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Abstract:
Numerical simulation results of laboratory tests on reinforced concrete beams subjected to four-point bending for a separate variation of the height and length were presented. Due to the lack of a geometrical similarity, two major failure mechanisms were observed: flexural failure mechanism with plastic yielding of reinforcement and shear failure mechanism with two different modes: brittle diagonal tension and brittle diagonal shear-compression. The shear strength increased with increasing effective height and decreased with increasing shear span-effective height ratio. In simulations, the finite element method was used, based on a coupled elasto-plastic-damage constitutive model for concrete under plane stress conditions. The constitutive model was enhanced by integral-type non-locality in the softening regime to yield mesh-independent results. The bond-slip law was assumed between concrete and reinforcement. Two-dimensional numerical calculations under plane stress conditions satisfactorily reproduced both experimental shear strengths and failure mechanisms with one set of input parameters. In addition, the effect of different material constants on strength and fracture was comprehensively studied. Advantages and shortcomings of the numerical approach were discussed.

Keywords:
size effect, finite element method, elasto-plasticity, damage mechanics, reinforced concrete, non-local theory

Abstract:
The paper presents results of laboratory experiments carried out on longitudinally reinforced concrete beams subjected to four-point bending. Beams of separately varying height and length were analyzed to investigate the size effect on nominal strength and post-critical brittleness. Beams were scaled in the height direction in the first test series and in the length direction in the second series. Due to lack of geometrical similarity, different failure mechanisms were exhibited. Load-deflection diagrams and crack paths were registered during experiments. The digital image correlation technique was applied to visualize strain localization on the concrete surface. The crack opening and crack slip displacements were also measured. The beam response was characterized by two non-dimensional parameters ηa = a/D and ηb = b/D defined as the ratios of shear and bending spans to the beam depth D assumed as the size parameter and the reinforcement position parameter ηc = c′/D. Two major failure mechanisms were observed: flexural failure in the central beam zone combined with plastic yielding of the reinforcement and the diagonal shear crack failure in external shear zones. Two distinct modes of shear failure can be specified depending on the dominance of crack opening or crack closure contact zones. Two different effective stresses associated with failure mechanisms were defined to specify the beam strength's dependence on ηa, ηb, ηc and D. Some analytical formulae specifying the critical shear stress dependence on ηa, ηc and strengths ratio of reinforcement and concrete were presented at the end of paper and compared with experimental data.

Keywords:
Concrete beams, Longitudinal reinforcement, Four-point bending, Failure modes, Height or length variation, Size effect

Abstract:
The paper presents a comprehensive experimental and numerical analysis of slender rectangular reinforced concrete beams with longitudinal BFRP bars without shear reinforcement subjected to 3-point bending. The experiments included 4 different beams which were similar in two directions. The main research objective was to investigate the size effect on the nominal shear strength of beams. The detailed experimental analysis of beam strength, failure mode and cracking evolution was presented and compared with previous test results on beams reinforced by ordinary steel bears. The experiments with BFRP bars were numerically reproduced using the 2D finite element method based on a coupled elastic-plastic-damage formulation. In order to describe strain localization in concrete, a non-local constitutive model was applied with account for a characteristic length of micro-structure developing in the softening regime. The numerical results were in satisfactory agreement with the experimental data. Advantages and disadvantages of BFRP reinforcement in concrete beams were next outlined.

Keywords:
BFRP bars, concrete, elastic-plastic-damage, non-local softening, shear, size effect, strain localization

Abstract:
The effect of fluctuating local tensile strength on a coupled energetic–statistical size effect in plain concrete beams under bending was numerically investigated. First, the influence of varying autocorrelation length of the random field describing a spatial variation of local tensile strength was studied. Next, the influence of the coefficient of variation of local tensile strength was analyzed. The numerical FE investigations were performed for unnotched concrete beams of similar geometry under quasi-static three point bending within elasto-plastic model assumptions, accounting for non-local softening. The FE analyses were carried out for four different beam sizes. In the calculations, the tensile strength took the form of spatially correlated random field described by a Gaussian distribution.

Keywords:
Autocorrelation length, Concrete beam, Elasto-plasticity, Non-local softening, Random field, Size effect, Strain localization, Tensile strength

Abstract:
The paper presents results of FE analysis of mechanical size effects in longitudinally reinforced concrete slender beams without shear reinforcement failing in shear mode. The simulations were performed under plane stress conditions for three beams of different sizes and a fixed shape (height/length ratio). The attention was focused on deterministic and statistical size effects related to the nominal beam shear strength. Concrete was assumed as an isotropic elasto-plastic material exhibiting non-local softening. The bond strength between concrete and reinforcement was assumed to depend on interface slip with both stable and softening responses. Statistical simulations were performed for spatially correlated Gaussian random fields of tensile strength using a stratified sampling reduction method. The FE numerical results were compared with the respective own experimental test results.

Keywords:
Elasto-plasticity, Non-local softening, Random fields, Reinforced concrete beams, Size effect, Strain localization

Abstract:
The numerical FE investigations of a deterministic and statistical size effect in unnotched concrete beams of similar geometry under quasi-static three point bending were performed within elasto-plasticity with non-local softening. The FE analyses were carried out with four different beam sizes. Deterministic calculations were performed with the uniform distribution of a tensile strength. In turn, in statistical calculations, the tensile strength took the form of spatially correlated random fields described by a truncated Gaussian distribution. In order to reduce the number of statistical realizations without losing the calculation correctness, Latin hypercube sampling was applied. The numerical outcomes showed a strong coupled deterministic and stochastic size effect which was compared with the size effect laws by Bazant and by Carpinteri.

Keywords:
Concrete beam, Elasto-plasticity, Latin hypercube sampling, Non-local softening, Stochastic fields, Size effect, Strain localization, Tensile strength

Abstract:
A quasi-static homogeneous drained triaxial compression test on cohesionless sand under constant lateral pressure was simulated using a three-dimensional DEM model. Grain roughness was modelled by means of symmetric clusters composed of rigid spheres imitating irregular particle shapes. The effect of grain roughness on shear strength, dilatancy, kinetic, elastic and dissipated energies was numerically analyzed. Some numerical results were compared with available experimental results.

Keywords:
Triaxial test, Granular material, Discrete element method, Grain roughness, Energy, Dissipation