA bone plates. The degree of crosslinkage varied depending on the crosslinker as shown by the rapid and almost complete leaching of NVP upon incubation in phosphate buffered saline at 37 ° C in 900 h and retention of 92% of HEMA. With the reinforced bone plates extracted for 72 h at room temperature methylene chloride, the extracted PLGA from NVP/PPF, NVP-EGDMA/PPF, and HEMA/PPF were 75.42% (w/w), 59.52% (w/w), and 30.86% (w/w), respectively. The flexural modulus and compressive strength of the crosslinked PPF reinforced bone plates were higher than that of the unreinforced bone plate. Atomic force microscopy showed that NVP/PPF reinforced PLGA bone plates eroded substantially (a mean surface roughness of 19.319 nm) whereas NVP-EGDMA-PPF reinforced bone plate showed a distinct crystalline organization (and a higher roughness, 43.525 nm). In conclusion, we propose the consideration of NVP-EGDMA/PPF reinforced PLGA as a biodegradable orthopedic implant material that has a lower likelihood of warping or failing catastrophically than the currently available materials.     

Deflections of concrete beams reinforced with FRP bars

The aim of this study is to experimentally and theoretically investigate the flexural behavior of concrete beams reinforced with fiber reinforced polymer (FRP) bars. In this research, three types of experiments were made. First, the tensile properties of FRP and steel bars were tested, then the bond-slip behavior between bars and concrete was tested on standard specimens and, in the end, three series of concrete beams reinforced with GFRP, CFRP and steel bars were tested up to failure. The theoretical model for calculating deflections was developed, which included bond-slip behavior of FRP bars. The theoretical results were compared to the test results of beam deflections, as well to deflection results obtained by theoretical models developed by other authors.     

Shear strength of reinforced concrete beams with stirrups

This study presents alternative shear strength prediction equations for reinforced concrete (RC) beams with stirrups. The shear strength is composed of the contribution of the nominal shear strength provided by stirrups and the nominal shear strength provided by concrete. For the concrete contribution, cracking shear strength values estimated by Arslan’s equations are almost same those obtained with ACI 318 simplified equation in terms of coefficient of variation (COV). However, mean values estimated by ACI 318 tend to be more conservative comparing to the mean values obtained with Arslan’s equations. Thus, for the consideration of concrete contribution to shear strength, Arslan’s equations are used. To obtain the shear strength of RC beams, shear strength provided by stirrups is added to the concrete shear strength estimated by Arslan’s equations. Results of existing 339 beam shear tests are used to investigate how accurate proposed equation estimates the shear strength of RC beams. Furthermore, ACI 318 and TS500 provisions are also compared to the aforementioned test results. It is found that proposed equations for beams with shear span to depth ratios (a/d) between 1.5 and 2.5 are also conservative with a lower COV than ACI 318 and TS500. However, when a/d ratios exceed 2.5 (both normal and high strength concrete beams), ACI 318, TS500 and proposed equations give similar COV value.     

Shear resistance of reinforced concrete beams with non-prismatic sections

The work presented herein is mainly concerned with an experimental investigation into the shear behaviour of reinforced concrete beams of varying depth, and the way in which they fail in shear, compared to primatic beams.Several beams were tested under the same loading conditions and the following controlling parameters were varied: main longitudinal reinforcement, presence of stirrups, depth of beams at supports and the slope of the beams.The paper describes the application of the British Code of Practice, the American A.C.I., Code and the Russian Code of Practice for the determination of the ultimate shear strength to non-prismatic beams. Beam details and experimental data obtained during the tests are also included.     

Shear strengthening of reinforced concrete beams with PBO-FRCM composites

The work analyses the functioning of reinforced concrete beams which are shear reinforced with FRCM (Fibre Reinforced Cementitious Matrix) composite mesh on mineral mortar. The analysis of previous tests provides an overview on the efficiency of the related FRP (Fibre Reinforced Polymers) system used for strengthening the beams against shearing forces. The information gathered on the functioning of FRCM strengthened concrete elements show that the key factor in this technology is ensuring appropriate carrying capacity of anchoring of the mesh. The tests were carried out on reinforced concrete beams with different design of lateral reinforcement. The beams differed in terms of the angle of inclination of the reinforcement to the longitudinal axis, as well as in terms of the manner of anchoring the mesh at the top and bottom surface of beams. The destruction mechanisms of beams were analysed and described. Value of deflection measured at the maximum load for all of the beams is significantly greater than the allowable value. The distribution of strains is presented as a function of the load, where the phases of operation in relation to shear of FRCM reinforced beams are visible.     

Numerical modelling of concrete beams reinforced with pre-stressed CFRP

In this paper, a numerical simulation is presented on the behaviour of concrete beams, reinforced with pre-stressed CFRP. The numerical results are compared to experimental results. Nonlinear material behaviour is considered, namely: the inelastic compressive concrete behaviour, the elasto-plastic behaviour of steel reinforced bars, the bond-slip relationship between the concrete and the internal steel reinforced bars, the mode-II fracture interface between the concrete and the pre-stressed CFRP and concrete cracking. Cracking in concrete is modelled according to a discrete crack approach: micro-cracking is assumed to localize at fictitious cracks with initial zero width. Two different approximations are adopted: (i) the fictitious cracks are embedded within the finite elements, giving rise to a discrete strong discontinuity formulation and (ii) main cracks, similar to the experimentally observed, are introduced, using interface elements, along the element boundaries, since the beginning of the analysis. A non-iterative sequentially-linear approach is adopted in order to avoid convergence problems. The aim of the present analysis is to try to better understand the failure mechanisms found in the experimental tests. Despite the complexity of the multiple nonlinear aspects of the behaviour of the structure, it is concluded that the numerical results are similar and are close to those observed experimentally.