Concrete is the most widely used man made material in the world. Reinforced with steel, it forms a key enabler behind our rapidly urbanising built environment. Yet despite its ubiquity, the failure behaviour of the material in shear is still not well understood. Many different shear models have been proposed over the years, often validated against sets of physical tests, but none of these has yet been shown to be sufficiently general to account for the behaviour of all possible types and geometries of reinforced concrete structures. A key barrier to a general model is that concrete must crack in tension, and in shear such cracks form rapidly to create brittle failure. Peridynamics (PD) is a non-local theory where the continuum mechanics equilibrium equation is reformulated in an integral form, thereby permitting discontinuities to arise naturally from the formulation. On the one hand, this offers the potential to provide a general concrete model. On the other hand, PD models for concrete structures have not focussed on applications with reinforcement. Moreover, a robust model validation that assesses the strengths and weakness of a given model is missing. The objectives of this paper are twofold: (1) to evaluate the benchmark tests involving shear failure for RC structures; and (2) to review the most recent PD theory and its application for reinforced concrete (RC) structures. We investigate these models in detail and propose benchmark tests that a PD model should be able to simulate accurately.

Whilst the origins of concrete can be traced back several thousand years, the addition of reinforcement to carry tension occurred only in the mid-nineteenth century [

Concrete presents quasi-brittle behaviour and is usually taken as a homogeneous material at the macroscale. Nevertheless, concrete is clearly a heterogeneous material at the microscale, as it is constituted by different materials such as sand, cement and aggregate. As the structure is loaded, there will be stress concentrations around the aggregates, which can vary due to their different dimensions. This can lead to high values of stresses in localised regions of the RC structure, and potentially causing its early failure.

The behaviour in failure of reinforced concrete (RC) structures can be classified in flexural and shear failure. While flexural failure is a well understood phenomenon, shear failure in RC is still an open problem [

Many researches have provided crucial insights for the mechanisms associated with shear failure. The ultimate shear capacity is governed by the combined resistance to shear force offered by (1) the uncracked compressive zone, (2) arch or direct strut action, (3) aggregate interlock, (4) dowel action, and (5) the residual tensile strength in the fracture process zone (FPZ). The contribution of each action to the overall shear resistance is related to parameters such as shear span, beam depth (size effect), and reinforcement ratio. There is no consensus on the relational theory between shear failure and the many influencing parameters. Size effect in shear failure was discussed by Bažant and Kazemi [

Numerical models have also been used in the study of RC structures. The first attempts on modelling the quasi-brittle behaviour have used the cohesive formulation developed by Dugdale [

Vecchio and Collins [

A large number of numerical models for RC structures have been obtained, and they have been validated using benchmarks such as other numerical solutions or experimental results. Model verification and validation are two independent processes required for assessing the suitability of a numerical model but they are commonly assumed to be the same process [

The issues with model verification and validation have not been explicitly addressed, but they have been an underlying problem in numerical models for RC structures. The International Federation for Concrete (

Diversity of theoretical approaches: non-linear, plasticity, fracture, damage mechanics;

Diversity of behaviour models: compression softening, tension softening, aggregate interlock, confinements effects, and many others;

Incompatibility of models and approaches: models are developed for specific assumptions/methodologies, and can not be calibrated to other assumptions easily;

Experience required: analyst experience also affects the results, as shown by [

Too much information: too much data generated in commercial softwares;

Incomplete knowledge: we still do not have a complete understanding on the behaviour in shear of RC structures;

Research philosophy: assumptions used in numerical models are not suitable for concrete (for instance, classic continuum mechanics).

Peridynamics (PD) is a non-local framework first introduced by Silling [

Javili et al. [

The objective of this paper is to review the works on PD for RC structures. We discuss the mechanisms of shear transfer, the different types of failure, some of the main results obtained in the field for shear failure and the importance of benchmark tests in Sect.

Understanding the behaviour in failure of RC is critical to ensure that the structure will safely withstand the designed load capacity. There are many nomenclatures when approaching failure of RC structures, but the failure modes are generally classified as flexural or shear failure.

Flexural failure is characterised by vertical cracks that appear when the RC structure is loaded. The vertical cracks form cantilever-shaped structures, with the longitudinal reinforcement at the end of these cantilevers. This configuration can resist the loading coming from the reinforcement bars, as have been reported by [

Shear failure in concrete occurs when the shear force exceeds the shear capacity of the section, leading to a sliding diagonal failure. Diagonal failure is usually analysed with a 4-point bending test (4PBT) with longitudinal reinforcement but without shear reinforcement, so diagonal cracks can form in the structure. There has also been works to study how shear failure can develop (or be prevented) due to the use of shear reinforcement. The 4PBT combines two different loading conditions, pure bending between the applied load locations, and constant shear force at the end of the section [

Typical 4-point bending test

There are several known mechanisms that influence shear failure, and they are briefly described in the next sections.

There are two main shear transfer actions that appear in RC beams with longitudinal reinforcement. When vertical cracks start to appear, that zone has a reduced capacity to carry loading. The remaining uncracked concrete forms an arch type compression zone, and it leads to strengthening of the remaining structure rather than weakening. This shear transfer is known as arch action. In this case, the overall behaviour of the structure can be represented by the Strut-and-Tie method [

On the other hand, if the shear-span-to-depth ratio is larger (typically values between 3 and 8, beams are also denominated slender beams), a different shear transfer mechanism is dominant. The vertical cracks form tooth-like arrangements as described by Kani [

Fenwick and Paulay [

Kani [

After flexural cracks have appeared in an RC beam, it is normally assumed that the remaining uncracked compressive zone can provide shear resistance [

The mechanism present in the uncracked compressive zone is not the same as the shear arch action, particularly for slender beams. The arch action will have little influence as the compressive zone is intersected by the diagonal crack, which cannot contribute to shear strength [

Crack surfaces are not smooth in concrete structures. The hardened cement matrix represents most of the crack surface, but an aggregate surface is also present. The aggregate surface can interlock with the crack surface, resisting to displacements along the crack plane, hence leading to shear stresses [

Fenwick and Paulay [

To the authors’ best knowledge, the aggregate interlock contribution is not explicitly defined in the design codes.

Mörsch [

Kani [

Olonisakin and Alexander [

The softening behaviour of the concrete is partly due to the residual tensile stresses arising at the crack tip. In this case, a fracture process zone (FPZ) can be defined, and the residual stresses in the FPZ are given by a softening law that depends on the crack opening. A common softening bi-linear law was defined by Hordijk [

Hillerborg et al. [

Kani [

For values of

The region delimited

For regions beyond the transition value

Kani shear valley

The values for

Kotsovos [

Type I: flexural failure only, the beam develop its full flexural capacity;

Type II: a diagonal crack initiates near the tip of the flexural crack closest to the support and propagates toward the load point. The crack can also propagate towards the support along the reinforcement. The flexural capacity reduces with the shear-span-to-depth ratio

Type III: a diagonal crack forms independently of the flexural cracks. In this case, the flexural capacity increases from the critical value in Type II to another value, also dependent upon the area of the reinforcement, until it reaches the full flexural capacity again;

Type IV: failure is caused by a diagonal crack joining the support to the load point, and the beam can develop its full flexural capacity.

Critical shear crack allowing full-arching action to develop;

Failures following a stable propagation of the critical shear crack;

Failure triggered by local loss of aggregate interlock capacity due to the propagation of an internal crack;

Failure triggered by the merging of a secondary flexural crack with a primary flexural crack.

Possible types of shear failure for different

The shear failure can be classified as [

Beam 1: shear-web failure (crack forms in a direct line between applied loading and support)

Beams 2 and 3: shear-compression failure (flexural cracks appear first, then propagates towards the applied loading; crushing also present; arch shear transfer is more dominant)

Beams 4, 5, 6, 7, 8: shear-flexural failure (flexural cracks appear first, then propagates towards the applied loading; uncompressive concrete zone may retard crack propagation for some

Beam 10: flexural failure (full flexural capacity attained)

Kani and his group analysed hundreds of beams, particularly beams with rectangular and T-shaped cross sections, with longitudinal and also web (shear) reinforcement [

Furthermore, 271 beams were investigated to quantify the influence of the web reinforcement and the results presented in [

Significant dowel forces may develop after the horizontal cracking occurs at the longitudinal reinforcement interface, if the web reinforcement are in the correct locations, i.e., where shear failure could appear. Leonhardt and Walther [

The concrete cover is important, if the cover is too large the effectiveness of the reinforcement is reduced;

Web reinforcement leads to many well-distributed cracks, rather than larger ones when the stirrups are widely spaced. Smaller cracks do not penetrate into the compression zone, maintaining the aggregate interlock.

Collins et al. [

Collins and co-workers also concluded that shear stresses are transmitted across the cracks due to aggregate interlock action [

The presented database contained 1098 tests for slender beams, where

Additionally, 1343 shear failures were associated to the beam action mechanism. As the specimen depth and the reinforcement stress increase, the shear strength estimation from the ACI 318-08 design code [

Size effects are also a factor that influences shear failure. The definition of size effects and design for quasi-brittle materials can be found in [

Jin et al. [

Other authors had different conclusions for the effect of statistical size effects. For instance, Syroka-Korol et al. [

Bažant et al. [

Bentz [

Uniaxial tests with HSV. Based on [

The size effect is most likely governed by the amount of concrete volume in tension, rather than a material or geometrical parameter [

For a volume in tension higher than

Traditionally, shear failure has been investigated using 4PBT with longitudinal reinforcement. Researchers have been able to gather the main aspects that dictate shear failure, such as the arch and beam mechanisms [

Benchmark tests are important to understand the behaviour of complex problems in a systematic manner. According to the

Level 1: model calibration with material properties, to ensure that the material properties (such as Young modulus, Poisson ratio, compressive rate in failure

Level 2: validation and calibration with systematically arranged element-level benchmark tests. At this level, we are interested in fundamental tests, where the test data represents a simple physical behaviour of the problem studied. Different experiments may be required in order to capture the behaviour of RC structures.

Level 3: validation and calibration at structural level. In this case, the numerical model is compared to complex behaviour of experiments, and compared with respect to the load-displacement diagram, shear capacity, deformation at peak load and deformation at failure, just to mention some of the relevant parameters in RC structures.

Collins et al. [

The experience of the analyst in RC structures can be important when validating a numerical model. For instance, mesh sensitivity can affect the crack patterns obtained by the end of the analysis. Ingraffea and Saouma [

Červenka et al. [

Sensitivity analysis should be performed to guarantee that the numerical analysis results are reliable. The FE mesh size itself can introduce a fixed length scale that will influence the results of the analysis [

In the next sections we discuss some of the common benchmark tests in concrete structures.

The 3-point bending test (3PBT) has been used to validate concrete models for many researchers [

Most approaches for theoretical modelling of concrete consider softening behaviour [

Softening laws for concrete

One should observe that

The 3PBT with initial notch has also been used to validate numerical models. In this case, the comparison is usually made with the crack propagation path and the load-displacement curves. Due to the size effect related to the fracture energy, different dimensions and loading conditions will provide different values for the fracture energy. In general numerical models do not take the size effect into account. Moreover, numerical models based on linear elastic fracture mechanics (LEFM) do not have the capability of modellng the unnotched 3PBT for instance, as crack initiation is a non-linear problem. A general numerical model for concrete should be able to consider both cracked and pristine structures. Khalipour et al. [

Analysis using commercial softwares should also be able to produce similar results for the same problem, but it is usually not the case, as reported in [

4PBT with longitudinal reinforcement is probably the most used experiment to study shear failure. The 4PBT furnishes a region in the centre with maximum moment, and constant shear force in the areas between the applied loading and the supports. For examples of 4PBT without reinforcement, the reader is referred to [

Swartz et al. [

4PBT used by Swartz et al. [

Swartz and Taha [

Influence of the axial force in the crack propagation paths [

4PBT also furnish information on the possible shear failure modes due to crack propagation, as defined by Kotsovos [

The behaviour of RC structures in shear failure cannot be evaluated by just evaluating a single RC structure. Leonhardt and Walther [

Hofbeck et al. [

Typical Hofbeck test specimen

In [

When a crack is present along the shear plane, there is a reduction of the ultimate shear transfer strength and increase of the slipping, and it is independent from the applied loading. The shear transfer strength depends on the reinforcement ratio and the compressive yield strength

The shear strength of the initially cracked specimens is not directly proportional to the amount of reinforcement. The data suggests that the slip in the shear plane is higher for the pre-existing crack cases.

An issue present in push-off tests is that there is no standardised test, so different authors make small modifications in the test in order to measure the influence of a particular mechanism. This issue contributes to the lack of unified understanding of shear failure in RC structures.

Mattock and Hawkins [

Walraven and Reinhardt [

Echegaray-Oviedo et al. [

In this section we reviewed important aspects of shear failure and different attempts to investigate this problem. The 4PBT is one of the most used experimental set-ups to measure shear capacity in RC structures. However, empirical and numerical models on shear failure are designed for particular cases, for instance, specific loading, geometries and reinforcement ratios [

Additionally, the known mechanisms involved in shear are not represented in the models. For instance aggregate interlock can generate friction at the diagonal cracks, which increases the loading capacity of the structure, but contact models are not normally employed. The bond-slip behaviour at the steel-concrete interface also plays an important role for short beams, where the arch action is the main responsible for the shear capacity. Some authors have modelled this behaviour by refining the discretisation around the reinforcement (for instance, [

Numerical models are based on assumptions coming from continuum and fracture mechanics, but these theories do not represent the behaviour of RC structures during shear failure. Concrete is a particulate material, being composed by sand, aggregates and cement, so a numerical model that does not enforce the same conditions seen in fracture mechanics could provide better results. In this sense, peridynamics (PD) is a framework that is based on continuum mechanics, but does not present discontinuities in the formulation, which permits to model pristine and cracked domains without special assumptions. PD is a particle-based method, so it can model the concrete internal structure easily. For these reasons, PD is a potential numerical method for obtaining a more general approach in modelling RC structures. Several researchers have already employed PD in concrete modelling. We review the PD theory and its application on concrete structures in the next sections.

PD was introduced by Silling [

Partial derivatives pose a difficulty in fracture mechanics, since cracks represent discontinuities in the geometry and also introduce singularities in the stress field at the crack tips in the LEFM theory. For this reason, special assumptions need to be taken in fracture mechanics problems, whether using analytical [

Bond-based is the original PD formulation, described in detail in Silling [

The area of influence (also called as family) of a particle is commonly defined as a circumference (in 2D) or a sphere (for 3D) with radius

Horizon of a particle in bond-based PD

Two particles interact with each other through a shared bond, which also contains the material properties (stiffness). The bond forces are aligned with the bond and have opposite directions. In the bond-based theory, a bond has the same analogy of a spring in classical continuum mechanics.

We assume that the scalar bond force

The local damage at a point is defined as:

As we have seen, there are only two parameters that define the PMB material, the spring constant

Other authors have proposed different constitutive models other than the PMB. For instance, a conical micro-modulus has been used by [

The material constants obtained in Eqs. (

Horizon of a particle in the bulk (

Le and Bobaru [

Other methods include the force density [

Another issue in PD arises on how to define the family of a particle, as the particles around the edge of the horizon

Family of a particle with symmetric circumference horizon

Assuming that partially contained particles belong entirely to a given particle family is a source for errors similar to the surface effects. To overcome this problem, a simple rule defined by [

Seleson [

Seleson and Littlewood [

The reader is referred to [

Table

Comparison of between RC structures models—bond-based PD

Reference | Material model | Formulation modifications | Benchmarks | Model validation |
---|---|---|---|---|

Demmie and Silling [ | Bond stiffness | 2D Impact of a plane in an RC panel | Qualitative analysis | |

Gerstle et al. [ | Bi-linear softening | Micropolar (introduce moments) | 2D plain concrete cantilever | Deformed shape is shown |

Gerstle et al. [ | Bond stiffness | PD implemented in EMU | 3D reinforced lap-splice modelled with ANSYS; 3 Experiments | Lap-splice: experimental crack pattern differ from numerical model (rate of loading very different) |

Oterkus et al. [ | PMB; critical stretch | Modified 4PBT with initial notch; 3D Reinforced panel | 4PBT: comparison of crack paths with experimental results; qualitative analysis of damage in concrete and reinforcement | |

Huang et al. [ | 2D plain cantilever | Good agreement with displacements obtained with ANSYS; crack propagation paths for pristine and damage cantilever presented, but no comparison provided | ||

Zaccariotto et al. [ | Bi-linear softening | 2D Double cantilever; 3PBT and 4PBT with initial notch | Double cantilever: Good agreement with load-displacement curves from experimental and numerical models for the double cantilever and 3PBT; crack propagation paths also as expected ; Qualitative analysis only for a double cantilever with eccentric crack and the 4PBT | |

Gu et al. [ | Improved PMB [ | Contact-impact model | Impact failure in a Brazilian disk (1D analysis) | Damage in PD model is similar to the one observed in experiments |

Yaghoobi and Chorzepa [ | Exponential softening | Micropolar | 2D 3PBT with initial notch | Good agreement with experiments varying the fibre reinforcement ratio and notch location; Crack paths also match experiments |

Miranda et al. [ | PMB with modified bond stretch | 3D clamped plain beam with uniform loading; 3D plain cantilever; 3D cubic compression, 3D 3PBT with longitudinal reinforcement | Deflection of the beams (midspan - clamped; end - cantilever) presented good agreement with Euler-Bernoulli theory; 3PBT with reinforcement crack pattern is similar to experiment; cubic compression failure similar to experiment | |

Aydin et al. [ | Tri-linear softening model (concrete); Perfect plastic material (steel) | Sequentially Linear Analysis (SLA) | Initial calibration with uniaxial tests; 3PBT with and without reinforcement; | Load displacement curve for uniaxial test matches well experimental curve; Reasonable agreement between 3PBT with reinforcement for numerical and experimental, assuming one particular choice of horizon; 3PBT with reinforcement crack patterns was very smeared |

Yang et al. [ | Prototype quasi-brittle material; Tri-linear softening | 2D 3PBT plain concrete | Good agreement with 8 load-displacement curves from experiments; final crack patterns matches experiments | |

Das et al. [ | PMB; critical stretch | 3D 3PBT with initial notch | Qualitative analysis of crack propagation | |

Sau [ | PMB until micro-yielding, then perfect plastic | Micropolar | 2D deep beam 3PBT; 2D 4PBT with (a) longitudinal reinforcement. (b) shear reinforcement, (c) no reinforcement | Comparison with ACI code for the 3PBT; 4PBT with reinforcement presented different crack patterns than experiments for the cases with reinforcement, test without reinforcement presented similar crack paths |

Nia et al. [ | PMB | PD coupled with FE mesh; SLA; critical stretch | Double cantilever; Single edge notch; 3PBT with initial notch and holes; Concrete plate with 3 holes; | Double cantilever used to calibrate the SLA; Single edge notch used to calibrate the model (grid spacing and horizon); 3PBT and concrete plate compared with experimental results and load-displacement curve, with good agreement |

Li and Guo [ | PMB; critical stretch | modelling of the meso-scale | 2D uniaxial plate in tension and compression; 3PBT with initial notch | Good agreement with 3 experimental curves for uniaxial tests; Qualitative analysis for 3PBT with analytical solution for crack propagation path |

Li and Guo [ | PMB with conical function | Dual-horizon; Coupled with FE [ | 2D shear test in a concrete beam with adhesive and FRP layer | 52 tests were available, good agreement with experimental results for ultimate load; good agreement with 3 load-displacement curves; |

Rossi Cabral et al. [ | Bi-linear softening | material length horizon | 1D calibration; 3D uniaxial test | uniaxial test was compared with experimental results |

Zhang et al. [ | PMB; critical stretch | 3 cases of 3PBT with initial notch | Comparison with load-displacement curve and crack propagation paths | |

Hai and Ren [ | PMB; critical stretch | RC slab under explosion | Qualitative analysis with FE solution | |

Zhao et al. [ | Conical micromodulus | IH-PD | concrete structure with single rebar (calibration); concrete structure with multiple rebars | Comparison with displacement field solution obtained with ANSYS; Qualitative analysis with experimental results for corrosion-induced cracking |

Tong et al. [ | Exponential softening; quasi-brittle bond | PD coupled with FE [ | 2D uniaxial rectangular block (calibration PD-FE); 2D 3PBT with initial notch; 2D double-edge notched plate; L-shaped specimen | Comparison with FE solution (calibration); Adjusting |

Gu and Zhang [ | PMB; critical bond energy | modified conjugated bond-based | 2D concrete dam under projectile impact | Comparison with displacement solution from ANSYS; Qualitative analysis of the crack pattern generated by the projectile impact |

Hobbs et al. [ | Bi-linear softening (concrete); Perfect plasticity (steel) | Stuttgart beam series (3D 4PBT with longitudinal reinforcement) | Shear strength comparison with experiments varies from +3% to -57%; Crack propagation paths match the ones observed from the experiments |

The first paper to tackle concrete structures using PD was from Gerstle et al. [

More recently, Sau et al. [

Yaghoobi and Chorzepa [

Miranda et al. [

Huang et al. [

Gu et al. [

In [

Aydin et al. [

Li and Guo [

In [

Yang et al. [

Bi-linear softening and tri-linear bond models

The crack widths

The history-dependent function used to calculate the damage is also modified to consider the softening behaviour and is defined as:

Rossi Cabral et al. [

The fracture energy

Rossi Cabral et al. [

The law descending branch

Zhang et al. [

Hobbs et al. [

Zhao et al. [

Tong et al. [

Gu and Zhang [

Other works on concrete structures include [

The PD formulation assumes that any pair of particles interacts only through a central potential which is independent of all the other particles surrounding it. This oversimplification has led to some restrictions of the material’s properties, such as the aforementioned fixed Poisson ratio of 1/4 for isotropic materials. Also, the pairwise force is responsible for modelling the constitutive behaviour of the material, which is originally dependent on the stress tensor. To overcome this limitation, Silling et al. [

Figure

Particle interaction in state-based PD

One of the particularities of the state-based PD framework is the definition of the vector state. From [

In the state theory, the equation of motion (

To ensure balance of linear momentum,

The deformation vector state field is stated as:

Silling et al. [

Differences between traction in the bonds in ordinary and non-ordinary state-based PD

The process of obtaining the equivalent material properties in the ordinary state-based formulation is not straightforward. Similar to the bond-based theory, there are no direct equivalences of stresses and strains in the ordinary framework. In this sense, a typical approach is to draw an equivalence between the strain energy for the continuum mechanics theory and the strain energy density in the PD framework. The strain energy density

The equivalence between strain energy densities from continuum mechanics and PD framework also poses another inconvenience, as seen in the bond-based theory. Surface effects are also a source for errors in ordinary state-based formulation. Le and Bobaru [

The forces of the bonds in the non-ordinary state-based are not restricted to the direction of the bond. It permits the model to be more general, but it also implies that the balance of linear and angular momentum is not automatically satisfied.

There are different ways to obtain the force-state in the non-ordinary framework, and one of the simplest methods is using correspondence models. In this case, a stress-strain relation from classical continuum mechanics is used as an intermediate step to obtain the traction state at the bonds. The main advantage of the correspondence model is that different material models such as plasticity, cohesive, amongst others, can be used without any modifications. Adopting these material models in bond-based or ordinary state-based require obtaining the equivalent of these classical continuum mechanics parameters into the PD framework. Another particularity of correspondence models is that surface effects correction are not required, since there is no equivalence with strain energy density from continuum mechanics. Silling et al. [

The traction state is defined explicitly as [

Some of the earlier works on the non-ordinary state-based PD were performed by Warren et al. [

Zero-energy modes were first reported by [

Several researchers have investigated the zero-energy modes. Silling [

Yaghoobi and Chorzepa [

Wu and Ren [

Chen et al. [

Bond-associated horizon

The particles within the horizon

A good approximation of the bond-associated strain energy density can be obtained from the total strain energy density

Table

Comparison between RC structures models—state-based PD

Reference | Formulation type | Material model | Benchmarks | Model validation |
---|---|---|---|---|

Gerstle [ | SPLM | Isotropic damage model | Brazilian test; 3PBT with longitudinal reinforcement (then shear reinforcement is added) | Qualitative analysis of the crack patterns; Comparison with predictions for shear/flexure failure from the ACI code [ |

Nikravesh and Gerstle [ | Improved SPLM | Two-spring damage model; hardening and softening plasticity model | 3D uniaxial tension and compression; Brazilian test | Qualitative analysis with previous formulation [ |

Lammi and Zhou [ | OSB | Multi-scale (pores and aggregate); critical bond energy [ | Impact in a 3D cylinder slab | Qualitative analysis under 3 possible fracture energies |

Yang et al. [ | OSB | Bi-linear; exponential softening | L-shaped panel; 3PBT and modified 4PBT with initial notch; Dam collapse | Good agreement with load-displacement curve from experiments; crack paths also match experiments |

Yaghoobi and Chorzepa [ | NOSB | Exponential softening; critical bond energy [ | 2D uniaxial tension (calibration); 2D fibre reinforced plate with semi-circle symmetric notches; 4PBT without notch; 3PBT with initial notch | Qualitative analysis of plate with semi-circle notches; 3PBT and 4PBT load-displacement curve compared with experimental data |

Yaghoobi and Chorzepa [ | NOSB | Mesoscale modelling (pores, aggregate, ITZ); exponential softening | 2D square plate under uniaxial tension | Model calibrated using different configurations for aggregate and pore distributions; Results are compared against one experimental result and another FE analysis; Discussion on sensitivity of the aggregate/pore distribution choices |

Lu et al. [ | NOSB | Drucker-Prager plasticity (plastic yielding); critical bond energy [ | 3D plain cantilever; 3D anchor bolt pullout | Compare displacements with PD with FE solution and analytical solution (cantilever); PD model does not agree with load-displacement curves from experiments (anchor pullout) |

Hattori et al. [ | NOSB | Isotropic material model; Mazars-based damage | 2D RC cantilever; 2D pull-out test | Qualitative analysis for deformed shape and crack propagation |

Wu et al. [ | NOSB | HJC [ | 3D 3PBT with initial notch under impact; 3D rectangular RC block under projectile impact | Good agreement with experimental results for crack propagation |

Wu et al. [ | OSB | HJC [ | 3D L-shaped specimen; 3D 3PBT with initial notch and double symmetric notches | Crack propagation paths agree with experiment and XFEM benchmark (L-shaped); Good agreement with some load-displacement curves from experiment and numerical benchmark (3PBT) |

In [

In [

Lammi and Min [

A model for cohesive crack growth in quasi-brittle materials was developed in [

Yaghoobi and Chorzepa [

Yaghoobi et al. [

The damage

Lu et al. [

To obtain the optimum grid spacing and horizon size, they used the model to obtain the solution of a 3D plain concrete beam subjected to a point load at the end. The deflection was in good agreement with the analytical solution and from a FE analysis. Next, they investigate an anchor bolt pullout problem. The steel reinforcement does not appear to be explicitly defined in the model, nor any particular changes in the model due to the large difference on the stiffness between steel and concrete. The load-displacement curves obtained by the PD model do not agree well to the experimental results. Not modelling the steel reinforcement and ignoring the friction generated by the pullout experiment are clear sources of error for this particular test.

Hattori et al. [

Another reason for the errors is the difference of wave speeds between steel and concrete. If a particle near the interface share bonds with both concrete and steel particles, the wave speed will propagate with very different speeds, leading to a surface effect at the materials interface.

In [

Wu et al. have analysed impact loading problems with ordinary state-based [

Recently, Madenci et al. [

It was stated in [

Gu et al. [

The PD theory [

The weak formulation can still suffer from instabilities, if the first derivative is used to obtain the internal forces, but the use of second-order derivatives prevents zero-energy modes. An alternative is to use the bond-associated deformation gradient from [

To the authors’ best knowledge, the PDDO has not been used for the analysis of concrete structures. Problems with different material properties have not yet been considered in the weak formulation. A generalisation of the weak form for different materials (for instance, anisotropic) would be the first step, then make changes to the formulation to consider more than one material in the same analysis.

The purpose of this paper was twofold: (1) to review the mechanisms responsible for shear failure and benchmark tests used in the RC structures research field, and; (2) to review the state-of-the-art on PD for concrete structures. We have reviewed the different types of framework for PD, and how they have been employed in modelling concrete structures. Most of the PD models only consider plain concrete problems. Moreover, the used benchmark tests usually include an initial notch, in order to validate with the fracture energy from the LEFM theory, but most models do not take advantage in using the crack initiation capabilities of the PD framework.

While failure due to flexural behaviour is well understood in RC structures, shear failure is still an open problem. We have discussed some of the most important aspects of shear failure in RC structures. The arch and beam mechanisms provide a good insight on how shear damage appears. The reduction of shear capacity demonstrated with experiments by Kani [

Different material models have been defined for bond-based and ordinary state-based, including bi-linear and tri-linear softening models. The use of softening models in the material model has improved the results, as observed in [

The majority of the PD models for concrete structures use the bond-based PD formulation. It appears that the choice on whether PD formulation to be applied in a particular problem depends on the user only. Nevertheless, these formulations have their own particularities, so a given PD framework may be more suitable for a specific type of problem. Bond-based is the simplest PD formulation but it has shown to provide good results for RC structures. Surface effects are present and can be reduced but not completely eliminated. On the other hand, the correspondence model can easily employ complex constitutive models but suffer from zero-energy modes. Only two papers tackled RC problems using state-based theory [

The known mechanisms of concrete structures are not enforced in the PD models, such as the friction generated by the aggregates on the crack surfaces and the relative slip between concrete and reinforcement. PD is a particle-based theory, that can incorporate well these aspects in the physical model. This will increase the likelihood to obtain a more general numerical model.

In this paper, four main challenges have been identified on the use of PD for RC concrete structures: (1) modelling the interface of the reinforcement and the concrete; (2) modelling of the real behaviour of RC structures that present shear failure; (3) ensure correct model validation, and (4) the choice of benchmark tests for validation/calibration.

The interface between the steel and the concrete has an important role for arising diagonal cracks. Little information can be found in the literature on modelling two materials in PD, and it is usually empirical. Aggregate interlock and dowel action are important mechanisms in shear failure that closely depend on the interface. We should investigate the issues arising in modelling the interface steel (ductile, continuum material) and concrete (quasi-brittle, particle-type material) in RC structures. Using bond breaking as the only damage parameter is sufficient for problems where mode I crack propagation is dominant, but it is not able to model possible shear stresses at the crack surfaces due to the aggregate interlock.

To model the relevant aspects in shear failure, we have to incorporate the known mechanisms for shear failure into the model. The residual tensile stresses has been included into PD models using softening laws (bi-linear, tri-linear, exponential), but it is not sufficient to capture the shear failure. The aggregate interlock is known to be an important factor that influences the shear failure in RC structures. It can be incorporated using a contact model at the formed crack surfaces, or by allowing a small number of bonds to heal, so there is resistance to sliding in the direction of the crack plane [

We need to define tests used for calibration of the model and different tests for validating the model. For instance, a Brazilian test can be used to verify that the material properties used in the numerical model lead to failure at the expected load, with crack propagation also matching experiments. Then a 4PBT such as the experiment reported in [

The choice of benchmark test is also important in order to capture the different aspects of behaviour in shear failure. 4PBT are popular since they are relatively easy to perform, and varying the shear-span-to-depth ratio for longitudinally reinforced beams permits to quantify how the model capture the shear transfer mechanisms defined in Sect.

Push-off tests are also suitable to analyse shear failure, but a standard test must first be defined by the research community. 3PBT with shear reinforcement can also provide additional information, since it will form an asymmetrical diagonal crack when failing in shear.

Figure

This work was received financial support from the Engineering and Physical Sciences Research Council (EPSRC) grant EP/M020908/1 (Concrete Modelled Using Random Elements - CMURE).

All data supporting this study is provided in this paper.

The authors declare they have no conflict of interest.

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