M ASONRY AS A HOMOGENEOUS MATERIAL

In Table 3.5, a comparison is made between continuous and discrete modelling strategies for masonry arch bridges. In general, both modelling strategies can co-exist because they have different application fields. Discrete modelling studies are necessary to give a better understanding of the local behaviour of masonry structures. Continuous modelling studies are more adequate for engineering applications.

Table 3.5 – Comparison between continuous and discrete modelling for masonry arch bridges (adapted from (Sarhosis et al., 2016).

Continuous modelling Discrete modelling

Assumptions

- Masonry assumed as homogenous isotropic or anisotropic material

- Stone block, mortar, and interface, are described as an equivalent continuum material;

- Masonry assumed as a composite of its individual components;

- Stone blocks and mortar in the joints are represented by continuum elements whereas the stone-to-stone interface or mortar-to-stone interface is represented by discontinuous elements;

Parameters and requirements

- Reduced time and memory requirements;

- Number of needed parameters to characterise masonry is high;

- Constitutive models for masonry with good accuracy;

- The geometry of the model needs to be represented in detail;

- A large number of material parameters is required;

- Large computational effort required;

Field of application

- Efficient for large scale models;

- Useful for large multi-span bridges for a preliminary assessment of the load-carrying capacity;

- Provide an understanding of the global behaviour of the structure;

- Used for both research and design purposes;

- Suitable for small size models;

- Better accuracy for determining the load-carrying capacity of bridges;

- Good applicability to investigate crack initiation and propagation up to failure;

- Provides a better understanding of local behaviour of the bridges. Including the exact location of maximum tension zones, cracks along the joints or through cross-section of the units;

Limitations

- Localised conditions such as cracks along the interfaces cannot be represented sufficiently enough;

- Some failure modes cannot be captured due to the simplicity of the model;

- Blocks are irregular and geometry not known;

- Too complex with backing and filling;

- Not efficient for large-scale models. Because of the complexity of the model, current computers cannot perform the analysis within economical times;

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works (Bayraktar et al., 2009); (Costa C. et al., 2015); (Costa et al., 2016); (Conde et al., 2017). Another option consists of a numerical homogenization, where the mechanical parameters of each component of the masonry determined experimentally are introduced into a numerical simulation of a masonry representative volume (Domede et al., 2013; Reccia et al., 2014). The parameters of the homogenized masonry constitutive law are then fitted. An application of this methodology to masonry, involving simulation of some material tests as well as a shear wall test, is provided by (Massart et al., 2007). Other approaches are purely analytical, by application of the mathematical theories of homogenization for periodic media (Pietruszczak & Niu, 1992); (Anthoine, 1995); (Cecchi & Sab, 2002). Such homogenization techniques provide continuum average results as a mathematical process that includes information on the microstructure. Many authors have applied this methodology under different mechanical frameworks (elasticity, plasticity, damage, fracture, limit analysis), and a review of the different strategies is provided by Lourenço et al. (2007).

Figure 3.31 – Homogenization process scheme (Lourenço et al., 2010).

Moreover, the interaction between the structural elements of a bridge and its material behaviour is the main aspect to be represented by structural modelling. At the material level, suitable constitutive models of the bridge materials defined and supported by experimental results are a crucial contribution to the validation of the structural numerical models. Typically, the damaged structural condition found in many masonry bridges requires adopting non-linear models for representing the material behaviour. Thus, FEM approaches have been used considering the non-linear stress-strain behaviour of the masonry, both in tension and compression. Typically, continuous damage-based and plasticity-based formulations are adopted.

3.4.1MODELS FOR MASONRY AND INFILL BASED ON PLASTICITY THEORY

The theory of plasticity, which was initially developed for the study of ductile materials, is currently well adapted to homogeneous and isotropic continuous finite elements. In this theory, conditions of yield and maximum strength are considered at the level of one Gauss point, which translates the experimentally observed material behaviour and is expressed as a function of the stress components (or their invariants) and material parameters (Costa, 2009). The yield criterion (yield surface) is usually represented by a stress/strain evolution that represents the uniaxial behaviour in terms of these quantities.

i) Mohr-Coulomb criterion

The quasi-brittle behaviour of materials such as rock, concrete or masonry is characterized by reduced tensile strength and shear yielding, which can be translated by a Mohr-Coulomb (M-C) failure criterion.

The M-C criterion is a set of linear equations in the principal stress space describing the conditions for which an isotropic material will fail, with any effect from the intermediate principal stress σ2 being neglected. It can be written as a function of the maximum σ1 and minimum σ3 principal stresses; or the

normal stress σ and shear stress τ on the failure plane. M-C’s condition can be based on a linear failure envelope to determine the critical combination of σ and τ that will cause failure on some plane.

The advantages of the M-C failure criterion are its mathematical simplicity, the clear physical meaning of the material parameters and a general level of acceptance. A limitation surrounds the numerical implementation of a failure criterion containing corners in the p-q plane (where p is the mean stress or hydrostatic stress and q is Von Mises equivalent deviatoric stress), as opposed to a smooth function, e.g., Drucker-Prager failure criterion. Also, a deformation analysis requires a flow rule, a relationship between strain increments and stress, such that the flow rule determines the orientation of the strain-increment vector for the yield condition, e.g., normal for an associated flow rule (Labuz & Zang, 2012).

ii) Drucker-Prager criterion

The Drucker–Prager (D-P) failure criterion was established as a generalization of the M-C criterion (Drucker & Prager, 1952). Alejano and Bobet (2012) mention that the advantages of the D-P criterion are its simplicity and its softness and, except for some modified criteria, the failure surface is symmetric in the stress-space, which favours its implementation into numerical codes. It can be expressed as:

(1) Where α and k are material constants, I1 is the first invariant of the stress tensor and J2 is the second invariant of the stress deviator tensor. The material parameters α and k can be determined from triaxial tests by plotting the results in the I1 and J2 space. Alternatively, the parameters can be obtained from standard compression triaxial tests and can be expressed in terms of internal friction angle and cohesion values:

(2) (3)

where c and ϕ are the cohesion and internal friction angle of the material, respectively. The Drucker–

Prager failure cone circumscribes the Mohr-Coulomb hexagonal pyramid (Figure 3.32).

Figure 3.32 – Drucker-Prager and Mohr-Coulomb criteria.

The plastic potential function has the same form as the yield surface, and is defined by the following equation:

𝑓(𝜎) = 𝛼𝐼₁+ 3𝐽₂− 𝑘

𝛼 = 2 sin ∅

√3(3 − sin ∅) 𝑘 = 6𝑐 cos ∅

√3(3 − sin ∅)

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(4) where,

(5) where ψ is the dilatancy parameter. This parameter should be less than or equal to the friction angle (ϕ), which makes the flow rule non-associated or associated respectively.

The D-P material model is available in almost all commercial software, including in ANSYS (2017), which offers three different versions, as illustrated in Figure 3.33. In the Ansys models library, it is available the classic version, Figure 3.33-a, with cohesion, friction angle and dilatancy parameters, and perfectly plastic behaviour. Another option available is the extended cap model, Figure 3.33-b. It is similar to the classic version but with the addition of two cap surfaces that truncate the yield surface in tension and compression regions. This model assumes elastic-plastic behaviour, with no softening representation, which is a limitation, particularly in the tension/shear envelope.

Another option is the D-P concrete model, Figure 3.33-c, with a combination of two yield surfaces, which offers a more realistic description of the large differences in the tensile and compressive behaviour of brittle materials like concrete or masonry. There is also, the possibility of combining a D-P surface for the compressive side and a Rankine surface for tension, which has advantages in modelling masonry material, which has a low or in some cases almost null tensile resistance, and the possibility to control tension limits for tension stresses.

a) b) c)

Figure 3.33 – Drucker-Prager model in Ansys: a) classic version; b) extended version; c) concrete version (adapted from Ansys).

iii) Crack models

The description of cracking and failure within finite element analysis of quasi-brittle structures and materials has led to two fundamentally different approaches: the discrete and the smeared one (Jendele et al., 2001). They represent two distinct viewpoints on the problem of modelling damage in quasi-brittle materials. Smeared model, which was first introduced by Rashid (1968) and Cervenka and Gerstle (1971), is based on the development of appropriate continuum material models, in which cracks are smeared over a distinct area, as a distributed effect over the damaged finite element or an area corresponding to an integration point of the finite element. It builds upon equivalent continuum concepts of elastic degradation and/or softening plasticity within the fixed mesh approach. Although a few commercial packages support nowadays a discrete crack model as well (at least in a simplified manner), the smeared model seems to retain its dominant role for engineering applications.

𝑔(𝜎) = 𝛾𝐼₁+ 3𝐽₂

𝛾 = 2 sin 𝜓 (3 − sin 𝜓)

The smeared model was first implemented in a finite element program Sbeta, and now is implemented in many commercial software packages. After crack initiation that is controlled by a bi-axial failure envelope (see Figure 3.34a), the original isotropic material formulation changes to an orthotropic one.

The unloading moduli for each direction are determined from an equivalent uniaxial diagram shown in Figure 3.34b. The smeared crack models are usually formulated in a stress-strain space.

Smeared crack models can appear in different versions and software packages: Gibbons and Fanning (2016) used a smeared crack model based on William-Warnke failure criterion, to assess the non-linear response of masonry arch bridges under service traffic loads. Franck et al. (2019) resorted to a smeared crack model to realistically reproduce the characteristic crack patterns in masonry arch bridges for the load-carrying capacity assessment using ATENA software (Červenka et al., 2007). Other plasticity formulations are also found in the literature based on combined models. The most frequent use the orthotropic Rankine multi-criterion for tension and an isotropic Drucker-Prager criterion for shear and compression failure models (Domede et al., 2013). Also, a Total Strain Crack model is reported, in which the tensile and compressive behaviour of the material is described by a single stress-strain relationship (Conde et al., 2017; Zampieri et al., 2021).

a) b)

Figure 3.34 – Sbeta material model: a) the biaxial failure envelope and b) the equivalent uniaxial stress-strain relationship (Jendele et al., 2001).

3.4.2MODELS FOR MASONRY BASED ON DAMAGE MODELS

One methodology based on continuum damage mechanics was developed by Faria (1994) for quasi-brittle materials like concrete material. This method was originally designed to reproduce large and plain concrete masses, but it is considered that the model has a wider field of application. One example of the application of this method to stone masonry structures can be found in Silva (2008).

In the continuum damage models, it is considered the stress tensor as a basic entity, which for many practical applications, could be considered as coincident with the elastic stress tensor, σ = D0ε, in which ε, denotes the strain tensor and D0, elastic stiffness tensor. Additionally are introduced scalar damage variables, d, being one for traction (d+) and another for compression (d-), that translate the damage evolution in the material. These variables assume values between 0 and 1, corresponding to virgin or collapse situations, respectively.

Merged with the concept of damage, there is also the concept of effective stress which can be divided into the tensile components 𝜎+ and compression 𝜎-, by the following expression:

𝜎 = (1 − 𝑑⁺)𝜎⁺+ (1 − 𝑑⁻)𝜎⁻ (6)

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The equation becomes readily noticeable when considering tensile or compression uniaxial tests where one of the components is zero:

- Tensile uniaxial test:

(7) - Compressive uniaxial test:

(8) The algorithm built for the integration of the concrete constitutive model becomes computationally very efficient, as it is highlighted with the help of Figure 3.35, which illustrates the basic aspects of the constitutive 1D model under tension and compression stresses. By the analysis of the figure, it is possible to conclude that this continuum damage model does not take into consideration the plastic behaviour in tension, and the unloadings are directed towards the origin, by contrast with the behaviour in compression, where non-zero plastic extension ε𝑝 is taken into account. The described continuum damage model was implemented in the structural analysis software Cast3M by Costa (2009).

a) b)

Figure 3.35 – Yield surfaces for the damaged model: a) uniaxial tensile, b) uniaxial compression (Faria, 1994).

Other works where continuum damage models are used for masonry can be found in some references.

For example, in Gambarotta and Lagomarsino (1997) a continuum damage model to represent the response of masonry walls with homogeneous material under seismic actions is presented, and in Zucchini and Lourenço (2004) where models involving homogenization techniques and damage models to represent the behaviour of old masonries are presented. More recently, based on a 3D FE model, Addessi et al. (2021) implemented an isotropic damage model in FEAP software (Taylor, 2017) to investigate the non-linear dynamic response of a masonry bridge under seismic actions. Also in FEAP software, Gatta et al. (2018) developed a damage-plastic model to investigate the mechanical response of masonry elements under static and dynamic actions. In Pantò et al. (2022) an isotropic damage model is implemented in Adaptic software (Izzuddin, 1991) and tested using macro and mesoscale bridge models.