Numerical Solution for Masonry Wall Using General Static Step with ABAQUS/Standard

There are several methods and scales for representing the mechanical behavior of masonry have been proposed due to the complexity of this kind of structure. Each modeling strategy has its own set he assumptions that, in general, can be applied to a specific kind of problem. The current modeling approaches for unreinforced masonry structures are divided into the following categories [12]:

a. Models based on blocks:

It represents the masonry behavior at the scale of the material's primary heterogeneity and is classified by blocks joined together by mortar joints, which consist of the failure response and the material's mechanical. Actually, the texture of the masonry can be taken into account, which has a major influence on the failure pattern and material's anisotropy [12]. Block-based models are divided into several kinds depending on how the action between blocks is represented: Extended finite element approaches [8], Contact-based approaches [13-15], Block-based limit analysis approaches [16], and Textured continuum-based approaches [17] and Interface element-based approaches [18]. Figure 1(a) shows an example of block-based models.

b. Macroelement models:

This method is employed to measure the global non-linear earthquake response of existing unreinforced structures (URM) simply. There is a general assumption that macro element models prevent the excitation of local failure modes, primarily connected with masonry wall behavior under out-of-plane [19]. It is favoured in professional practice because only a few mechanical parameters are needed, and structural analysis can be carried out quickly and at a low-cost thanks to a low computational burden. The structure was idealized as an assemblage of piers (vertical) and spandrel (horizontal) elements. Typically, the spandrels and piers are schematically represented as two-node elements and are linked to each other via infinitely rigid beams or rigid offsets to illustrate the coupling among elements that are deformable [19]. This method includes two approaches: The equivalent beam-based approach [19], and the Spring-based approach [20]. Figure 1(b) shows an example of macro element models.

c. Models based on geometry:

In this method, the structure was modeled as a rigid body. The geometry of the structure is essentially the major input, other than the loading condition. The model of Heyman's rigid no-tension and limit analysis was usually utilized [12, 21], and divided into two approaches: Static Theorem‑Based Approaches (which are suited to the study of equilibrium states in masonry vaulted structures) [22, 23], and Kinematic Theorem‑Based Approaches (which employed in recent years for quick and effective evaluation of existing masonry structures, but they are unable to offer information on masonry structures’ displacement capacity [24]. Figure 1(c) shows an example of geometry-based models.

d. Continuum models:

For analyses that are large and practice-oriented, where a balance of efficiency and accuracy was required, this method was a viable approach. Because the material is treated as an orthotropic homogenous continuum, it offers significant computational benefits in terms of decreased time and memory needs, in addition to user-friendly mesh generation [25]. In terms of global structure behavior, the units and mortar interaction are usually negligible. Material is classified as an anisotropic composite, and average masonry strains and stresses relation is established. The orthotropic material must next be reproduced using a comprehensive macro model with different strengths of the tensile and compressive along the axes of material [26]. The application of suitable and useful homogenous constitutive laws for masonry can be accomplished in one of two ways:

• Direct approaches: To give the total masonry’s mechanical response approximately, these processes use calibrated mechanical properties like elastic parameters or any other properties, from experimental trials or any other data like domains of analytical strength determined experimentally [26-28].

• Homogenization procedures: Using a homogenized procedure, a material-scale model is connected to a structural-scale model (which represents the major heterogeneities) in order to obtain the constitutive law of the material [29-31].

Figure 1(d) shows an example of a continuum model.

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Figure 1. Examples of modeling strategies

In this part, the specifics of the URM walls' Finite Element modeling in the ABAQUS program are discussed.

• The geometry of the model

The simulation and analysis of the unreinforced brick wall in this paper were conducted using a continuum strategy. This was accomplished by simulating the wall's geometry using the experimental specimen's global dimensions. Regarding the Abaqus model, only one part of the geometry (brick masonry wall) is defined in the part module by using SI (mm) dimensions.

• Constitutive material models

The main mechanical characteristics of a brick wall are its elastic modulus (E_m), Poisson’s ratio ($v$), density, compressive and tensile strengths, and stress-strain curves. These characteristics are necessary for modeling. The wall parameters from experimental tests have been adjusted according to studies [34, 35] to reproduce the elastic and plastic properties of the continuum brick wall. For the elastic properties, the modulus of elasticity (E_m) can be derived from Eq. (1) [34]. Table 1 shows the elastic properties of the wall.

$E_m=550 f_m^{\prime} M P a$         (1)

where,

$f_m^{\prime}$: Ultimate brick masonry prism compressive strength in MPa.

Following a basic classification of all material implementations in Abaqus, Concrete Damaged Plasticity (CDP) the most appropriate model for the material was chosen to describe the nonlinear compression and tension behaviour in the plastic range [36]. CDP models were proposed by Lubliner et al. and Lee and Fenves are the theory references for this implementation [37]. This model gives a general ability for concrete analysis and is appropriate for brick masonry or any other quasi-brittle materials in all types of constructions [37]. The uniaxial compressive and tension responses are shown in Figure 3. Other parameters are required for CDP for quasi-brittle materials in ABAQUS. These parameters are tabulated in the Table 2.

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Figure 3. Typical uniaxial loading response in the (a) tension; and (b) compression [37, 38]

Masonry's compressive strength ($f_m^{\prime}$) is determined by bricks and mortar strengths. The nonlinear behavior between the two strengths was discovered empirically from Eq. (2) [34], with the effects of the individual strength on the total masonry strength.

$f_m^{\prime}=0.63 f_b^{0.49} f_j^{0.32}$        (2)

The parabolic brick masonry stress-strain curve, adopted in this study, was shown in Figure 4 [34] and can be described in expressions of stress and strain ratios in a non-dimensional form as:

$\frac{f_m}{f_m^{\prime}}=2 \frac{\varepsilon_m}{\varepsilon_m^{\prime}}-\left(\frac{\varepsilon_m}{\varepsilon_m^{\prime}}\right)^2$                   (3)

where,

$\varepsilon_m^{\prime}=C_j \frac{f_m^{\prime}}{E_m^{0.7}}$

$f_m^{\prime}$: Ultimate masonry prism compressive strength in MPa.

f_b: Ultimate brick unit compressive strength in MPa.

f_j: Ultimate mortar compressive strength in MPa.

f_m: Compressive stress in masonry (MPa).

ε_m: Compressive strain in masonry.

C_j: Factor based on the strength of the mortar used in prism =$\frac{0.27}{f_j^{0.25}}$.

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Figure 4. Compressive stress-strain curve [34]

Eq. (4) defines the elastic strain in compression [39]. Then the inelastic deformation, plastic strain, and damage parameters (d_c) can be found in Eqs. (5)-(7), respectively [39].

$\varepsilon_{c o}^{e l}=\frac{\sigma_c}{E_0}$        (4)

$\varepsilon_c^{i n}=\varepsilon_c-\varepsilon_{c o}^{e l}$        (5)

$\varepsilon_c^{p l}=\varepsilon_c^{i n}-\frac{d_c}{\left(1-d_c\right)} \cdot \frac{\sigma_c}{E_o}$        (6)

$d_c=1-\frac{\sigma_c}{\sigma_c^{\prime}}$     (7)

where,

$\sigma_c^{\prime}$: The brick masonry has compressive strength.

The unreinforced brick masonry tensile strength is estimated from Eq. (8), which suggests that the masonry tensile strength is 5.5% of its compressive strength [35]. Figure 5 showed an empirically estimated tensile curve [35].

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Figure 5. Empirically estimated tensile curve [35]

$f_t=5.5 \% \cdot f_m^{\prime}$        (8)

where, f_t: Brick masonry tensile strength in (MPa).

The mathematical expression for the tensile behavior of brick masonry material is illustrated by Eqs. (9)-(10).

$\sigma_t^1=E_c \varepsilon_t$      (9)

$\sigma_t^2=f_{c r}\left(\frac{\varepsilon_{c r}}{\varepsilon_t}\right)^c$           (10)

where,

f_cr: The cracking strength (MPa).

ε_cr: The strain under fracture strength.

c: the stiffening parameter that determines the model curve՚s sharpness post-cracking=0.2 [35].

Similar to compression strain, elastic strains in tension can be determined using Eq. (11), while inelastic deformation, plastic strain, and damage parameters (d_t) can be found in Eqs. (12)-(14), respectively.

$\varepsilon_{t o}^{e l}=\frac{\sigma_t}{E_0}$       (11)

$\varepsilon_t^{i n}=\varepsilon_t-\varepsilon_{t o}^{e l}$       (12)

$\varepsilon_t^{p l}=\varepsilon_t^{i n}-\frac{d_t}{\left(1-d_t\right)} \cdot \frac{\sigma_t}{E_o}$       (13)

$d_t=1-\frac{\sigma_t}{\sigma_t^{\prime}}$      (14)

where,

$\sigma_t^{\prime}$: The brick masonry tensile strength.

All the details for the compressive and tensile values and the damage parameters are tabulated in Tables 3 and 4.

Table 1. Bricks wall’s elastic properties in Abaqus

Table 2. Plastic properties for the brick wall in Abaqus

Table 3. Compressive behaviour properties for the bricks wall in Abaqus

Table 4. Tensile behavior properties for the brick wall in Abaqus

• Loading and boundary conditions

In the assembly module, two reference points were put at the centres of the top and bottom wall faces. These points were kinematic coupling constraints to the faces in the interaction module.

The numerical investigation can be achieved by executing two major steps. In the first one, vertical compression stress is put. During the second step, an incremental application of the horizontal in-plane monotonic load was performed under displacement control while the vertical and out-of-plane horizontal displacement were tied at the top of the wall. These two steps will maintain the experiment boundary condition. Figure 6 shows the compression load and boundary conditions on the wall. The period of this analysis was one second with details shown in Table 5.

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Figure 6. Compression load and boundary conditions for the wall

Table 5. Details of step module

• Analysis algorithm

In this work, a general non-linear static method has been used for following the Newton-Raphson algorithm solution, which repeated solves for equilibrium in increments, and there is no imposed time scale in static problems, so time step choice is only based on accuracy in modelling nonlinear effects as discussed in Abaqus Theory Guide.

• Mesh generation and element type

30954 elements resulted after meshing the wall in global seeds with a 3D stress element (C3D8R) with an approximate size equal to15mm as shown in Figure 7. The reason for using (C3D8R) is because a linear brick element of the first order is commonly used for large models like brick and mortar with a simpler and faster calculation time. However, the whole purpose of developing a reduced integration element was to increase computational efficiency without losing accuracy.

7.png

Figure 7. Mesh of the numerical model

The current study aims to evaluate the performance of unreinforced masonry shear walls without openings under in-plane loading. This evaluation is achieved by comparing an earlier experimental study with a contemporary numerical model. The continuum representation technique has been utilized to predict the wall's global behavior using the non-linear general static method in ABAQUS/Standard. Upon analysis using the ABAQUS software, it was found that the results closely align with the experimental model. This suggests that this numerical representation can be effectively applied to more complex analysis conditions in future investigations.