Size Effect Law Research Articles

Size Effect on Nominal Strength of Circular Stirrup-Confined RC Columns under Axial Compression: Mesoscale Study

AbstractMost previous studies have made great progress in the size effect of concrete materials and several classic size effect laws (SELs) have been proposed. However, there is a lack of size effe...

Measurement of fracture toughness of copper via constant-load microscratch with a spherical indenter

Fracture toughness can be measured by microscratch test from objective force and depth measurement, which is much convenient than indentation approach and conventional means, since initial notch or pre-crack is not required. Micro-scratch test was conducted on copper under a spherical probe of radius 105 μm and constant applied normal load with the aim of characterizing fracture toughness of copper by sliding technique with a sphere. Scratch variables such as normal load, lateral load, friction coefficient, and penetration depth were analyzed, and different methodologies for calculating fracture toughness were compared. Linear elastic fracture mechanics was found to be inapplicable to scratch of elastic-plastic material with sufficient plasticity under a spherical probe, while energetic size effect law can be successfully applied to characterize fracture toughness by microscratch test with a sphere abrading metallic material.

Numerical study on shear failure and the corresponding size effect of interior RC beam-column joints

The shear failure of the core zone of reinforced concrete (RC) beam-column joint has brittle characteristics, so the size effect of shear strength can be caused. Based on the existing experimental research results, the numerical test method is used to analyze the structural size, axial compression ratio, and stirrup ratio on the effect of failure mechanism and failure mode of the RC beam-column joint, and reveal their influence on the size effect of shear strength. The maximum cross-sectional dimension of the joint is 900 mm × 900 mm. The results show that: (1) under the monotonic loading, the beam-column nodes exhibit brittle shear failure in the core region, and the nominal shear strength has a significant size effect; (2) the increase in the axial compression ratio can enhance the shear capacity of the middle joint and also strengthen the size effect of shear strength; (3) an increase in the stirrup ratio can increase the shear capacity of the joint, but it can also weaken the size effect of the shear strength. Lastly, the classical size effect law proposed by Bažant can reflect the size effect of shear failure of the middle node, but it cannot describe the quantitative influence of the axial compression ratio and stirrup ratio.

The effect of reinforcement bridging on the elastic fracture energy of concrete

Failure of structures may be analyzed based on strength criterion or fracture mechanics criterion. Strength criterion deals with material resistance beyond the critical conditions such as yield, and ultimate strength. In fracture mechanics the critical condition is defined by toughness which can be stress intensity factor (K) or fracture energy (G). This study explores the critical point during loading up to the limit of elasticity based on linear elastic fracture mechanics – LEFM. However, nonlinearity frequently appears due to the existence of a relatively large fracture process zone – FPZ located at the crack-tip. The assumption of LEFM in quasi – brittle material such as concrete is therefore limited to large size structures only. The well-known approach to obtain the fracture energy Gf for infinite large structures is size effect law – SEL. Gf is defined as the specific energy, i. e. energy per unit crack plane area. This is elastic energy which is linear. To answer a question whether reinforcement affect the elastic fracture energy, a research was conducted following the principles of work-of-fracture – WOR using the RILEM Specification Test Method. Three-point-bend beam specimens were made of normal concrete, subjected to monotonic load P, until they failed. P–u relationship prevails the real work. The elastic fracture energy based on WOR that is was found by divided the area under P–u curve of elastic range with ligament. The result is, equal to 397.87 N/m for beam A1 (bending failure l/h=5.5), 133.6 N/m for beam B1 (Shear failure, l/h=5.0). 101.93 N/m for beam B2 (shear failure, l/h=5.0), 153.75 N/m for beam B3 (shear failure, l/h=5.0). The greater value comparing to Gf implies that reinforcement affect significantly the elastic fracture energy.

Size effect in shear failure of RC beams with stirrups: Simulation and formulation

The objectives of the present study are to discover the influences of stirrup ratio and shear-span ratio on the shear failure and size effect of reinforced concrete (RC) beams having different sizes, and to build a theoretical formula for quantitatively describing the size effect on shear failure of RC beams. A three-dimensional meso-scale simulation method for modelling the failure of RC beams is established, and it is utilized to explore the failure behavior of geometrically-similar RC beams. In addition, the influences of shear-span ratio and stirrup ratio on the size effect of shear failure was examined. Finally, a novel size effect law (SEL) for predicting the nominal shear strength of RC beams having different sizes is established and verified. The results indicate that, (1) the shear-span ratio has a great influence on the shear capacity of RC cantilever beams, but has ignorable effect on the size effect; (2) the increase of stirrup ratio increases the shear capacity of RC cantilever beams and weakens the size effect of the shear strength; (3) stirrups cannot increase the shear capacity of RC beams indefinitely, and the increase of shear strength is obvious under the condition of small stirrup ratio; (4) the developed SEL for shear failure of RC beams can well describe the quantitative influence of stirrup ratio and shear-span ratio, and it is consistent with the available test date.

Size effect of axial-loaded concrete-filled steel tubular columns with different confinement coefficients

Numerous size effect laws of concrete materials have been proposed by scholars, while the size effect laws for concrete filled steel tubular members are still under study. A three-dimensional meso-scale numerical model that can consider the heterogeneities of concrete and concrete-steel tube interactions was established to study the size effect in concrete filled steel tubular columns. The failure behavior of axial-loaded circular/square concrete filled steel tubular columns having different structural sizes under different lateral constraints (represented by confinement coefficients) were investigated based on the meso-scale method. The influence of lateral confinement on nominal axial compressive strength for concrete columns and corresponding size effect were also explored. As the simulation results indicating, with increasing the specimen size, the nominal axial compressive strength decreases, and size effect is weakened as the confinement coefficient increases. Moreover, combining with the influencing mechanism of steel tube on compressive strength for axial-loaded concrete column, a theoretical formula that can describe the influence of lateral constraint on size effect was established according to the available size effect law for concrete materials. Comparing with the simulation results and reliable test data, the accuracy and rationality of the theoretical formula is well verified.

Effect of grading type of aggregate on fracture parameters of concrete

Experimental studies have revealed that fracture parameters of concrete are mainly influenced by four material parameters: compressive strength, maximum aggregate size, water/cement ratio and aggregate type (river or crushed). It has been noted that the fracture parameters of concrete can also be affected by other material parameters such as the type of cement, aggregate/sand ratio, porosity and curing conditions. Aggregates can be dense gap-graded or continuously graded in concrete technology. However, the effect of aggregate gradation (continuously or gap-graded) on the fracture parameters of concrete has not been examined up until now. In this study, the notched and un-notched cubes produced by both gap and continuously graded batches were tested under splitting loading. The results are discussed based on the three most popular fracture approaches, namely the size effect law, the multi-fractal size effect law and the two-parameter fracture model. Although concretes made with an aggregate of either type of grading have similar mechanical properties such as compressive and tensile strength according to the literature, this study emphasises that the mixes with gap-graded aggregates indicate a more ductile behaviour than those with continuously graded aggregates for the same maximum aggregate size.

Relationship between residual flexural strength and compression strength in steel-fiber reinforced concrete within the new Eurocode 2 regulatory framework

Annex L of the next Eurocode 2 establishes a classification of the steel-fiber reinforced concrete (SFRC) by means of classes of residual flexural strength for two values of crack mouth opening displacement, 0.5 mm and 2.5 mm. Nevertheless, this indexing does not completely describe the resistance behavior of the SFRC since it does not include the residual compression strength of the material, defined as the stress resisted when the deformation is three times that reached under the maximum load [Hormigón y Acero 69:S1 (2018) 75–80], the value of which is not negligible when steel fiber is included in the concrete. The principal objective of this research consists of determining the relationship that exists between compressive and flexural residual strengths by applying the Response Surface Methodology (RSM) as an instrument of analysis. It has been performed on a combination of experimental data collected in two databases, one with compression tests and the other with flexural tests, with 484 and 197 pieces of data respectively. The classification of SFRC could be completed when the value of residual compression strength corresponding to each class of concrete given by the RSM analysis is included, thereby completely characterizing the material for the practical effects of analysis and structural design. In addition, the RSM analysis provides numerous insights on the relations of the residual strengths with the parameters of the base concrete and the fiber. The paper also provides a new cylinder/cube conversion formula based in the results of Viso, Carmona and Ruiz [Cement and Concrete Research 38 (2008) 386–395] and in the Bažant’s size-effect law, which can be used for cubes of any edge length and of any strength grade.

On shear in members without stirrups and the application of energy‐based methods in light of 30 years of test observations

AbstractIn a recent paper in Structural Concrete, the authors Dönmez and Bažant explain that the theoretical background of the Model Code 2010 equations for one way and punching shear are not sufficiently grounded in theory and should instead use an energy‐based size effect law in their formulation to match behavior. To support this claim, finite element simulations were presented. In this paper the basic assumption that an energy‐based method must govern the shear failure of beams without stirrups is questioned. These questions are shown to be based on the hardening behavior of aggregate interlock tests and the inability for slip strains to localize during shear failure. In addition to these theoretical arguments finite element analyses were conducted with a constitutive model that is energy‐based but that also does an appropriate job at modeling aggregate interlock, an aspect that appears to be lacking in the analyses of Dönmez and Bažant. These new results are shown to better model the test results and confirm that aggregate interlock is important in explaining shear strength and therefore the size effect in shear for slender members. As such any concerns about the safety of the Model Code shear equations appear unwarranted.

Mode I and II Interlaminar Fracture in Laminated Composites: A Size Effect Study

This work investigates the mode I and II interlaminar fracturing behavior of laminated composites and the related size effects. Fracture tests on geometrically scaled double cantilever beam (DCB) and end notch flexure (ENF) specimens were conducted. The results show a significant difference between the mode I and mode II fracturing behaviors. The strength of the DCB specimens scales according to the linear elastic fracture mechanics (LEFM), whereas ENF specimens show a different behavior. For ENF tests, small specimens exhibit a pronounced pseudoductility. In contrast, larger specimens behave in a more brittle way, with the size effect on nominal strength closer to that predicted by LEFM. This transition from quasi-ductile to brittle behavior is associated with the size of the fracture process zone (FPZ), which is not negligible compared with the specimen size. For the size range investigated in this study, the nonlinear effects of the FPZ can lead to an underestimation of the fracture energy by as much as 55%. Both the mode I and II test data can be captured very accurately by the Bažant’s type II size effect law (SEL).

Dynamic size effect of concrete under tension: A numerical study

The macro nonlinearity and size effect of concrete are essentially attributed to the internal heterogeneity of concrete. Considering the heterogeneities and the strain rate effect of meso-components, the concrete was modelled as a three-phase composite composed of aggregate particles, mortar matrix and the interfacial transition zones (ITZs). Taking double-edge notched concrete specimens as examples, a meso-scale numerical model for the simulation of the failure and size effect of concrete under dynamic tensile loading with different strain rates from 10−5/s to 100/s was established. The effect of structural sizes on the dynamic tensile strength of concrete was explored. The numerical results indicate that dynamic size effect on tensile strength of concrete has an obvious discrepancy with the static one. There exists a critical strain rate (ε˙cr = 1/s), and at the strain rate the dynamic tensile strength turns to be independent on the structural size of the specimen. For the case less than the critical strain rate, the tensile strength decreases with increasing the specimen size, and the size effect on the dynamic tensile strength is weakened or suppressed as the strain rate increases. For the strain rate larger than the critical one, there is no obvious difference in the dynamic tensile strength of the concrete specimen having different structural sizes, and the corresponding size effect completely disappears. According to the influencing mechanism of strain rate effect and size effect, a Static and Dynamic unified Size Effect Law (i.e. SD-SEL) for tensile strength of concrete was built. The proposed SD-SEL was also verified by the mesoscopic numerical simulation results.

Fracture Parameters of Cement Mortar with Different Structural Dimensions Under the Direct Tension Test.

In this paper, we measured the fracture properties of cement mortar—which is composed of sand and has a nearly constant diameter—using a direct tension test. Four double-notched mortar bar specimens with different structural dimensions were assessed. The failure load, load-crack mouth opening displacement, and elongation of the gauge length were measured under direct displacement control. The fractured surfaces were scanned and measured so that we could calculate the tensile strength accurately and determine the fracture energy and characteristic length. The average ratio of total fracture energy (GF) to specific fracture energy (Gf) was 1.94; this was lower than the typical value for concrete, of 2.5. The direct tension test showed that the double-notched mortar specimens had a smaller fracture processing zone after the initiation of tensile cracks, so the tail portion of the softening branch was small. This decreased the GF/Gf ratio. We verified this result based on a nonlinear fracture mechanics simulation and found that it agreed well with our experimental results. We also investigated the size effects of four different scaled specimens while holding the ratio of structural dimension, d, and notch length, a is constant, so that there was no shape effect. The traditional linear elastic fracture mechanics (LEFM) prediction and Bažant’s size effect law yield a gradient closer to 1/2 in the case of relatively large specimens. In the case of our cement mortar specimens, this prediction was not supported, where the value of the slope was 1/0.727. This was unexpected because LEFM predicts strong size effects. One possible explanation for this result is that the size effects of concrete are most often evaluated using a bending test; also, concrete has a larger maximum aggregate size than mortar. Due to the random heterogeneities in aggregate distribution, higher tail energies may be seen for concrete, leading to differences in the GF/Gf ratio. At the same time, the peak tensile stress could be affected by the relationship between structural dimensions and aggregate size.

Size effect study on compressive strength of SCLC

In the present study, effect of size and placement of cubic specimens on compressive strength of self-compacting lightweight concrete (SCLC) were considered. To do so, 81 specimens of different sizes (50 mm, 75 mm, 100 mm, and 150 mm) were prepared by using three different mixes of SCLC. Results of the cured specimens were then used in regression analyses to find predictive equations with regard to both the placement direction and the size. Test results showed that the strength ratio in cases in which the direction of loading and placement were parallel, were higher than those specimens, whose configurations were normal between loading and placement. In addition, strength ratios in SCLC mixes were slightly higher than those are for self-compacting normal weight concrete. In order to analyze the effect of size on compressive strength the conventional size effect law as well as the modified size effect law (MSEL) were used. Besides, the convergence criterion of nonlinear regression process of size effect study has been discussed. Analyses of the results showed that the unconstraint nonlinear regression in size effect study of SCLC mixes could lead to erroneous results.

Granite strength and toughness from small notched three-point-bend specimens of geometry dissimilarity

Maximum fracture loads (Pmax) of small notched granite specimens under three-point-bending (3-p-b) conditions can be easily measured with any notch/size ratio. In this study, we report a simple closed-form solution of a non-Linear Elastic Fracture Mechanics (non-LEFM) model emphasizing the influence of average grain size G on quasi-brittle fracture of granite. This simple analytical solution containing the grain size G can be conveniently used to determine granite tensile strength ft and fracture toughness KIC from Pmax measurements of small notched 3-p-b specimens of geometry dissimilarity. The span/width (S/W) ratios of small 3-p-b specimens can vary, e.g. 2.5 or 4. The notch/width (a0/W) can also vary, e.g. the notch a0 can be as short as the average grain size (G), or close to width W. However, specimens with α-ratio (=a0/W) around 0.2 are recommended to minimize the boundary influence/effect from both the front and back specimen boundaries as proven by the Boundary Effect Model (BEM). Blue granite with the average grain size around 2 (mm) was selected to test the new method. Total 64 granite samples from four different groups (different 3-p-b sample designs) were tested, with W = 27, 40 and 70 (mm), S/W = 2.5, 4.0 and a0 = 4, 6 and 8 (mm). The tensile strength ft and fracture toughness KIC estimated from every group is fairly close to the values determined from the entire population of 64 tests. Therefore, tests from any specimen group of given geometry and size are sufficient. Estimations for G = 1.5 and 2.5 (mm) were also provided and compared with those for G = 2 (mm) to show the grain size influence. Advantages and disadvantages of BEM and well-known SEL (size effect law) are also discussed using the granite results.

Critical comparison of the boundary effect model with cohesive crack model and size effect law

For several decades it has been clear that the size effect on structural strength, exhibiting a major non-statistical component, is a quintessential property of all quasibrittle materials. However, progress in design codes and practice for these materials has been retarded by protracted controversies about the proper mathematical form and justification of the size effect law (SEL). A fresh exception is the American Concrete Institute which, in 2019, becomes the first concrete code-making society to adopt the SEL based on quasibrittle fracture mechanics. This article begins by discussing several long-running controversies that have recently abated, and then focuses critically on the so-called Boundary Effect Model (BEM), promoted for concrete relentlessly for two decades, in ever-changing versions, by Xiaozhi Hu et al. The BEM is here compared to the quasibrittle SEL based on asymptotic matching. Its errors, weaknesses and inconsistencies are identified—including incorrect large- and small-size asymptotic size effects, conflicts with broad-range comprehensive test data and with the cohesive crack model, incorrect aggregate-size dependence of strength, illogical dependence on ligament stress profile, inability to capture the statistical part of size effect at large sizes, simplistic effect of boundary proximity, and lack of distinction between Type 1 and 2 size effects. In contrast to the SEL, the BEM is not applicable to mixed and shear fracture modes and to complex geometries of engineering structures, and is not transplantable from concrete to other quasibrittle materials.The purpose of this critique is to help crystallize a consensus about the proper size effect formulation, not only for concrete structures but also, and mainly, for other quasibrittle materials and structures, including airframes made of fiber composites, ceramic components and micrometer-scale devices, and for failure assessments of sea ice, rock, stiff soils, bone, and various bio- or bio-mimetic materials, for all of which the non-statistical size effect is yet to be widely accepted in practice.

Critique of critical shear crack theory for fib Model Code articles on shear strength and size effect of reinforced concrete beams

The size effect of Muttoni et al.'s critical shear crack theory (CSCT) is shown to be quite close (with differences up to 15%) and asymptotically identical to the energetic size effect law (SEL), which has been extensively verified experimentally and theoretically (and is adopted for the 2019 ACI Code, Standard 318, for both beam shear and punching). However, the CSCT derivation and calculation procedure obfuscates the mechanics of failure. It is shown to rest on six scientifically untenable hypotheses, which would have to be taught to students as an article of faith. They make CSCT untrustworthy outside the testing range; ditto for beams with T, I and box cross section, or for continuous beams. The present conclusions are supported by experimentally calibrated finite element simulations of crack path and width, of stress distributions and localizations during failure, and of strain energy release. The simulations also show the CSCT to be incompatible with the “strut‐and‐tie” model, which is (for 2019 ACI Code) modernized to include the size effect in the compression strut. Finally, further deficiencies are pointed out for the Modified Compression Field Theory (MCFT), currently embedded in the Model Code.

Determination of Fracture Properties of Concrete Using Size and Boundary Effect Models

Tensile strength and fracture toughness are two essential material parameters for the study of concrete fracture. The experimental procedures to measure these two fracture parameters might be complicated due to their dependence on the specimen size or test method. Alternatively, based on the fracture test results only, size and boundary effect models can determine both parameters simultaneously. In this study, different versions of boundary effect models developed by Hu et al. were summarized, and a modified Hu-Guan’s boundary effect model with a more appropriate equivalent crack length definition is proposed. The proposed model can correctly combine the contributions of material strength and linear elastic fracture mechanics on the failure of concrete material with any maximum aggregate size. Another size and boundary model developed based on the local energy concept is also introduced, and its capability to predict the fracture parameters from the fracture test results of wedge-splitting and compact tension specimens is first validated. In addition, the classical Bažant’s Type 2 size effect law is transformed to its boundary effect shape with the same equivalent crack length as Koval-Gao’s size and boundary effect model. This improvement could extend the applicability of the model to infer the material parameters from the test results of different types of specimens, including the geometrically similar specimens with constant crack-length-to-height ratios and specimens with different initial crack-length-to-height ratios. The test results of different types of specimens are adopted to verify the applicability of different size and boundary effect models for the determination of fracture toughness and tensile strength of concrete material. The quality of the extrapolated fracture parameters of the different models are compared and discussed in detail, and the corresponding recommendations for predicting the fracture parameters for dam concrete are proposed.

Mesoscopic numerical simulation of dynamic size effect on the splitting-tensile strength of concrete

Under static loadings, concrete has been demonstrated to exhibit obvious size effect due to the heterogeneity of mesoscopic composition. Under dynamic loadings, the strain rate effect has a significant influence on the failure of concrete. It is therefore of great importance to explore the size effect of concrete under dynamic loadings. In this study, the focus is on the size effect in dynamic splitting-tensile strength of concrete. A mesoscopic numerical model for the simulation of the splitting-tensile failure and size effect at different strain rates (10−5/s ∼ 200/s) was established. The mesoscopic simulation results indicate that the size effect on the dynamic splitting-tensile strength of concrete has an obvious discrepancy with the static one. There is a critical strain rate in dynamic splitting-tensile strength of concrete. As the applied strain rate below the critical strain rate the size effect behavior is restrained and weakened gradually with the addition of strain rate. As the applied strain rate exceeds the critical strain rate the dynamic strength enhances linearly as the structural size increases. Furthermore, considering the contribution of strain rate on the dynamic size effect, a Static and Dynamic unified Size Effect Law (i.e. SD-SEL) for splitting-tensile strength of concrete was developed. The proposed SD-SEL was verified by the numerical results and the available test data.

Probabilistic size effect law for mode II fracture from critical lengths in snow slab avalanche weak layers

ABSTRACTFrom field observations, dry snow slab avalanche initiation is associated with fracture within relatively thin weak layers under stronger, cohesive slabs. For risk-based avalanche prediction, it is important to understand the fracture properties of alpine snow. Alpine snow is a quasi-brittle material with a fracture mechanical size effect on nominal shear strength meaning that strength decreases with increasing specimen size. A related size effect is the critical length required for rapid propagation of a shear fracture. In that case, the probability of fracture increases with increasing crack length. In this paper, 45 sets of field-measured critical lengths are presented based on 591 individual tests. From analysis, a probabilistic size effect law based on critical lengths is derived analogous to the deterministic size effect law for nominal shear strength related to fracture mechanics. The new size effect law may be useful in applications, particularly since the critical length is easily measured in the field and it is a principal component of weak layer fracture toughness.

EFFECT OF STRUCTURAL SIZE ON SEISMIC PERFORMANCE OF REINFORCED CONCRETE SHORT COLUMNS—A MESO-SCALE STUDY

Under the action of earthquake, reinforced concrete (RC) short columns often undergo brittle compression-shear failure due to the combined effects of vertical and horizontal loadings. The brittle failure may exhibit obvious size effect of the columns. Considering the meso-scale structural characteristics of concrete, the heterogeneity of concrete and the nonlinear interaction between steel bars and surrounding concrete, a three-dimensional meso-scale numerical model is established to study the mechanical behavior and seismic performance of reinforced concrete short columns with different structural sizes. The meso-scale simulation results are in good agreement with the available experimental results. Moreover, the effect of the axial compression on the seismic performance of RC short columns with different sizes, especially large-sized ones is investigated, including the shear bearing capacity, hysteretic behavior, ductility and energy dissipation capacities. Finally, based on the classical Size Effect Law (SEL) proposed by Bažant, a modified formula is proposed to reflect the size effect on the shear capacity of reinforced concrete short columns.