Mechanical behaviour of corroded reinforced concrete beams—Part 2: Bond and notch effects

This paper deals with the effect of steel cross-section and bond strength reduction on the mechanical behaviour of corroded RC-beams. In the case of corroded reinforced concrete members, those effects are always coupled and a previous study (part one) has shown that it is not realistic to forecast the behaviour of corroded beams merely in terms of steel cross-section reduction. The object of the study is thus to understand the separated and coupled effects of the reduction in bond strength and steel cross-section. These investigations are carried out in order to be able to model the behaviour of corroded structural members and to predict how and when repairing is necessary. Different experimental simulations of corrosion were made. The results show the significant impact of coupling between reduction of bond strength and steel cross-section.

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Résumé Cet article traite de l’influence de la réduction de section d’acier et de l’adhérence acier béton en partie tendue sur le comportement mécanique des poutres corrodées. L’étude précédente (partie 1) semble montrer qu’il n’est pas réaliste de vouloir prédire le comportement mécanique des éléments de structures corrodées en ne tenant compte que de la réduction de section des aciers tendus. Par conséquent, l’objectif de ce travail est de mieux comprendre et quantifier les effets couplés et découplés de ces deux paramètres. Pour cela, plusieurs simulations expérimentales des effets de la corrosion ont été réalisées sur poutres ou sur échantillons d’armature non corrodés. Les expérimentations mécaniques sont réalisées en service et à rupture. Les résultats obtenus confirment largement qu’une prédiction réaliste du comportement mécanique résiduel en service des poutres corrodées, ne sera obtenue qu’en prenant en compte de l’effet couplé de la réduction de section d’acier et de l’adhérence acier béton en partie tendue.

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Editorial Note Prof. Ginette Arliguie is a RILEM Senior Member. She works at LMDC (Laboratoire Matériaux et Durabilité des Constructions), a RILEM Titular Member.     

Round-Robin analysis of over-reinforced concrete beams—Comparison of results

Following the Round-Robin test on compressive softening, RILEM TC 148-SSC (“Strain Softening of Concrete”, see [1]) has proposed a second Round-Robin on the application of strain-softening data from simple uniaxial compression tests for the analysis of over-reinforced beams. The beams were tested at Aalborg University in Denmark in 1996. The Round-Robin is carried out in collaboration with ACI-ASCE committee 447 (“Finite Element Analysis of Reinforced Concrete Structures”). The results of the beam tests have been kept confidential, and researchers were invited to analyze the beams given the compressive and tensile properties of the concrete and the reinforcement. In total six contributions were submitted, which were compared to the experimental results in a workshop at the 3rd International Conference on Fracture Mechanics of Concrete and Concrete Structures, in Gifu, Japan in October 1998. In this report the outcome of the different analyses are summarized and the results are compared to the experimentally obtained peak loads, ductility and size effect.     

Flexural–torsional postbuckling analysis of beams of arbitrary cross section

In this article, the postbuckling analysis of axially compressed elements of arbitrary cross section is presented taking into account moderately large displacements, moderately large angles of twist and employing nonlinear relationships between bending moments and curvatures. The elements are supported by the most general boundary conditions including elastic support or restraint. Based on Galerkin’s method and approximating the displacement field of the element by polynomial expressions the governing differential equations lead to a nonlinear algebraic system. The geometric, inertia, torsion and warping constants of the arbitrary beam cross section are evaluated employing the boundary element method. The proposed formulation does not stand on the assumption of a thin-walled structure and therefore the cross section’s torsional rigidity is evaluated exactly without using the so-called Saint–Venant’s torsional constant. Both the Wagner’s coefficients and the shortening effect are taken into account, while their influence is examined and discussed. Numerical examples are worked out to illustrate the efficiency, the accuracy and the range of applications of the developed method.     

Dynamic buckling of elastic–plastic beams including effects of axial stress waves

Buckling of axially compressed elastic–plastic beams is discussed. The load is applied instantaneously and remains unaltered during the motion. The effect of stress waves travelling along the beam is taken into account. It is assumed that the material of the beam has linear-strain hardening. A method of solution, based on the Galerkin technique, is proposed; this method is applicable to an arbitrary number of degrees of freedom. Numerical examples are presented.     

Post-cracking strength and ductility of glass–GFRP composite beams

This paper presents results of experimental and numerical investigations on the structural behaviour of composite beams made of annealed glass panes and glass fibre reinforced polymer (GFRP) pultruded profiles. The main goal of the transparent structural solutions presented here is to increase the post-cracking residual strength and ductility of glass by using GFRP strengthening laminates. The experimental programme included (i) tensile tests on double lap joints between glass and GFRP pultruded laminates, bonded with different types of structural adhesives, and (ii) full-scale flexural tests on glass beams and glass–GFRP composite beams, with different strengthening geometries and structural adhesives. Results obtained in this study show that, unlike glass beams, in glass–GFRP composite beams it is possible to obtain relatively ductile failure modes, with a significant increase of both strength and deformation capacity after the initial cracking of glass. The stiffness of the structural adhesive used, together with the geometry of the GFRP strengthening element, have a major influence on the structural response of the composite beams. Finite element models were developed for all tested beams, allowing to simulate their serviceability behaviour (prior to glass cracking) with fairly good accuracy, namely in what concerns the degree of shear interaction at the bonded interfaces.     

Propagation properties of Airy–Gaussian beams in centrosymmetric photorefractive media

Based on the biased centrosymmetric photorefractive media, we investigate numerically the propagation and interaction properties of Airy–Gaussian beams. The single Airy–Gaussian beam forms the one-component breather with the help of the photorefractive nonlinearity. The interaction properties of two Airy–Gaussian beams can be controlled by adjusting the relative parameters, such as photorefractive nonlinearity, transverse distance and relative phase of two incident beams. The two-component breather with ladder structure can be observed for both the in-phase and out-of-phase cases when the self-acceleration property is balanced by the photorefractive nonlinearity. The one- or three-component breathers can be observed for the in-phase case only when the transverse distance takes a certain range.     

Prediction of thermal post buckling and deduction of large amplitude vibration behavior of spring–hinged beams

A general energy formulation to predict the thermal post buckling behavior of uniform isotropic beams is presented in this paper. The hinged ends of the beam contain elastic rotational restraints to represent the actual practical support situation. The large amplitude vibration behavior of beams is deduced from the post buckling results. The classical hinged and clamped conditions can be obtained as the limiting cases of the rotational spring stiffness. The numerical results, in the form of the ratios of the post buckling to buckling loads for various maximum deflection ratios, are presented in the digital form. An alternate independent formulation, based on the nonlinear finite element formulation, is also used in this paper to validate the numerical results of the present work. Further, the results for the large amplitude vibrations, deduced from the thermal post buckling results are also presented and these results compare very well with the finite element results, available in the literature, for the large amplitude vibration problem. These comparisons show an excellent agreement not only for the present work on the proposed thermal post buckling formulation but also on the deduced results for the large amplitude vibration of beams with the ends elastically restrained against rotation (spring–hinged beams). The numerical results presented confirm the efficacy of the proposed methodology used for predicting the post buckling behavior and deducing the large amplitude vibration behavior of the spring–hinged beams.     

Effective radius of curvature of partially coherent Hermite–Gaussian beams propagating through non-Kolmogorov turbulence

Based on the extended Huygens–Fresnel principle and non-Kolmogorov spectrum, the analytical expression for the effective radius of curvature of partially coherent Hermite–Gaussian (PCHG) beams propagating through non-Kolmogorov turbulence is derived, and the relative effective radius of curvature is used to describe the effect of turbulence on the effective radius of curvature. It is shown that the effective radius of curvature of PCHG beams depends on the beam and non-Kolmogorov turbulence parameters and on the propagation distance. The variation of relative effective radius of curvature with increasing generalized exponent parameter α of non-Kolmogorov turbulence is non-monotonic. The longer the propagation distance is, the larger the effect of turbulence on the effective radius of curvature of PCHG beams is. The effective radius of curvature of PCHG beams with shorter wavelength, smaller beam order, larger beam waist width or better spatial coherence is more affected by the non-Kolmogorov turbulence. The results are interpreted physically.     

Numeral modeling of fracture failure of recycled aggregate concrete beams under high loading rates

Fracture tests of recycled aggregate concrete (RAC) beams of different sizes were conducted under high loading rates. In order to characterize the effect of high loading rate on the behavior of RAC beams, two new material models were used together with the commercial finite element software \(\hbox {ABAQUS}^{\mathrm{R}}\). One model is a viscoelastic model that can predict the increase of stiffness (modulus of elasticity) of RAC with increasing loading rate, and the other model is a multiphase composite model that can determine the effective stiffness of RAC taking into account the special internal structure of recycled aggregate. Two different cases were considered in the numerical simulation. Case 1 is for fixed beam size under different loading rates, and Case 2 is for fixed loading rate with different beam sizes. For Case 1, the simulation results of the maximum loads under three different strain rates agreed with test data quite well. The Force-CMOD curves of the numerical simulation and test data showed similar trends. The higher the strain rates, the wider the high stresses spread in the crack propagation zone. The good agreements with the test data indicated that the two new material models can characterize the effect of high loading rate on RAC beams very well. For Case 2, three beam sizes and one loading rate was studied. The post-peak Force versus CMOD curves from the simulation follow the same trend of the test data. The stress distributions in the beams of different sizes are similar. On the other hand, the maximum loads predicted by the numerical model did not agree very well with test data. This is due to the fact that the maximum forces of RAC notched beams exhibited size effect, which was not considered in the fracture criteria adopted in \(\hbox {ABAQUS}^{\mathrm{R}}\) and not in the two new material models. This will be a topic for future research.     

Propagation properties of rectangular Laguerre–Gaussian-correlated Schell-model beams in atmospheric turbulence

Based on the extended Huygens–Fresnel principle and the second-order moments of the Wigner distribution function (WDF), the analytical expressions for the cross-spectral density (CSD) and the propagation factor of a rectangular Laguerre–Gaussian-correlated Schell-model (LGCSM) beam propagating in atmospheric turbulence are derived. The statistical properties, such as the average intensity, the spectral degree of coherence (SDOC) and the propagation factor, of a rectangular LGCSM beam in free space and atmospheric turbulence are comparatively analysed. It is illustrated that a rectangular LGCSM beam exhibits self-splitting and combing properties on propagation in atmospheric turbulence, and the self-splitting properties of such beam are closely related to its beam orders m and n, which is quite different from other self-splitting beams. In addition, the rectangular LGCSM beam has an advantage for reducing the turbulence-induced degradation compared with the conventional partially coherent beams.     

Representation and focusing properties of higher-order radially polarized Laguerre–Gaussian beams

A method based on higher-order Poincaré sphere was proposed to represent the states of polarization of higher-order radially polarized Laguerre–Gaussian (LG) vectorial beams (VBs). And the focusing properties of such LG beams of different radial orders focused by a high-NA lens were discussed. By tuning the ratio of the pupil radius to the waist of the incident beams, some cage-like or needlelike electric intensity field is generated in the focal region for several specific LG VBs with high order. Modulated by diffractive optical elements, the shape of the focal field shows novel behaviors such as splitting of cage-like modes, which provides potentially a more flexible control over micro-particles.     

Behaviour of hybrid FRP–UHPC beams in flexure under fatigue loading

The fatigue behaviour of innovative hybrid FRP–UHPC beams under flexural loading is investigated in this paper. The beams were made up of a pultruded GFRP hollow box section beam with a cast-in-place UHPC layer on top and either a CFRP or SFRP sheet bonded along the bottom. Four hybrid beams were tested under variable amplitude loading, to determine the effect of cyclic loading on flexural strength and stiffness. Analysis also included the development of a modified S–N curve and evaluation of fatigue damage using the Palmgren–Miner rule. It was found that insignificant loss in strength and stiffness occurred for all beams, where the fatigue damage estimated using the Palmgren–Miner rule overestimated the fatigue life. It was postulated, by comparison, that the hybrid beams reinforced with CFRP sheets may perform better under fatigue loading than the beams reinforced with SFRP sheets.     

Behaviour of hybrid FRP–UHPC beams subjected to static flexural loading

This paper provides the experimental results of a new hybrid beam intended for use in bridge applications. The hybrid beams were made up of pultruded Glass Fibre Reinforced Polymer (GFRP) hollow box section beams strengthened with a layer of Ultra-High-Performance-Concrete (UHPC) on top and either a sheet of Carbon FRP (CFRP) or Steel FRP (SFRP) on the bottom of the beam. Four hybrid FRP–UHPC beams were tested along with one control GFRP hollow box beam under four-point static flexural loading. Two types of beams were tested (Phase I and Phase II), which incorporated different connection mechanisms at the GFRP–UHPC interface. It was concluded that the hybrid beams had higher flexural strength and stiffness than the control beam, where the beams reinforced with SFRP showed greater percent cost effectiveness than beams reinforced with CFRP. In addition, the improved connection mechanism used in Phase II beams was found to provide adequate interface bond strength to maintain full composite action until ultimate failure.     

Impact failure of beams using damage mechanics: Part II—Application

The analytical model developed in Part I (Int J Impact Eng 2002;27(8):837–61) using continuum damage mechanics is used in this paper to predict the displacement to failure of beams subjected to static or dynamic loads producing large plastic strains. The beams, which were made of a ductile, strain rate sensitive material, were loaded at different positions on the span, by two tup geometries travelling with initial impact velocities up to 15 m/s. A reasonable correlation is obtained between the theoretical predictions and the experimental results for the displacements to failure.     

Shear deformation effect in flexural–torsional vibrations of beams by BEM

In this paper, a boundary element method is developed for the general flexural–torsional vibration problem of Timoshenko beams of arbitrarily shaped cross section taking into account the effects of warping stiffness, warping and rotary inertia and shear deformation. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved employing a boundary integral equation approach. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. All basic equations are formulated with respect to the principal shear axes coordinate system, which does not coincide with the principal bending one in a nonsymmetric cross section. To account for shear deformations, the concept of shear deformation coefficients is used. Six boundary value problems are formulated with respect to the transverse displacements, to the angle of twist, to the primary warping function and to two stress functions and solved using the Analog Equation Method, a BEM based method. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy.     

Reinforcement corrosion and the deflection of RC beams––an experimental critique of current test methods

This paper presents an experimental critique of the current test methods used to assess the effects of reinforcement corrosion on the serviceability deflections of reinforced concrete beams. Importantly, the work reported here highlights the weakness of tests aimed at assessing the deflection behaviour of beams in which the corrosion of the steel and the application of the service loads are undertaken as two separate and sequential processes.In the present series of tests, the central deflections of beams subjected to 23% and 34% of the design ultimate load, under 4-point loading subjected to simultaneous accelerated corrosion, were monitored over a period of approximately 30 days. Uncorroded beams were used as control samples and tested in parallel with the corroded samples.The results show the importance of assessing the structural effects of reinforcement corrosion under simultaneous load and corrosion conditions, as would occur in situ. In this situation, when 6% of the mass of steel is corroded, beam deflections are increased by 40–70% relative to the deflection of the control samples.     

The vibrations of pre-twisted rotating Timoshenko beams by the Rayleigh–Ritz method

A modeling method for flapwise and chordwise bending vibration analysis of rotating pre-twisted Timoshenko beams is introduced. In the present modeling method, the shear and the rotary inertia effects on the modal characteristics are correctly included based on the Timoshenko beam theory. The kinetic and potential energy expressions of this model are derived from the Rayleigh–Ritz method, using a set of hybrid deformation variables. The equations of motion of the rotating beam are derived from the kinetic and potential energy expressions introduced in the present study. The equations thus derived are transmitted into dimensionless forms in which main dimensionless parameters are identified. The effects of dimensionless parameters such as the hub radius ratio, slenderness ration, etc. on the natural frequencies and modal characteristics of rotating pre-twisted beams are successfully examined through numerical studies. Finally the resonance frequency of the rotating beam is evaluated.     

Impact failure of beams using damage mechanics: Part I—Analytical model

A simple theoretical method, which is based on ductile damage mechanics and which retains strain rate effects, is presented for predicting the failure of beams made from a perfectly plastic material and subjected to impact loads. For this class of materials, the strains can be estimated by defining a hinge length. The definition adopted here leads to reasonable predictions for the plastic strains and the strain rate, as shown by comparing the results with numerical calculations and experimental data. The equivalent strain and the strain rate can be used in the damage model to predict the failure of beams, as shown in a companion paper (Alves, Jones, Int J Impact Eng 2002;27(8):863–90).