Size independent fracture energy evaluation for plain cement concrete

The present work deals with the investigation of a robust analytical scheme to assess the size‐independent fracture energy of concrete. The study involves the numerical modelling of three‐point bend (TPB) concrete beams that are geometrically similar, having constant length to depth ratio with varying range of notch to depth (a/W) ratios. The unique nonlinear behaviour of concrete 1material is incorporated through fracture energy‐based strain‐softening model in the finite‐element numerical simulation. The International Union of Laboratories and Experts in Construction Materials, Systems and Structures (RILEM) fracture energy values are evaluated through numerical simulation of several set of experimentally observed load‐load line displacement response. The RILEM fracture energy values associated with geometrically similar beams have been utilised to develop a methodology for assessment of the size‐independent fracture energy of concrete. The numerically predicted and experimentally evaluated size‐independent fracture energy using the RILEM fracture energy values are found to be in close agreement.     

用能量方法研究混凝土断裂过程区的力学性能

准脆性混凝土自由裂缝前缘断裂过程区的发展与其非线性断裂特征及尺寸效应现象密切相关。它的物理力学行为的量化分析对理解混凝土断裂破坏机理和建立适用于混凝土结构裂缝稳定分析和安全评估断裂准则尤为重要,一直是混凝土断裂力学研究的核心问题。该文依据Hillerborg给出的断裂能定义,给出了计算单位长度断裂过程区发展能量耗散的通用表达式。以三点弯曲梁为例,采用非线性软化本构关系,进一步给出了计算此平均能量耗散的具体步骤及对应的公式。在根据实测的三点弯曲梁的断裂能回归拟合了特征裂缝张开位移w0后,计算了每个试件整个断裂全过程中不同荷载时刻断裂过程区耗能的平均值。结果表明:随着裂缝扩展,断裂过程区能量耗散的变化和试件尺寸无关,可描述断裂过程区混凝土材料的力学性能。     

Determination of the kink point in the bilinear softening model for concrete

The characterization of the softening curve from experimental results is essential for predicting the fracture behavior of quasi-brittle materials like concrete. Among various shapes (e.g. linear, exponential) to describe the softening behavior of concrete, the bilinear softening relationship has been extensively used and is the model of choice in this work. Currently, there is no consensus about the location of the kink point in the bilinear softening curve. In this study, the location of the kink point is proposed to be the stress at the critical crack tip opening displacement. Experimentally, the fracture parameters required to describe the bilinear softening curve can be determined with the “two-parameter fracture model” and the total work of fracture method based on a single concrete fracture test. The proposed location of the kink point compares well with the range of kink point locations reported in the literature, and is verified by plotting stress profiles along the expected fracture line obtained from numerical simulations with the cohesive zone model. Finally, prediction of experimental load versus crack mouth opening displacement curves validate the proposed location of the kink point for different concrete mixtures and also for geometrically similar specimens with the same concrete mixture. The experiments were performed on three-point bending specimens with concrete mixtures containing virgin coarse aggregate, recycled concrete coarse aggregate (RCA), and a 50-50 blend of RCA and virgin coarse aggregate. The verification and validation studies support the hypothesis of the kink point occurring at the critical crack tip opening displacement.     

Numerical characterization of the nonlinear fracture process in concrete

The growth and development of the fracture process zone in plain concrete has been investigated. A fictitious crack model based noniterative numerical scheme is developed to study the fracture characteristics of specimens of different sizes and geometries. Results from numerical studies on four different geometrically similar specimen sizes and two different specimen geometries are reported and discussed. The finite element program developed accommodates linear as well as nonlinear softening laws for the fracture process zone in concrete. It is observed that the process zone reaches a steady state length which is specimen size as well as specimen geometry dependent. As long as the process zone is allowed to develop to its steady state length, the energy absorbed in the process zone appears to be size and geometry independent. Results from tests on three-point bending specimens and compact tension specimens reported in the literature have been compared with the numerical solutions obtained in this investigation. Specimen size and geometry dependence generally observed in these fracture experiments have been duplicated. The numerical model also successfully reproduces many of the other experimentally observed characteristics in the fracture of plain concrete.     

Boundary effect on the softening curve of concrete

The apparent fracture energy of concrete experimentally determined on the basis of the work of fracture in bending or wedge splitting tests becomes larger with increasing specimen dimensions. This experimental observation may be attributed to the varying local fracture energy along the crack path. When the crack tip approaches the specimen boundary, the size of the fracture process zone will be reduced and, consequently, only a portion of the fracture energy is activated; i.e., the local fracture energy is getting smaller. The influence of this boundary effect diminishes with increasing specimen size resulting in the size dependence of the apparent fracture energy determined by the work-of-fracture method as an average value in the ligament. With varying local fracture energy, the local softening curve will also show variations. The latter are subject of the present study. Wedge splitting tests with different specimen sizes as well as inverse analyses of these experiments were carried out. For the inverse analyses, the cohesive crack model was adopted and an evolutionary optimization algorithm has been used. The boundary effect on the local fracture properties was taken into account and, as a result, the variation of the softening curve along the crack path could be determined. It was found that the tail of the softening curve is shortened and lowered due to the boundary effect whereas the initial slope of this curve appears to be not affected.     

On the fracture behaviour of thin-walled SFRC roof elements

The paper presents an experimental and a numerical investigation on precast, prestressed reinforced concrete (RC) and steel fibre reinforced concrete (SFRC) roof elements. The element investigated has a complex geometry, because it is characterized by a thin-walled open cross-section and a long span. In order to reduce the total weight of the traditional RC element and favour an industrialized production process, the structure can be made of fibre reinforced concrete. This composite presents a significant toughness after cracking that can substitute the diffused reinforcement made of common steel-welded meshes, conserving the longitudinal prestressed reinforcement. The mechanical characterization of SFRC material has found recently a shared design approach that starts with the identification of the uniaxial tension constitutive law obtained from a standardized bent notched specimen. Nevertheless, for defined casting procedures of the structure, like in prefabrication, the identification of the uniaxial tension constitutive law can be performed by a four point bending tests on suitable unnotched specimens, able to take into account the effective fibre orientation in the structure and the real nominal thickness of the critical portion of the element. The latter two different experimental test procedures (on notched or unnotched specimens) lead to significant differences in the tension softening response. For this reason SFRC tension softening relations, coming from the previously mentioned experimental tests, are analyzed in this paper in order to evaluate their effects on the structural response of this large-scale roof element. The results of the experimental tests on the roof element presented in this paper show that second-order effects drastically anticipated the achievement of the longitudinal bending moment resistance calculated following the beam theory and neglecting transverse equilibrium and in-plane cross section deformation. Two numerical models are proposed in this paper to evaluate second-order effects in the resistance assessment of the precast structure. The first one is based on a plane section approach (PSA), while the second one is based on a non-linear finite element analysis (NLFEA). Both second-order effect and uniaxial tension constitutive relationship roles are examined in relation to the global response of the structure up to failure. The final remarks, coming from a careful comparison between experimental and numerical results, highlight that the failure is mainly led by a structural behaviour, because second-order effects prevail on non-linear response of SFRC materials adopted.     

Fracture properties of concrete reinforced with steel–polypropylene hybrid fibres

This research discusses polypropylene fibres and three sizes of steel fibres reinforced concrete. The total fibre content ranges from 0% to 0.95% by volume of concrete. A four-point bending test is adopted on the notched prisms with the size of 100×100×500 mm³ to investigate the effect of hybrid fibres on crack arresting. The research results show that there is a positive synergy effect between large steel fibres and polypropylene fibres on the load-bearing capacity and fracture toughness in the small displacement range. But this synergy effect disappears in the large displacement range. The large and strong steel fibre is better than soft polypropylene fibre and small steel fibre in the aspect of energy absorption capacity in the large displacement range. The static service limitation for the hybrid fibres concrete, with “a wide peak” or “multi-peaks” load–CMOD patterns, should be carefully selected. The ultimate load bearing capacity and the crack width or CMOD at this load level should be jointly considered in this case. The K_IC and fracture toughness of proper hybrid fibre system can be higher than that of mono-fibre system.     

A model for cracking in fibre-reinforced concrete structures

After a quick review concerning the probabilistic discrete cracking model relative to non-steel fibre reinforced concrete (Rossi probabilistic discrete cracking model) and a critical analysis of its extension to steel fibre reinforced concrete, a new Rossi probabilistic discrete cracking model applied to steel fibre reinforced concrete is proposed. In this model, cracking is modelled through contact elements which will have a perfect elastic-plastic behavior with brittle fracture. The plastic step as well as the postcracking energy are uncorrelated random variables. Numerical work still remains, consisting in adapting the existing contact elements to the case of perfect elastic-plastic behavior with brittle fracture, as well as work on a suitable resolution algorithm.     

Correlation of tensile and flexural responses of strain softening and strain hardening cement composites

Closed form equations for generating moment–curvature response of a rectangular beam of fiber reinforced concrete are presented. These equations can be used in conjunction with crack localization rules to predict flexural response of a beam under four point bending test. Parametric studies simulated the behavior of two classes of fiber reinforced concrete: strain softening and strain hardening materials. The simulation revealed that the direct use of uniaxial tension and compression responses under-predicted the flexural response for strain softening material while a good prediction for strain hardening material was obtained. The importance of strain softening range on the flexural response is discussed using non-dimensional post-peak parameters. Results imply that the brittleness and size effect are more pronounced in the flexural response of brittle materials, while more accurate predictions are obtained with ductile materials. It is also demonstrated that correlations of tensile and flexural results can be established using normalized uniaxial tension and compression models with a single scaling factor.     

Experimental Investigations of Fracture Process Using DIC in Plain and Reinforced Concrete Beams under Bending

The fracture behaviour of concrete and reinforced concrete beams under quasi‐static three‐point bending was comprehensively investigated with experiments at laboratory scale. The eight various concrete mixes were tested. The influence of the shape, volume and size of aggregate particles and reinforcement on concrete fracture under bending was studied. Displacements on the surface of concrete beams were measured by means of the digital image correlation (DIC) technique. Attention was paid to the formation of a localized zone and its characteristics. In order to avoid the effect of the search patch size and the cut‐off value at displacement and strain profiles, a consistent method was proposed to determine uniformly and accurately the width of a localized zone. Measured surface displacements from DIC were fitted by the error function ERF, whereas surface strains calculated from displacements were fitted by the usual normal distribution (Gauss) function. The width of a localized zone preceding a macro‐crack grew strongly with increasing maximum aggregate size and slightly with diminishing aggregate volume. It did not depend on the aggregate roughness and reinforcement presence.     

The effect of CFRP and CFRP/concrete interface models when modelling retrofitted RC beams with FEM

Concrete structures retrofitted with fibre reinforced plastic (FRP) applications have become widespread in the last decade due to the economic benefit from it. This paper presents a finite element analysis which is validated against laboratory tests of eight beams. All beams had the same rectangular cross-section geometry and were loaded under four point bending, but differed in the length of the carbon fibre reinforced plastic (CFRP) plate. The commercial numerical analysis tool Abaqus was used, and different material models were evaluated with respect to their ability to describe the behaviour of the beams. Linear elastic isotropic and orthotropic models were used for the CFRP and a perfect bond model and a cohesive bond model was used for the concrete–CFRP interface. A plastic damage model was used for the concrete. The analyses results show good agreement with the experimental data regarding load–displacement response, crack pattern and debonding failure mode when the cohesive bond model is used. The perfect bond model failed to capture the softening behaviour of the beams. There is no significant difference between the elastic isotropic and orthotropic models for the CFRP.     

基于扩展有限元法的钢筋混凝土梁复合断裂过程模拟研究

针对钢筋混凝土梁裂纹扩展问题,基于扩展有限元法,建立了预置裂纹的简支混凝土梁三维模型,用粘聚裂纹模型描述裂纹面间的力学行为,采用线性的软化曲线表示裂纹尖端断裂过程区的应变软化行为,分别对素混凝土梁和钢筋混凝土梁的复合断裂过程进行模拟,分析了纵向钢筋对裂纹扩展路径、荷载-挠度和荷载 -CMOD (裂缝开口处张开位移)曲线的影响,并与文献中的试验结果进行对比,计算结果与试验结果吻合良好,展示了扩展有限元法在结构断裂破坏分析方面的独特优势。     

The effect of specimen size on the fracture energy and softening function of concrete

This paper examines the effect of specimen size on the fracture energy of concrete, G_F, as measured using the three-point bending test on pre-notched beams prescribed by RILEM TC-50 [1]. The concept of partial fracture energy is introduced and used to explain the observed size effect. The opening displacement at the top of the notch in the test specimen at the end of the test, ω, is affected by the size of the specimen, which in turn affects the measured value of the concrete fracture energy. In theory, when the specimen is large enough to allow the fracture process zone to develop fully,w _f will reach its critical value,w _c , and the effect of specimen size onG _F will be eliminated. The experimental results included here show that in reality the size of the specimen does affect the measurement ofG _F , even when the size is such that the fracture process zone develops fully. This may be due to local plastic deformation in the area around the loading point, which is particularly significant in larger specimens. It may also be due to differences in the influence of the boundary conditions of the test for different specimen sizes. In addition, a procedure is outlined for the determination of the softening function for concrete based on the fracture energy measured in RILEM tests, in which the specimens are small enough to ensure that the energy measured is actually due to fracture and not plastic deformation.     

The fracture and fatigue performance in flexure of carbon fiber reinforced concrete

The fracture parameters and fatigue performances of carbon fiber reinforced concrete is investigated by three point bending tests. In comparison with the results of quasi-static tests where no pre-cyclic loading is applied, the influence of pre-cyclic loading history on fracture parameters was researched by using compliance calibration. The test results show that the fracture parameters of carbon fiber reinforced concrete and plain concrete will be reduced if the pre-cyclic loading stress levels are higher than a certain threshold, and this threshold value for carbon fiber reinforced concrete is higher than that of plain concrete. The critical effective crack length for carbon fiber reinforced concrete is significantly larger than that of plain concrete and independent of the pre-cyclic loading history and fatigue life. Carbon fiber reinforced concrete has a considerable beneficial effect on the behaviour of concrete subjected to flexure fatigue loading.     

Fracture stability and residual strength assessment of reinforced concrete beams

In conventional analysis and design procedures of reinforced concrete structures, the ability of concrete to resist tension is neglected. Under cyclic loading, the tension-softening behavior of concrete influences its residual strength and subsequent crack propagation. The stability and the residual strength of a cracked reinforced concrete member under fatigue loading, depends on a number of factors such as, reinforcement ratio, specimen size, grade of concrete, fracture properties, and on the tension-softening behavior of concrete. In this work, a method is proposed to assess the residual strength of reinforced concrete beams subjected to cyclic loading. The crack extension resistance based approach is used for determining the condition for unstable crack propagation. The effect of reinforcement is modeled as a closing force counteracting the effect of crack opening produced by the external moment. The effect of percentage reinforcement and specimen size on the failure of reinforced beams is studied. Finally, the residual strength of the beams are computed by including the softening behavior of concrete.     

Fracture toughness characterisation of a glass fibre‐reinforced plastic composite

By examining the state of the art, it can be realised that few research works have been done on the fracture behaviour of plastic composites reinforced with continuous glass fibres. Therefore, the present paper deals with the fracture toughness of a unidirectional glass fibre‐reinforced plastic (GFRP), such a parameter being analytically determined by means of the modified two‐parameter model (MTPM). The input data of the MTPM are obtained from an experimental campaign related to three‐point bending tests on single edge‐notched specimens characterised by different sizes. The novelty of this research work is that the MTPM, originally proposed for isotropic materials, is here employed to estimate the fracture toughness of GFRPs characterised by orthotropic mechanical properties.     

Structural response of hyperstatic concrete beams reinforced with GFRP bars: Effect of increasing concrete confinement

The design of concrete structures reinforced with glass fibre reinforced polymer (GFRP) bars is influenced by their reduced stiffness and brittleness. In hyperstatic structures, the methodology used in force analysis depends on the ductility of the structural systems, which in this case, being essentially provided by the concrete, can be potentially increased by confining concrete in critical zones. This paper presents experimental and numerical investigations about the flexural behaviour of continuous beams reinforced with GFRP bars, namely of their service and failure responses, and the effect of increasing concrete confinement in critical cross-sections. A calculation procedure to quantify the confinement effect in beams due to the reduction of the spacing between shear stirrups is first presented. The experimental investigations comprised a comparative study in which two-span concrete beams reinforced with either GFRP or steel bars were tested in bending. In the former, the effect of reducing the shear stirrups spacing was analyzed together with the under- and over-reinforcement at the central support and midspan cross-sections, respectively. The development of a crack hinge in the continuity support zone highlighted the better performance of beams under-reinforced on the top layer with GFRP bars compared to “equivalent” beams reinforced with steel, namely at the resistance level. In addition, the confinement at critical zones increased significantly the strength and ductility. The numerical investigations included the development of non-linear finite element models for all beams tested - numerical results are in good agreement with test data and seem to confirm the confinement effect observed in the experiments.     

A design model for fibre reinforced concrete beams pre-stressed with steel and FRP bars

This paper presents a design oriented model to determine the moment–curvature relationship of elements of rectangular cross section failing in bending, made by strain softening or strain hardening fibre reinforced concrete (FRC) and reinforced with perfectly bonded pre-stressed steel and fibre reinforced polymeric (FRP) bars. Since FRP bars are not affected by corrosion, they have the minimum FRC cover thickness that guaranty proper bond conditions, while steel bars are positioned with a thicker FRC cover to increase their protection against corrosion. Using the moment–curvature relationship predicted by the model in an algorithm based on the virtual work method, a numerical strategy is adopted to evaluate the load–deflection response of statically determinate beams. The predictive performance of the proposed formulation is assessed by simulating the response of available experimental results. By using this model, a parametric study is carried out in order to evaluate the influence of the main parameters that characterize the post cracking behaviour of FRC, and the pre-stress level applied to FRP and steel bars, on the moment–curvature and load–deflection responses of this type of structural elements. Finally the shear resistance of this structural system is predicted.     

The size effect expression for an elastic-softening material

Earlier work has led to the derivation of a size effect expression which relates the maximum sustainable load for an elastic-softening material, such as concrete, to a solid's dimensions, provided these are large. The derivation depends on assumptions made with regard to the behaviour of the elastically equivalent softening zone size and the fracture energy, as the maximum load state is approached. The present paper provides underpinning for the size effect expression as it applies to a positive geometry, and also as it applies to a negative geometry when the expression gives the load required for a softening zone to be fully developed.