Size effect on failure of overreinforced concrete beams

The results of full-scale failure of singly reinforced four-point-bend beams of different sizes containing deformed longitudinal reinforcing bars are reported. The tests consisted of four groups with one, two and three different size combinations. The specimens were made of concrete with a maximum aggregate size of 10 mm. The beams were geometrically similar in one, two and three-dimensions, and even the bar diameter and cover thicknesses were scaled in proportion. The reinforcement ratio was 3%. The results revealed the existence of a significant size effect, which can approximately be described by the size effect law previously proposed by Bazant. The size effect is found to be stronger in two-dimensional similarities than for one and three-dimensional similarities.     

Influence of loading rate on concrete cone failure

Three different effects control the influence of the loading rate on structural response: creep of bulk material, rate dependency of growing microcracks and structural inertia. The first effect is important only at extremely slow loading rates whereas the second and third effects dominate at higher loading rates. In the present paper, a rate sensitive model, which is based on the energy activation theory of bond rupture, and its implementation into the microplane model for concrete are discussed. It is first demonstrated that the model realistically predicts the influence of the loading rate on the uniaxial compressive behaviour of concrete. The rate sensitive microplane model is then applied in a 3D finite element analysis of the pull-out of headed stud anchors from a concrete block. In the study, the influence of the loading rate on the pull-out capacity and on the size effect is investigated. To investigate the importance of the rate of the growing microcracks and the influence of structural inertia, static and dynamic analyses were carried out. The results show that with an increase of the loading rate the pull-out resistance increases. For moderate loading rates, the rate of the microcrack growth controls the structural response and the results of static and dynamic analysis are similar. For very higher loading rates, however, the structural inertia dominates. The influence of structural inertia increases with the increase of the embedment depth. It is shown that for moderately high-loading rates the size effect becomes stronger when the loading rate increases. However, for very high-loading rate the size effect on the nominal pull-out strength vanishes and the nominal resistance increases with an increase of the embedment depth. This is due to the effect of structural inertia.     

Size effect and inverse analysis in concrete fracture

This paper summarizes the basic experimental and numerical results supporting an easy procedure to determine up to two fracture parameters based on numerically computed size effect curves. Furthermore, it supplies closed-form expressions to determine the initial linear segment approximation of the (stress vs. crack opening) softening curve of cohesive crack models for concrete, based only on the peak loads determined in splitting-tension (Brazilian) tests and in three-point-bending test on notched specimens. Knowledge of the initial segment, although not enough to describe all the fracture process of concrete structures, is enough to predict the fracture behavior of unnotched concrete structures prior and around the peak load. The same is true for notched structures provided their size is less than a limiting size, approximately defined in the paper. This revised version was published online in July 2006 with corrections to the Cover Date.     

Equivalent localization element for crack band approach to mesh‐sensitivity in microplane model

The paper presents in detail a novel method for finite element analysis of materials undergoing strain‐softening damage based on the crack band concept. The method allows applying complex material models, such as the microplane model for concrete or rock, in finite element calculations with variable finite element sizes not smaller than the localized crack band width (corresponding to the material characteristic length). The method uses special localization elements in which a zone of characteristic size, undergoing strain softening, is coupled with layers (called ‘springs’) which undergo elastic unloading and are normal to the principal stress directions. Because of the coupling of strain‐softening zone with elastic layers, the computations of the microplane model need to be iterated in each finite element in each load step, which increases the computer time. Insensitivity of the proposed method to mesh size is demonstrated by numerical examples. Simulation of various experimental results is shown to give good agreement. Copyright © 2004 John Wiley & Sons, Ltd.     

Size effect of concrete samples on the kinetics of external sulfate attack

The external sulfate attack (ESA) of concrete is a disease related to expansive sulfate hydrate formation in a hardened cement matrix. The aim of this research is to study how the choice of a concrete sample size can impact on the kinetics of ESA, by exposing different types of specimen to constant immersion in a solution dosed with 5% Na₂SO₄⋅10H₂O. Monitoring involves mass, dynamic modulus and expansion measurements. It is concluded that 4 × 8 cm concrete cylinders (cored from 11 × 22 cm concrete cylinders) are more quickly damaged by ESA than usual sample types (11 × 22 cm concrete cylinders and 4 × 4 × 16 cm mortar prisms). For all sample types, damage is always limited to the periphery of the sample in the short run. The thickness of the damaged zone is in the region of the size of the largest aggregates. For 4 × 8 cm concrete cylinders, this periphery corresponds to the entire sample because the maximum aggregate size is of the order of the size of the specimen. In this situation, the percolating crack network resulting from swelling is assumed to dramatically damage the cement matrix and to give sulfate solution access to the whole sample. Hence, by using this original type of cored samples, the concrete resistance to sulfate attack can be studied under reliable conditions (concrete formulations and not mortar ones, good sensitivity to ultrasonic tests) and advantage can be taken of the increased kinetics of degradation.     

Size effect in uncracked and pre-cracked reinforced concrete beams shear-strengthened with composite jackets

The paper deals with the size effect on shear behaviour of reinforced concrete beams strengthened with fiber reinforced polymer jackets. Continuous U-jackets were made of glass or carbon fiber fabrics and epoxy composite materials. Twelve uncracked or pre-cracked strengthened reinforced concrete beams and six beams without strengthening, all of them in 6 different sizes, were tested. The results indicate that fabric-epoxy continuous U-jackets have reduced the brittleness of the shear failure of beams, tensile strains in stirrups, and, in a significant way, also the width of shear cracks at the failure state. Although similar strengthening was used for both, uncracked and pre-cracked beams, activation of jackets significantly differed. While jacket strains and their strengthening effectiveness were affected by the sizes of uncracked, retrofitted beams, they remained almost constant in pre-cracked, repaired beams of varying sizes. In contrast to repaired beams, stirrups in retrofitted beams did not yield at failure. Degree of strengthening, defined as the ratio of strengthened-to-unstrengthened beam shear capacities, was studied. It was found out that consideration of the degree of strengthening would provide relations reflecting real behaviour of reinforced concrete beams strengthened with fiber reinforced polymer U-jackets or U-jacketed strips.     

Size effect in shallow and deep notched quasi-brittle structures

The nominal strength of a quasi-brittle structure is known to vary with its size. If the structure undergoes large stable crack growth prior to failure or if it contains a large pre-existing crack, then the failure load is known to approach the asymptotic limit of linear elastic fracture mechanics (LEFM) for large structures from below. In this paper, the size effect is studied on a particular structural geometry containing a crack which can be relatively shallow or deep. The study is conducted within the framework of the fictitious crack model for the fracture of quasi-brittle materials. By allowing for the redistribution of the stresses in the fracture process zone (FPZ), the essential result of the size effect is confirmed. However, it is shown that this result can only be obtained from tests on specimens whose size exceeds a certain minimum value depending on the material, so that at failure the fully developed FPZ is contained wholly within the test specimen. Moreover, the minimum size of the test specimen is shown to increase as the depth of the pre-crack is reduced, thus requiring specimens of very large sizes to obtain valid results from tests on specimens with very shallow pre-cracks. This revised version was published online in July 2006 with corrections to the Cover Date.     

Size effect in the strength of concrete structures

This paper reports on the range of applicability of the various size effect formulae available in the literature. In particular, the failure loads of three point bend (TPB) beams are analysed according to the size effect formulae of Bažant and of Karihaloo for notched beams and according to those of Bažant and of Carpinteri for unnotched beams, and the results of this analysis presented. Improvements to Karihaloo’s size effect formula are also proposed.     

Modeling pull-out resistance of corroded reinforcement in concrete: Coupled three-dimensional finite element model

Aggressive environmental conditions, such as exposure to the sea climate or use of de-icing salts, have considerable influence on durability of reinforced concrete structures due to reinforcement corrosion-induced damage. In the present paper, a recently developed coupled three-dimensional chemo-hygro-thermo-mechanical model for concrete is discussed [1], [2]. The model takes into account the interaction between non-mechanical processes and mechanical properties of concrete (damage). The mechanical part of the model is based on the microplane model. It is validated through a 3D transient finite element analysis of a pull-out of corroded steel reinforcement from a concrete beam-end specimen, which was exposed to aggressive environmental conditions. For the corrosion phase, the influence of the anode and cathode position on the electric potential, current density, corrosion rate and corrosion induced damage is investigated. Moreover, the effect of corrosion on the pull-out capacity of reinforcement and the influence of transport of corrosion products through cracks are studied.     

Size effect on evaporation temperature of nanocrystals

Based on our model for size-dependent cohesive energy, the size-dependent evaporation temperature of nanocrystals has been modeled without any adjustable parameter. The model predicts a decrease of the evaporation temperature of nanocrystals with decreasing size. The model predictions are in good agreement with available experimental results for Ag, Au and PbS nanocrystals.     

Lattice modelling of size effect in concrete strength

This paper uses a recently improved lattice network model to study the size effect in the strength of plain concrete structures. The several improvements made to the lattice network model are: (i) tension softening of the matrix phase is included in the material modelling; (ii) the structural response is modelled by incrementing the deformation rather than the load. This eliminates the need for introducing arbitrary scaling parameters in the beam element failure criteria and; (iii) a square rather than a triangular lattice beam network is found to be adequate for modelling concrete, thus greatly reducing the computational time.The improved square lattice network has been used to simulate the complete load-deformation response of notched three-point bend beams of different sizes with a view to checking the validity of several size effect models available in the literature. Lattice simulation was found to identify microcracking, crack branching, crack tortuosity and bridging, thus allowing the fracture process to be followed until complete failure. The improved lattice model predicted smooth structural response curves in excellent agreement with test results.The simulated nominal strengths also correlated very well with the test results, apart from that for the smallest beams (depth 38.1 mm). However, even in the relatively broad range of sizes (1:8) of the test beams, there was no clear evidence that one size effect model is superior to the other. In fact, rather surprisingly the test data would appear to be equally well described by all the available size effect models. The lattice simulations however indicated a trend which is better predicted by the multifractal scaling model.     

Size and geometry dependent tensile behavior of ultra-high-performance fiber-reinforced concrete

This research investigated direct tensile stress versus strain response of ultra-high-performance fiber-reinforced concrete (UHPFRC) with various sizes and geometries. The UHPFRC in this research contained 1% macro twisted and 1% micro smooth steel fibers by volume. The effects of gauge length, section area, volume and thickness of the specimens on the measured tensile response of the UHPFRC were experimentally discovered. The different sizes and geometries of specimens did not generate significant influence on the post cracking strength of UHPFRC whereas they produced clear effects on the strain capacity, energy absorption capacity and multiple cracking behavior of UHPFRC. The strain capacity, energy absorption capacity and the number of multiple micro cracks within unit length obviously decreased as the gauge length, section area and volume of UHPFRC specimens increased. In contrast, as the thickness of the specimen increased, different tendency was observed.     

Microplane finite element analysis of tube‐squash test of concrete with shear angles up to 70°

Finite element analysis of the response of concrete structures to impact events such as missile penetration, explosive driving of anchors, blast, ground shock or seismic loading, requires knowledge of the stress–strain relations for concrete for finite strain at high pressures. A novel type of material test achieving very large shear angles of concrete at very large pressures, called the tube‐squash test, can be used to calibrate a concrete model taking into account plastic deformation at extreme pressures. A finite element analysis of such a test is performed by using a finite strain generalization of microplane models for concrete and steel. The results obtained are in good agreement with those previously obtained with a simplified method of analysis. Thus, they provide a validation of the microplane model, which is shown to be capable of reproducing the response of concrete not only for small strains at small pressures, which is predominantly brittle, but also for high pressures and large finite strains, which is predominantly frictional plastic. Copyright © 2001 John Wiley & Sons, Ltd.     

Modelling the response of reinforced concrete panels under blast loading

A finite element model is developed for the simulation of the structural response of steel-reinforced concrete panels to blast loading using LS-DYNA. The effect of element size on the dynamic material model of concrete is investigated and strain-rate effects on concrete in tension and compression are accounted for separately in the model. The model is validated by comparing the computed results with experimental data from the literature. In addition, a parametric study is carried out to investigate the effects of charge weight, standoff distance, panel thickness and reinforcement ratio on the blast resistance of reinforced concrete panels.     

Damage distribution and size effect in numerical concrete from lattice analyses

Size effect on structural strength of concrete prisms subjected to three-point bending has been studied using the lattice model, which has been extended and now contains a realistic aggregate structure of concrete. The aggregate structure was obtained from CT-scans of real concrete prisms and overlaying the obtained image with a 3-dimensional hcp-lattice. The numerical analyses show that a size effect on structural strength exists for all studied aggregate densities and aggregate shapes. The size effect can be approximated with a Weibull model, where the main parameter, the Weibull modulus, depends on the concrete composition. The crack size distributions have been calculated and show a similar distribution as hypothesized before for fracture in ceramics. The results from the crack size distribution are helping to provide insight into the nature of the fracture process, which seems to differ from that hitherto assumed in cohesive crack models. After a weakening of the material through a multitude of microcracks, at peak load a single large crack propagates while loading continues in the softening regime. The presumed ‘cloud of microcracks’ advancing ahead of the macro-crack tip has not been found. Instead an alternative macroscopic model strategy, referred to as the 4-stage fracture model, is proposed.     

Influence of local fracture energy distribution on maximum fracture load of three-point-bending notched concrete beams

The maximum fracture load of a notched concrete beam has been related to the local fracture energy at the cohesive crack tip region analytically in this paper, and then the correlation between the size effects on the maximum fracture loads and the RILEM specific fracture energy is established. Two extreme conditions have been established, namely zero crack-tip bridging with zero local fracture energy and maximum crack-tip bridging with the maximum size-independent fracture energy. It is concluded that the local fracture energy at the crack tip region indeed varies with the initial crack length and the size of specimen. The tri-linear model for the local fracture energy distribution is confirmed by using the proposed simple analytical solution.     

Determination of concrete fracture parameters based on two-parameter and size effect models using split-tension cubes

The notched beam specimens have been commonly used in concrete fracture. In this study, the splitting-cube specimens, which have some advantages - compactness and lightness - compared to the beams, were analyzed for the effective crack models: two-parameter model and size effect model. The linear elastic fracture mechanics formulas of the cube specimens namely the stress intensity factor, the crack mouth opening displacement, and the crack opening displacement profile were first determined for different load-distributed widths using the finite element method. Subsequently, four series of experimental studies on cubic, cylindrical, and beam specimens were performed. The statistical investigations indicated that the results of the split-cube tests look viable and very promising.     

Size effect on the fiber strength of composite pressure vessels

In this study, experimental tests and an analytical approach are conducted to verify the size effect on the fiber strength of a composite pressure vessel. As an analytical method, the Weibull weakest link model and the sequential multi-step failure model are considered and mutually compared. In the case of carbon fiber tensile strength, there is no large difference between the analytical methods for the volumetric size effect. To verify the validity of the analytical approach, experimental tests were performed using fiber strand specimens, unidirectional laminate specimens and composite pressure vessels. Good agreement for fiber strength distribution was shown between the test data and predicted results. The volumetric size effect shows the clearly observed tendency towards fiber strength degradation with increasing stressed volume. Because the volumetric size effect depends on material and processing factors, the reduction of fiber strength due to the stressed volume shows different values according to the variation of material and processing conditions.     

Boundary effect on concrete fracture and non-constant fracture energy distribution

This paper extends the local fracture energy concept of Hu and Wittmann [29] and [30], and proposes a bilinear model for boundary or size effect on the fracture properties of cementitious materials. The bilinear function used to approximate the non-constant local fracture energy distribution along a ligament is based on the assumption of the proportionality of the local fracture energy to the fracture process zone (FPZ) height and characterises the FPZ height reduction when approaching a specimen back boundary. The bilinear function consists of a horizontal straight line of the intrinsic fracture energy G_F and a declining straight line that reduces to zero at the back boundary. It is demonstrated that using the bilinear model, the size-independent fracture energy G_F can be estimated from the fracture energy data measured on laboratory-size specimens, and the intersection of these two linear functions, defined as the transition ligament, represents the influence of the back boundary on the fracture properties. It is also demonstrated that the specimen size alone is not sufficient to characterise the size effect in the fracture properties observed on laboratory-size specimens.     

Size effect on buckling strength of eccentrically compressed column with fixed or propagating transverse crack

The strength and size effect of a slender eccentrically compressed column with a transverse pre-existing traction-free edge crack or notch is analyzed. Rice and Levy’s spring model is applied to simulate the effect of a crack or notch. An approximate, though accurate, formula is proposed for the buckling strength of the column of variable size. Depending on the eccentricity, the crack at maximum load can be fully opened, partially opened or closed. The size effects in these three situations are shown to be different. The exponent of the power-law for the large-size asymptotic behavior can be −1/2 or −1/4, depending on the relative eccentricity of the compression load. Whether the maximum load occurs at initiation of fracture growth, or only after a certain stable crack extension, is found to depend not only on the column geometry but also on its size. This means that the definition of positive or negative structural geometry (as a geometry for which the energy release rate at constant load increases or decreases with the crack length) cannot be extended to stability problems or geometrically nonlinear behavior. Comparison is made with a previous simplified solution by Okamura and coworkers. The analytical results show good agreement with the available experimental data.