New Model for the Analysis of Size-Scale Effects on the Ductility of Reinforced Concrete Elements in Bending

The well-known cohesive crack model describes strain localization with a softening stress variation in concrete members subjected to tension. An analogous behavior is also observed in compression, when strain localization takes place in a damaged zone and the stress reaches the compression strength with surface energy dissipation. In the present paper, we propose the new concept of overlapping crack model, which is analogous to the cohesive one and permits us to simulate material interpenetration due to crushing. The two aforementioned elementary models are merged into a more complex algorithm able to describe both cracking and crushing growths during loading processes in reinforced concrete members. A numerical procedure based on elastic coefficients is developed, taking into account the proposed constitutive laws in tension and compression. With this algorithm, it is possible to effectively capture the flexural behavior of reinforced concrete beams by varying the reinforcement percentage and/or the beam depth.     

Analytical Model for Concrete Confined with Fiber Reinforced Polymer Composite

An analytical model to predict the behavior of concrete confined with fiber reinforced plastic (FRP) composites subjected to axial compressive loads was developed. First, a constitutive model for plain concrete was formulated from past experimental results obtained from triaxial compression tests of concrete, in which concrete specimens were maintained under constant confining stresses. This was an orthotropic constitutive model based on the concept of equivalent uniaxial strain. Subsequently, in the analytical model for FRP confined concrete, the proposed constitutive model for concrete materials was incorporated. The FRP was assumed to be a linear elastic material. Force equilibrium and strain compatibility between the concrete and the FRP as well were satisfied. When the proposed model was applied to FRP confined concrete, the model overestimated the axial stress. To rectify this, a subsequent maximum strength criterion was introduced to control the maximum strength in the postpeak region when confining stress was continuously increased. The proposed analytical model with the addition of the subsequent maximum strength criterion is in good agreement with the experimental results.     

Fracture Mechanics Model for Analysis of Plain and Reinforced High-Performance Concrete Beams

In developing a one-dimensional analysis and design procedure for reinforced concrete structures, research is generally based on yield phenomena and the plastic flow of steel in tension and concrete in compression. The ability of concrete to resist tension is considered in the form of tension stiffening or is completely disregarded. This procedure does not account for the influence of structural size in changing the failure mode and the stress distribution across the uncracked or cracked ligament. The key factor affecting this stress distribution is found to be the strain-softening modulus. This paper presents an improved model that is based on the fundamental equilibrium equation for the progressive failure of plain concrete beams. The concrete stress-strain relationship in tension is derived by calculating the peak tensile stress and softening modulus for different depths of beams on the basis of the fracture parameters obtained with the size effect law. Thus, the proposed model uses the peak tensile stress and the softening modulus, which vary depending on the size of the beam. To study the effect of the strength of high-performance concrete (HPC) on the concrete tensile stress-strain relationship, the experimental load-deflection plots of different-sized beams are compared with those obtained by using the proposed analytical model for eight different mixes made with locally available fly ash and slag. The model is also extended for lightly reinforced concrete beams, and the results are compared with those in the literature and are found to be in good agreement.     

3D Finite-Element Analysis of Substandard RC Columns Strengthened by Fiber-Reinforced Polymer Sheets

Numerical analyses are performed to predict the stress–strain behavior of square reinforced concrete columns strengthened by fiber-reinforced polymer (FRP) sheet confinement. The research focuses on the contribution of FRP sheets to the prevention of elastic buckling of longitudinal steel bars under compression, in cases of inadequate stirrup spacing. A new Drucker–Prager-type plasticity model is proposed for confined concrete and is used in constructed finite-element model. Suitable plasticity and elasticity models are used for steel reinforcing bars and fiber-reinforced polymers correspondingly. The finite-element analyses results are compared against published experimental results of columns subjected to axial compression, to validate the proposed finite-element model. Stress concentrations in concrete core and on FRP jacket are investigated considering circular or square sectioned, plain or reinforced concrete columns. Geometry of the section as well as the presence of steel bars and stirrups affect remarkably the variation and magnitude of stress on FRP as percentage of its tensile strength.     

On Mechanical Degradation of Reinforced Concrete Affected by Alkali-Silica Reaction

In this paper a constitutive relation is developed for reinforced concrete subjected to a continuing alkali-silica reaction. The chemo-mechanical interaction is described within the framework of chemo-plasticity. The formulation invokes the assumption that the formation of expansive phases results in degradation of mechanical properties of the material. The approach employs a constitutive model for reinforced concrete, as recently developed by the writers. An extensive numerical analysis is conducted examining the effect of the reaction on mechanical characteristics in axial compression/tension for different reinforcement intensities. Some of the key results are compared with the available experimental data.     

Modeling Residual Displacements of Concrete Bridge Columns under Earthquake Loads Using Fiber Elements

Nonlinear dynamic analysis with fiber-element models is now widely used to assess the seismic response of bridge structures. The ability of such models to accurately simulate response parameters for characterizing the postearthquake condition of bridges, namely residual displacements, is assessed by comparison of analyses of dynamically loaded reinforced concrete bridge columns to experimental data. The models are unable to capture residual displacements, and the cause of the inability to capture residual displacements is investigated through dynamic analysis of fiber-element and single-degree-of-freedom (SDOF) models. A certain type of pinching present in the numerical hysteretic response shape is found to lead to poor residual displacement simulation both in the SDOF models and in fiber-element models. When eliminating this pinching, improvements to residual displacement simulation are found. A modified concrete constitutive model representing damage accumulation from cyclic loading is implemented for the fiber-element analysis that incorporates changes to reloading behavior when moving from high tensile strain back to compression. Analysis using the modified concrete constitutive model leads to improvements in the ability of the fiber-element model to capture residual displacements.     

New Approach to Strain Rate Sensitivity of Concrete in Compression

A constitutive law to model the strain-rate-dependent compression behavior of concrete is proposed. It is derived from a mechanical model, where inertia effects influence damage formation. To verify this model an experimental method based on the split Hopkinson bar (SHB) technique is described, one that considers the special properties of concrete specimens. Compressive stress waves with varying peak values and durations were realized using an extended SHB device. Dynamic stress-strain relations—which differ considerably from the corresponding static relations—are derived for the investigated load histories and are modeled with the proposed dynamic constitutive law. While a number of compression load histories are sustained by the specimens, others result in failure. So a dynamic failure criterion is established, connecting failure occurrence, peak stress, and the time to reach the peak stress.     

Closed-Form Solutions for Flexural Response of Fiber-Reinforced Concrete Beams

A constitutive law for fiber-reinforced concrete materials consisting of an elastic perfectly plastic model for compression and an elastic-constant postpeak response for tension is presented. The material parameters are described by using Young’s modulus and first cracking strain in addition to four nondimensional parameters to define postpeak tensile strength, compressive strength, and ultimate strain levels in tension and compression. The closed-form solutions for moment-curvature response are derived and normalized with respect to their values at the cracking moment. Further simplification of the moment-curvature response to a bilinear model, and the use of the moment-area method results in another set of closed-form solutions to calculate midspan deflection of a beam under three- and four-point bending tests. Model simulations are correlated with a variety of test results available in literature. The simulation of a three- and four-point bending test reveals that the direct use of uniaxial tensile response underpredicts the flexural response.     

Global Constitutive Model for Reinforced Concrete Plates

A new constitutive model for reinforced concrete plates using global variables is discussed in this work. It includes the modeling of concrete cracking (through damage) and the plastic yielding of steel. The yield surface, derived from limit analysis, generalizes the Johansen’s criterion by taking into account both membrane and bending behaviors in reinforced concrete plates. Compared to three-dimensional models, this stress resultant model gives reliable results and can be applied to the study of large shell structures.     

Acoustic Measurements of the Anisotropy of Dynamic Elastic and Poromechanics Moduli under Three Stress/Strain Pathways

A series of triaxial compression experiments have been conducted to investigate the effects of induced stress on the anisotropy developed in dynamic elastic and poroelastic parameters in rocks. The measurements were accomplished by utilizing an array of piezoelectric compressional and shear wave sensors mounted around a cylindrical sample of porous Berea sandstone. Three different types of applied states of stress were investigated using hydrostatic, triaxial, and uniaxial strain experiments. During the hydrostatic experiment, where an isotropic state of stress was applied to an isotropic porous rock, the vertical and horizontal acoustic velocities and dynamic elastic moduli increased as pressure was applied and no evidence of stress induced anisotropy was visible. The poroelastic moduli (Biot’s effective stress parameter, α) decreased during the test but also with no evidence of anisotropy. The triaxial compression test involved an axisymmetric application of stress with an axial stress greater than the two constant equal lateral stresses. During this test a marked anisotropy developed in the acoustic velocities, and in the dynamic elastic and poroelastic moduli. As axial stress increased the magnitude of the anisotropy increased as well. The uniaxial strain test involved axisymmetric application of stresses with increasing axial and lateral stresses but while maintaining a zero lateral strain condition. The uniaxial strain test exhibited a quite different behavior from either the triaxial or hydrostatic tests. As both the axial and lateral stresses were increased, an anisotropy developed early in the loading phase but then was effectively “locked in” with little or no change in the magnitude of the values of the acoustic velocities, or the dynamic elastic and poroelastic parameters as stresses were increased. These experimental results show that the application of triaxial states of stress induced significant anisotropy in the elastic and poroelastic parameters in porous rock, while under the uniaxial strain condition the poromechanics, Biot’s effective stress parameter, exhibited the largest variation among the three test conditions.     

Model for Reinforced Concrete Members under Torsion, Bending, and Shear. I: Theory

A model has been developed that can predict the load-deformation response of a reinforced concrete (RC) member subjected to torsion combined with bending and shear to spalling or ultimate capacity. The model can also be used to create interaction surfaces to predict the failure of a member subjected to different ratios of applied torsion, bending, and shear. The model idealizes the sides of an reinforced concrete member as shear “wall panels.” The applied loads are distributed to the wall panels as uniform normal stresses and uniform shear stresses. The shear stress due to an applied torsional moment and shear force are summed over the thickness of the shear flow zone. Stress-strain relationships are adopted for tension stiffening and softened concrete in compression. The crack alignment rotates to remain normal to the principal tensile stress and the contribution of concrete in shear is neglected. The model has been validated by comparing the predicted and experimental behavior of members loaded under torsion combined with different ratios of bending and shear. The torque-twist behavior, reinforcement stress, and concrete surface strain predicted by the model were in agreement with experimental results.     

Modeling Cracking in Shell-Type Reinforced Concrete Structures

A three-dimensional (3D) hypoelastic material model for modeling material properties of cracked reinforced concrete is proposed. Material properties of multidirectionally cracked reinforced concrete are represented by the material properties of intact concrete and a number of uniaxially cracked concrete with their coupling solids. Cracking effects due to multiple nonorthogonal cracks are traced in each uniaxially cracked concrete. Tension softening and aggregate interlock occurring at the crack interface as well as tension stiffening and compression softening initiated in concrete between cracks due to multiple nonorthogonal cracks are all incorporated explicitly. RC panels under in-plane loading and RC slab under pure torsion have been analyzed. The developed 3D hypoelastic material model has been proved to be efficient and effective in modeling the material behaviors of cracked reinforced concrete in shell-type RC structures. The deformational response, the ultimate strength, and failure mode can be captured reasonably well.     

Stress-State dependence of strain-hardening behavior in 2014 Al/15 vol Pct Al2O3 composite

The purpose of this investigation was to examine if the effective stress-strain function for discontinuously reinforced aluminum (DRA) matrix composites is independent of stress state, as they are for aluminum alloys. The rationale for such work is provided by the need to develop constitutive equations for applications in metal forming and forging problems. Experimental effectiveas curves at room temperature were determined for a particulate-reinforced composite, 2014 Al/15 vol pct A1₂O₃, and the matrix material, 2014 Al, under a variety of stress states. The tests consisted of uniaxial tension, equibiaxial tension (bulge test), and compression tests. To eliminate the effects of prior precipitation, all samples were given a solution-heat-treatment prior to tests. It was found that for the composite the effective yield stress in uniaxial tension was higher than that in equibiaxial tension but slightly lower than that in compression. However, the effective yield stresses for the matrix material in uniaxial tension and equibiaxial tension were nearly the same. The strain-hardening rate of the composite under equibiaxial tension was higher than that under either uniaxial tension or compression. It is suggested that nondeformable dead zones can develop around the particles during deformation whose shape changes with the applied stress state, and this is partly responsible for the observed differences in behavior. Formerly Graduate Student, the University of MichiganSenior Engineer     

Creep Deformation of Fiber Reinforced Plastics-Plated Reinforced Concrete Tensile Members

The problem of long-term creep deformation of reinforced concrete tensile elements strengthened by external fiber reinforced plastic (FRP) plates is studied. Formation of discrete cracks in concrete under tension is taken into account. A kinematic model is used, where relative slips between concrete, steel bars, and FRP plates are considered, governed by viscous interface shear stress–slip laws. Bazant’ solidification theory and exponential algorithm are used to obtain incremental constitutive equations for concrete as well as for steel-concrete and FRP-concrete interface laws. Moreover, cohesive normal stresses across transverse cracks in concrete are considered. The incremental differential system of equations is transformed into a nonlinear algebraic system by a finite difference discretization with respect to axial coordinate. Several numerical examples are presented, concerning both short-term and long-term loadings. It is shown that reinforcing by means of FRP plates or sheets has significant beneficial effects on the behavior of reinforced concrete elements under service loadings because (1) it increases concrete tension stiffening effect and (2) it strongly reduces crack width. The present study shows that these beneficial effects are preserved also in the case of long-term loadings.     

High-Order Analysis of Reinforced Concrete Slabs Strengthened with Circular Composite Laminated Patches of General Layup

A high-order model for the analysis of reinforced concrete (RC) slabs strengthened with externally bonded composite laminated patches of a general layup is presented. The model follows the concepts of the high-order theory and it is based on variational principles, equilibrium, and compatibility requirements. The classical lamination theory is adopted for the composite patch and it yields a set of coordinate dependent constitutive relations. The governing equations form a set of partial differential equations with variable coefficients. The solution procedure adopts the Galerkin and the multiple-shooting methods in the circumferential and radial directions, respectively. The proposed model is used for the numerical study of a square RC slab strengthened with a circular cross-ply laminated patch. The results focus on the overall behavior of the slab and the localized shear and vertical normal stresses near the edge of the bonded patch. Comparison with results obtained using a simplified axisymmetric model is also presented and discussed. The study reveals that the anisotropy of the bonded patch affects the overall and the localized response of the strengthened slab. It also shows that the simplified axisymmetric analysis tends to underestimate the stresses and stress resultants, and thus may be considered unsafe.     

Effect of loading condition and stress state on damage evolution of silicon particles in an Al-Si-Mg-Base cast alloy

Damage evolution of Si particles in a Sr modified cast A356(T6) Al alloy is quantitatively characterized as a function of strain under tension, compression, and torsion. The fraction of damaged Si particles, their size distributions, and orientation distribution of particle cracks are measured by image analysis and stereological techniques. Silicon particle cracking and debonding are the predominant damage modes. Particle debonding is observed only under externally applied tensile loads, whereas particle cracking is observed under all loading conditions. The relative contributions of Si particle debonding and fracture to the total damage strongly depend on stress state and temperature. For all loading conditions and stress states studied, the average size of damaged Si particles is considerably larger than the bulk average size. The rate of damage accumulation is different for different loading conditions. At a given strain level, Si particle damage is lowest under compression and highest under torsion. The anisotropy of the damage is highly dependent on the deformation path and stress state. Under uniaxial tension, the cracks in the broken Si particles are mostly perpendicular to the loading direction, whereas in the compression test specimens they are parallel to the loading direction. The Si particle cracks in the torsion and notch-tension test specimens do not exhibit preferred orientations. The quantitative microstructural data are used to test damage evolution models.     

Simple Single-Surface Failure Criterion for Concrete

This technical note presents a new single-surface failure criterion for concrete that is derived using an approach developed for geologic materials. The main advantage of the proposed failure criterion is its simplicity and ease of implementation into nonlinear finite-element programs compared to existing criteria based on tension and compression meridians and a third function defining the deviatoric trace between meridians. The new failure criterion is able to closely describe concrete strength under general three-dimensional stress states and it gives improved predictions of concrete strength under plane stress conditions compared to existing single function models. A method for deriving the strength parameters is presented so that practitioners may consider adapting the function for materials such as high-strength or steel fiber-reinforced concrete. The work summarized herein is considered novel practical information that should be of use to practitioners wanting to implement simple phenomenological models for concrete in their finite element codes.     

Coupled Environmental-Mechanical Damage Model of RC Structures

The evaluation of strength reduction of RC structures subjected to mechanical damage process and chemical attack is carried out, with regard to concrete deterioration and steel corrosion. A coupled environmental-mechanical damage model, developed as an extension of that previously published is presented. Two independent scalar mechanical damage parameters are introduced, each of them representing the degradation mechanisms occurring under tensile and compressive stress conditions. The stiffness recovery upon loading reversal, which is manifest when passing from tension into compression, is fully captured by the proposed model. The environmental damage is strongly related to the diffusion process, as well as to the evolution of the chemical reaction between pollutant and cementitious constituents. An enhanced local method is proposed to regularize the problem of nonobjectivity of the finite-element solution due to the strong strain softening behavior of concrete material. The splitting test of a concrete specimen and a static analysis of an RC frame subjected to mechanical loads and chemical attacks are carried out, and the damage evolution is analyzed in detail.     

Predicting Shrinkage Stress Field in Concrete Slab on Elastic Subgrade

Concrete is a material that changes volumetrically in response to moisture and temperature variations. Frequently, these volumetric changes are prevented by restraint from the surrounding structure, resulting in the development of tensile stresses. This paper provides a method for computing the stress and displacement fields that develop in response to this restraint by considering the concrete slab as a plate resting on an elastic foundation. The interface between the slab and the foundation is capable of simulating all cases between complete perfect bond and perfect compression∕zero tension bond to permit debonding. In addition, stress relaxation is considered in the concrete to account for the reduction in stress due to creep∕relaxation-related phenomena. For this reason, the stress-strain relationship and equilibrium equations have been considered in the rate or differential form. The history-dependent equilibrium equations are obtained by integrating the differential equations with respect to time. An example is presented to illustrate the favorable correlation that exists between the predicted behavior of the plate model and finite-element modeling.     

Identification and Validation of a Discrete Element Model for Concrete

The use of a three-dimensional discrete element method (DEM) is proposed to study concrete structures submitted to dynamic loading. The aim of this paper is to validate the model first in the quasistatic domain, and second in dynamic compression, at the sample scale. A particular growing technique is used to set a densely packed assembly of arbitrarily sized spherical particles interacting together, representing concrete. An important difference from classical DEMs where only contact interactions are considered, is the use of an interaction range. First, the correct identification of parameters of the DEM model to simulate elastic and nonlinear deformation including damage and rupture is made through quasistatic uniaxial compression and tension tests. The influence of the packing is shown. The model produces a quantitative match of strength and deformation characteristics of concrete in terms of Young’s modulus, Poisson’s coefficient, and compressive and tensile strengths. Then, its validity is extended through dynamic tests. The simulations exhibit complex macroscopic behaviors of concrete, such as strain softening, fractures that arise from extensive microcracking throughout the assembly, and strain rate dependency.