[J014] Hygro-Thermo-Chemical Modeling of High Performance Concrete. II: Calibration and Validation.

Authors: G. Di Luzio and G. Cusatis.

Reference: Cement and Concrete Composites. 2009, 31(5), pp. 309-324.

Abstract: This paper presents the numerical implementation, calibration, and validation of the hygro-thermochemicalmodel formulated theoretically in the companion paper. Calibration and validation are based on experimental datagathered from the literature and relevant to hydration, self-desiccation, and drying of standard and high-performance concrete with and without silica fume.

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[J013] Hygro-Thermo-Chemical Modeling of High Performance Concrete. I: Theory.

Authors: G. Di Luzio and G. Cusatis.

Reference: Cement and Concrete Composites. 2009, 31(5), pp. 301-308.

Abstract: This study deals with the formulation, calibration, and validation of a new hygro-thermo-chemical modelfor high-performance concrete (HPC) suitable for the analysis of moisture transport and heat transfer atthe early age and beyond. In Part I of this study the theoretical formulation is presented and discussed indetail. Classical macroscopic mass and energy conservation laws are written by using humidity and temperatureas primary variables and by taking into account explicitly various chemical reactions, such ascement hydration, silica fume reaction, and silicate polymerization. The effect of cement hydration ismodeled through the classical concept of hydration degree. Silica fume reaction and silicate polymerizationare modeled by introducing the degree of silica fume reaction and the concentration of silicatepolymers, along with their evolution laws. The present model can simulate early age phenomena, suchas self-heating and self-desiccation, with great accuracy. Numerical implementation, calibration andvalidation of the model by comparison with experimental test data are postponed to Part II of this study.

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[J012] Cohesive Crack Analysis of Size Effect.

Authors: G. Cusatis and E. A. Schauffert.

Reference: Engineering Fracture Mechanics. 2009, 76, pp. 2163-2173.

Abstract:This paper deals with the analysis of size eect in concrete. An extensive campaignof accurate numerical simulations, based on the cohesive crack model, is performedto compute the size eect curves (CSEC) for typical test congurations. The resultsare analyzed with reference to the classical Bazant’s size eect law (SEL) to inves-tigate the relationship between CSEC and SEL. This analysis shows that the SEL represents the asymptote of the CSEC, and that the SEL parameter known as theeective fracture process zone length is a material property which can be expressedas a function of the cohesive crack law (CCL) parameters. Finally, the practicalimplications of this study are discussed in relation to the use of the CSEC or theSEL for the identication of the CCL parameters through the size eect method.

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[J011] Identification of Concrete Fracture Parameters through Size-Effect Experiments.

Authors: L. Cedolin and G. Cusatis.

Reference: Cement and Concrete Composites. 30 (2008) 788-797.

Abstract: This paper deals with the identification of concrete fracture parameters through indirect methods based onsize-effect experiments. These methods utilize the size effect curve (structural strength versus structuralsize), associated with a certain specimen geometry, to identify the tensile strength and the initial fractureenergy. These two parameters, in turn, are typically used to characterize the peak and the initial post-peak slope of the cohesive crack law. In the literature two different approaches can be found for the calculation of the size effect curve: (a) an approach based on the polynomial interpolation of numerically calculatedstructural strengths of geometrically similar specimens of different sizes, and (b) the classical approachbased on Equivalent Elastic Fracture Mechanics, which gives rise to the well-known Bažant’s size effectlaw (SEL). In this paper, the two approaches are first reviewed, the relationship between them isinvestigated, and a new procedure to identify the tensile strength using the SEL is proposed. Then severalsets of experimental results, recently performed at the Politecnico di Milano, are analyzed with bothapproaches in order to assess their range of applicability and accuracy in the identification of the twofracture parameters specified above.

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[J010] Capacity of Rectangular Cross Sections Under Biaxial Eccentric Loads.

Authors: L. Cedolin, G. Cusatis, S. Eccheli, and M. Roveda.

Reference: ACI Structural Journal, 2008, 105(2), pp. 215-224.

Abstract: This paper deals with the design of (short) columns with rectangular cross section under combined axial load and biaxial bending. The method presented is based on an analytical approximation of the moment contours, which represent horizontal sections of the interaction surface at failure for assigned axial loads. The analytical expression used is of the type suggested by Bresler, in which however the power exponent is calculated on the basis of the determination of the interaction diagram for load eccentricity along a diagonal. It is demonstrated that this interaction diagram in dimensionless form coincides with that of an equivalent square cross section under uniaxial bending. The method is checked to be very accurate and conservative. Dimensionless analytical expressions15 of the required interaction diagrams according to the ACI Code provisions are reported.

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[J009] Spectral Stiffness Microplane Model for Quasi-Brittle Composite Laminates: II. Validation and Calibration.

Authors: A. Beghini, G. Cusatis, and Z.P. Bažant.

Reference: Journal of Applied Mechanics, ASME, 2008, 75(2), pp. (021009)1-9.

Abstract: The spectral stiffness microplane (SSM) model developed in the preceding part I of this study is verifiedby comparisons with experimental data for uniaxial and biaxial tests of unidirectional and multidirectional laminates.The model is calibrated by simulating the experimental data on failure stress envelopes analyzed in the recent so-called“World Wide Failure Exercise”, in which various existing theories were compared. The present theory fits the experiments as well as the theories that were best in the Exercise. In addition, it can simulate the post-peak softening behavior andfracture, which is important for evaluating the energy-dissipation capability of composite laminate structures. The postpeaksoftening behavior and fracture are simulated by means of the crack band approach which involves a material characteristic length.

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[J008] Spectral Stiffness Microplane Model for Quasi-Brittle Composite Laminates: I. Theory.

Authors: G. Cusatis, A. Beghini, and Z.P. Bažant.

Reference: Journal of Applied Mechanics, ASME, 2008, 75(2), pp. (021010)1-6.

Abstract: The paper presents the spectral stiffness microplane (SSM) model, which is a general constitutive modelfor unidirectional composite laminates, able to simulate the orthotropic stiffness, pre-peak nonlinearity, failure envelopesand, in tandem with the material characteristic length, also the post-peak softening and fracture. The framework ofmicroplane model is adopted. The model exploits the spectral decomposition of the transversely isotropic stiffness matrixof the material to define orthogonal strain modes at the microplane level. This decomposition is a generalization of thevolumetric-deviatoric split already used by Baˇzant and coworkers in microplane models for concrete, steel, rocks, soils andstiff foams. Linear strain-dependent yield limits (boundaries) are used to provide bounds for the normal and tangentialmicroplane stresses, separately for each mode. A simple version, with an independent boundary for each mode, cancapture the salient aspects of the response of a unidirectional laminate, although a version with limited mode couplingcan fit the test data slightly better. The calibration of model parameters, verification by test data, and analysis ofmultidirectional laminates are postponed for the subsequent companion paper.

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[J007] Cohesive Fracture in Concrete: Theoretical Aspects and Experimental Evidence.

Authors: L. Cedolin, and G. Cusatis.

Reference: Studies and Researches – Politecnico di Milano, ed. by A. Migliacci, P.G. Gambarova, and F. Mola, publ. by Starrylink (Brescia, Italy), 2007, Vol. 27, pp. 167-192.

Abstract: The classical Hillerborg’s “fictitious crack” law, which consists in a relationship between stress and crack opening displacement, is a simple but very effective model for the simula-tion of mode I fracture of concrete. Its experimental calibration through direct methods (di-rect tension tests), however, is quite problematic due to the complexity and stability of this kind of tests. For this reason, in the recent past, indirect methods have been proposed. One of the most appealing is the “size effect” method in which the size effect curve (structural strength versus structural size) is used to identify the tensile strength and the initial fracture energy. These two parameters, in turn, are typically used to characterize the peak and the initial post-peak slope of the softening curve. In the literature two different approaches can be found for the calculation of the size effect curve: 1) an approach based on the polynomial interpolation of the ultimate nominal stresses (normalized failure loads or structural strengths) of geometrically similar specimens of different sizes numerically calculated on the basis of a linear stress-separation relationship, and 2) the classical approach based on nonlinear fracture mechanics, which gives rise to the well-known Bažant’s size effect law. In this paper, the two approaches are first reviewed and the relationship between them is investigated. Then, several sets of experimental results, relevant to three point bending tests recently performed at the Politecnico di Milano, are analyzed with both approaches, in order to assess their accuracy in the identification of the the tensile strength and the initial fracture energy. Finally, the physical interpretation of the results of this analysis is discussed with reference to both an experimental investigation on the local displacement field in the proximity of the fracture process zone and the predictions of a meso-scale numerical model of concrete behavior.

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[J006] Biaxial bending of concrete columns: an analytical solution.

Authors: L. Cedolin, G. Cusatis, S. Eccheli and M. Roveda.

Reference: Studies and Researches-Politecnico di Milano, ed. by A. Migliacci, P.G. Gambarova, and F. Mola, publ. by Starrylink (Brescia, Italy), 2006, Vol. 26, pp. 163-192.

Abstract: This paper deals with the computation of the failure envelope of rectangular reinforced concrete cross sections subjected to biaxial bending and axial load at the ultimate limit state. Due to the non-linearity of the constitutive laws for both steel and concrete it is practically impossible to find the exact analytical solution. Although the problem can be solved numerically and suitable numerical algorithms are indeed available in the literature, an analytical solution would be highly desirable in practice. In this study an approximated solution is proposed in the form of a power representation (Bresler curve) of the failure envelope, whose ingredients are the ultimate bending moments along the axes of symmetry and the power exponent. Accurate numerical simulations show that the power exponent is dependent on the axial load, on the reinforcement ratio and on the dimensionless cover (the ratio between the rebar cover and the length of the side of the cross section). This dependence can be computed by constraining the Bresler curve to match a particular failure point of the interaction diagram, for which the neutral axis is parallel to one of the diagonals of the cross section. The computation of the coordinates of this point is straightforward if one realizes that, in the plane of the dimensionless load and moment terms, it corresponds to uniaxial bending of an equivalent square cross section along a diagonal. Based on this observation, explicit analytical expressions are derived for the interaction diagram in this direction, which, together with the interaction diagrams in the principal directions, provide the data for the calculation of the power exponent of the Bresler curve. Accurate numerical simulations are performed in order to test the accuracy of the approximate analytical formulation.

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[J005] Two-scale Analysis of Concrete Fracturing Behavior.

Authors: G. Cusatis and L. Cedolin. <\p>

Reference: Engineering Fracture Mechanics, 2006, 74, pp. 3-17.

Abstract: The cohesive crack model is, nowadays, widely used for the analysis of propagating cracks in concrete and other quasi-brittle materials. So far the softening curve, whichgives the stress transmitted between two adjacent crack surfaces as a function of thecrack opening, has been considered a material property and the parameters definingthat curve, the tensile strength ft0, the initial fracture energy Gf (area under theinitial tangent of the softening curve) and the total fracture GF (total area underthe softening curve), have been identified by testing relatively small laboratory specimens.In this study typical experimental procedures used for the characterization of the fracture behavior of concrete, namely single notched specimens subjectedto tensile loading, three-point bending tests, bending of unnotched specimens, andsplitting (brazilian) tests are analyzed on the basis of the Confinement-Shear Lattice(CSL) model, recently developed by the authors. A two-scale procedure is outlinedin order to infer an equivalent macroscopic cohesive crack law from the meso-levelresponse. The analysis of the lattice model response is analyzed in order to definethe actual macroscopic material properties of concrete fracture. The results showPreprint submitted to Elsevier Science 13 July 2005that the tensile strength and the initial fracture energy are material properties andthat they can be identified from the analysis of laboratory specimens. On the contrary,the total fracture energy identified from usual experiments is size-dependentand boundary-condition-dependent.

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