1. Introduction
Over the years, increasing research efforts have been developed focusing on analyzing the structural behavior of masonry walls. Masonry is defined as a set of stone units that are connected using mortar joints that are organized to form a regular pattern [
1]. Masonry is commonly used in monuments, masonry arches, and also in low-cost houses. For these structural systems, the low tensile resistance of masonry or mortar interfaces may lead to a compromised response when in- and out-of-plane lateral forces reach high values.
Among several loading conditions, ongoing research aims to investigate the impact of blasting forces on masonry structures. In particular, research on this type of loading focuses on mining activities using blasting operations, which comprise the first phase of the production cycle in most of the mining processes. Blasting is used to fragment the rock overlying the coal seams in most mines. When the explosives are detonated, most of the energy is consumed in rock fragmentation [
2]. According to [
3], energy not used to break rock radiates out from the blast site in the form of ground vibrations and air blasts. Additionally, when explosives are ignited in rock, a shock wave is produced that breaks the rock and then a force in the form of gas pressure is formed [
4]. An explosion or blast activity is defined as the release of a significant amount of energy that takes place in a short time period.
Computer advancement in the past decades has enabled researchers to model masonry with its complexities using finite element analysis. The finite element (FE) method is one of the advanced numerical techniques that is commonly applied to analyze complex structural engineering problems. Research work presented in [
5,
6,
7,
8,
9] and others indicate that by using the FE method, the failure modes that occur in masonry due to blast loading can be successfully analyzed.
According to [
10], the collapse modes of masonry walls that are exposed to blast actions may include flexural failure, direct shear failure, and flexural–shear failure. Collapse modes are further discussed in this section and elaborated in the analysis section of this article. It is noted that these collapse modes were used for the validation of the proposed numerical model.
D’Altri et al. [
11] considered a masonry wall with dimensions of 1190 mm × 795 mm with a brick size of 112 mm × 53 mm × 36 mm. The boundary conditions were taken as fixed on all four sizes of the wall. The wall was loaded with a 20 KN/m
2 out-of-plane load. Their research aimed to assess the effectiveness of the micro-modeling approach and assess the out-of-plane response of the masonry walls. As defined by Lourenço [
12], micro-modeling is where “masonry units and mortar joints are represented by continuum elements, where the unit-mortar interface is represented by a discontinuous constitutive description”. A quasi-static (transient dynamic) procedure was used for the numerical study. Furthermore, the brick–mortar bond failures were accounted for using brick–mortar nonlinear cohesive interfaces. The failure pattern in the wall indicated that the maximum displacement often occurs at the center of the wall.
The discrete element method was used in [
13] to investigate the behavior of masonry structures under blast actions. A 2400 mm × 2400 mm wall, fixed on all sides, was simulated, and typical modes of failure, including out-of-plane failure, were observed. Furthermore, the study depicted the complete failure of the wall under a load of 810 kg TNT explosive weight at a standoff distance of 37 m. According to Masi et al. [
13], the geometry of the blocks and the interfaces may be directly modeled using the discrete element method. Their study was conducted using 3DEC software and the empirical model CONWEP to simulate the blast action. They used a soft-contact technique to simulate joint interactions between adjacent blocks. It is worth mentioning that the magnitude of the wall failure is dependent on various factors such as standoff distance, wall dimensions/properties, and boundary conditions.
Hao [
5] conducted a numerical analysis of a 2880 mm × 2820 mm masonry wall subject to blast load corresponding to a TNT explosive weight W = 2000 kg using AUTODYN software. In that study, the four sides of the wall were modeled as fixed, with a mortar layer between the fixed boundary and the masonry units of the wall, which, in turn, was assigned homogenized material properties. It was shown that for higher explosive weight and shorter standoff distances, the wall would collapse, and the center portion of the wall failed out-of-plane as one brick flew out as a single piece. The wall was also observed to be damaged near the boundary.
Shamim et al. [
14] conducted a numerical study investigating the effect of a blast on a 3000 mm × 3000 mm × 230 mm masonry wall, which had a reinforced concrete frame of 230 mm × 235 mm cross-section dimensions. In their macro-approach, masonry units, mortar joints, and the brick–mortar interface were modeled as a single material. They investigated the effect of 100 kg TNT explosive weight over 20 m, 30 m, and 40 m distances from the wall. Furthermore, their study considered a wall without an opening as well as a wall with a window opening at its center. The boundary conditions were defined such that the top of the wall was restrained in the direction parallel to the blast, simulating the restrain obtained from a slab due to its high in-plane stiffness. The results for the wall without the window showed that the peak values of displacements are found at mid-span. The peak displacement values were equal to 267.8 mm, 95.1 mm, and 59.9 mm for the three mentioned standoff distances between the blast source and the wall, respectively. For the wall with the window, the values of peak displacement at the top of the opening when out-of-plane failure arose were equal to 353.6 mm, 121.9 mm, and 73.2 mm, respectively. Overall, they observed that peak values on the wall with the window were higher than those of the wall without the window; however, the standoff distances were not the same when the opening was considered.
In a similar investigation presented in [
15], it was shown that a wall subjected to blast actions developed the highest displacement in the midsection of the masonry infill panel, while the reinforced concrete frame remained undamaged. It was shown that when the blasting source was close to the wall, the masonry panel collapsed completely, depicting displacements greater than the thickness of the wall (>230 mm). Their study also looked at the effect of changing blast load sizes, considering a TNT equivalent weight of charge equal to W1 = 25 kg, W2 = 50 kg, W3 = 75 kg, and W4 = 100 kg for a constant standoff distance of 20 m. It was observed that peak displacement increases with increasing weight of charge (at constant standoff distance of 20 m) and decreases with increasing distance.
In [
16], a numerical study was conducted on a masonry wall with dimensions of 1700 mm × 1550 mm × 100 mm. The model was constructed with 23 courses of solid clay bricks and analyzed using a simplified micro-modeling approach within finite element analysis. The simulation was implemented in steps, involving vertical displacements and cyclic out-of-plane actions. The failure mode was due to the formation of diagonal cracks caused by in-plane loading. As derived from the mentioned literature, the type of failure modes of masonry walls under in-plane and blasting, out-of-plane loading, are influenced by various characteristics—such as the load application, geometry, boundary conditions, and the quality of materials.
Some recent efforts aim at investigating the response of different types of reinforced masonry walls under blast actions. In [
17], a masonry wall connected with two transverse walls, one at each end, was numerically tested using the micro-modeling technique within non-linear finite element analysis. The work proposed numerical models to reinforce the wall using CFRP wrapping and a steel angle-strip system. In [
18], the behavior of unreinforced masonry walls with CFRP wrapping and mild welded steel wire mesh, under blast with low standoff distance, was investigated using non-linear finite element analysis. In [
19], a fragility analysis of masonry walls was proposed, illustrating the vulnerability of the structures against blast load, focusing on different types of unreinforced masonry walls and reinforced walls, using finite element analysis. In [
20], for masonry walls made of autoclaved aerated concrete and polymer-reinforced concrete that are subjected to heavy TNT explosive loads, both experimental and numerical testing were provided. In the numerical models, non-linear finite element analysis was used with cohesive zone models to depict damage to the wall.
Based on this short review of recent results, it seems that there is still space for more research investigating the collapse modes of masonry walls under blast actions. In particular, one of the goals of this article, which also highlights its innovative points, is to provide further insight into the way in-plane failure modes, such as diagonal cracking and out-of-plane damage, may appear in masonry walls subjected to blast actions. From another point of view, this article proposes a modeling technique using non-linear constitutive descriptions, incorporating opening-sliding failure modes adopting contact mechanics, as well as compressive/tensile damage, using continuum damage laws, all within finite element analysis. The proposed models can be implemented in commercial software.
Within the given framework, a numerical investigation of the mechanical response of masonry walls under blast actions, with and without openings, is presented. Non-linear finite element models are proposed to simulate all the joints between masonry units by introducing unilateral contact–friction interfaces. For the simulation of the blast action, an empirical model is used, and explicit dynamic analysis is adopted implementing this loading type. Various loading cases are tested, resulting in different failure modes.
In
Section 2 of this article, failure modes of masonry walls are provided and modeling approaches that can be used to capture these modes are briefly discussed. In
Section 3, all the details of the numerical model that is proposed in this article are presented. Among others, the details of the blast load simulation, the material constitutive description, and the geometry of the walls are given in this section. In
Section 4, a validation of the proposed model is conducted using a comparison of some results with published output. In
Section 5, results and discussions derived from the suggested approach are provided, and in
Section 6, the conclusions of this investigation are presented.
3. The Numerical Model Proposed in the Present Article
A non-linear finite element model is proposed for this study to simulate the response of masonry walls to blasts. For the evaluation of the failure response that is derived from the masonry unit interfaces, a unilateral contact and friction constitutive description is assigned to these interfaces. Thus, both in-plane and out-of-plane opening and/or sliding between the masonry units can be depicted with the proposed model. Since all the contact conditions between the blocks in the wall are simulated using principles taken from contact mechanics, it can be stated that the micro-modeling approach is used.
The unilateral contact law, provided in Equations (1)–(3) for a single degree of freedom system, is assigned in the interfaces between masonry blocks. Equation (1) is the non-penetration relation, Equation (2) states that only compressive stresses (
tn) can be developed in the interfaces, and Equation (3) is the complementarity relation, stating that either contact takes place (
u −
g = 0, where
u is a single degree of freedom and
g is an initial gap) or separation in the interface occurs (
tn = 0).
For the response in the tangential direction of the interfaces, a static version of Coulomb’s friction law is considered. Thus, sliding in the interfaces is initiated when the shear stress
tt reaches the critical value
τcr, according to Equation (4):
where
μ is the friction coefficient and
tn the normal stress (contact pressure) in the interfaces.
To represent the failure response of the masonry units, a continuum concrete damage plasticity model is used. Compressive and tensile failure modes developed at the masonry blocks are then depicted. In the following sections, the details related to the implementation of the blast loading, the material properties, and the dimensions of the walls that are studied in this article are provided.
3.1. Blast Shock Wave Modeling
An explosion loading wave is defined by three parameters, namely, the shape of a wave, the maximum pressure (
Pro), and the positive wave duration (
to), which is the time that pressure reaches zero [
24]. Various research efforts have shown that depending on the source of the explosion, the generated waves are divided into shock and pressure waves. In a shock wave, the pressure of gasses from the explosion or blasting is developed by emission from the source of the explosion [
24,
25]. The pressure increases to the maximum value
Pro and decreases to the environmental pressure, as shown in
Figure 4. Mining activities involving blasting generate blast pressures on neighboring structures. The pressure distribution from a blasting source at a particular distance is considered nearly consistent over a normal reflecting surface. According to [
26], a close-in explosion produces a pressure distribution that changes significantly in magnitude over the reflecting surface. This creates more complexity due to the non-uniform of pressure.
To determine the magnitude of peak overpressure, two major parameters are used: the charge weight and the distance between the blast source and the structure. By observing the pressure–time diagram depicted in
Figure 4, two main phases can be identified. The positive part of the diagram is called the positive phase and has a duration
to, as shown in
Figure 4, while the negative part is called the negative phase and has a duration
to−, also shown in
Figure 4. According to [
13], when a primary shock strikes a target, the reflected overpressure
Pr instigates. The negative phase exists for a longer duration with lower intensity pressure than the positive phase. As the standoff distance increases, it can be noted that the duration/period of the positive blast wave phase increases, and that results in lower amplitude and a significantly longer-duration shock pulse.
Using the Friedlander equation, the time evolution of the positive phase of the reflected pressure is analyzed (Friedlander, 1946):
where
H[t*] represents the step function,
d is the exponential decay coefficient, and
t* = t − tA, where
tA is depicted in
Figure 4. According to Rigby et al. [
28], the impulse
iro or
ir associated with the positive phase, which symbolizes the area under the pressure curve, can be formulated as:
One of the most effective means of representing a blast impact is the use of the CONWEP model. According to [
29], CONWEP is a model used to simulate the effects of a collection of conventional weapons, including air blast routines, breach, cratering, ground shock, and fragment and projectile penetration. The CONWEP charge property parameter is used in this study to simulate an air-based explosion using empirical data [
30]. Furthermore, according to this consideration, a time history diagram of the pressure loading is built. In order to utilize this empirical model, one would need to define the equivalent TNT (trinitrotoluene) mass of the explosive as well as the source point (i.e., where the explosive is located). The initial process in calculating the explosive wave from a blast source other than TNT is to convert the charge mass to TNT equivalent mass [
31].
Therefore, the CONWEP charge property is used in this study within commercial finite element software to simulate an air-based explosion by developing a time history pressure loading, similar to the one shown in
Figure 4. The data, which were entered to define the blast charge properties, include the equivalent mass of TNT, a multiplication factor to convert from that mass unit into kilograms, and multiplication factors to convert from the standoff distance, time, or pressure to meter, second, or pressure in Pascals, respectively.
3.2. Continuum Damage Law for the Masonry Units
A concrete damaged plasticity law is used to represent damage on masonry units. Rate independence is claimed for this law, which is based on incremental plasticity theory. According to Lubliner et al. [
32], Lee and Fenves [
33], Tapkın et al. [
34], and Daniel and Dubey [
35], this constitutive description is appropriate for the analysis of quasi-brittle materials such as concrete and masonry. It relies on the concept of isotropic damaged elasticity for the representation of the irretrievable damage or failure that occurs during the cracking process for materials under fairly low pressure. The concrete damage plasticity law uses a non-associated potential plastic flow, which is in turn the implementation of the Drucker–Prager hyperbolic function for flow potential [
36].
The common failure mechanisms that can be illustrated with this law are, namely, tensile cracking and compressive crushing. When unloading takes place, the elastic stiffness of the material is deemed damaged. This damage is implemented by introducing two damage variables as functions of the plastic strain, one for tension and the other for compression. A zero value of the damage variable indicates undamaged material, while a value equal to one indicates a total loss of strength. The corresponding uniaxial stress–strain relations, representing tension and compression, are provided below:
In the above equations, E0 is the preliminary elastic stiffness of the material and dt and dc are the tensile and compressive damage variables, respectively.
The compressive and tensile stress–strain curves used in this work to define the compressive and tensile failure response of the masonry units on the numerical models, as well as the corresponding damage variables diagrams, are provided in the figures below. The uniaxial stress–strain behavior of concrete is modeled utilizing a Hognestad-type parabola [
37], as per
Figure 5 below.
Figure 6,
Figure 7 and
Figure 8 provide the compressive damage parameter as well as the tensile stress–strain law and the tensile damage parameter used for this model [
37].
The uniaxial tensile damage and uniaxial compressive damage parameters were developed using the post-failure stress as a function of cracking strain. The cracking strain is equal to the total strain minus the elastic strain of the undamaged material [
32].
Some additional material properties used within the concrete damage plasticity law are provided in
Table 1. The material properties for each masonry unit are provided in
Table 2.
3.3. The Geometry of the Masonry Walls
The dimensions of each masonry unit considered in this study are equal to 430 mm × 140 mm × 190 mm. The size of each unit is as per the Concrete Manufactures Association [
39]. Low-cost housing in South Africa often uses concrete masonry blocks and clay bricks. This paper focuses on the use of concrete blocks, and the following limitations are noted:
- -
A single-leaf wall is considered, and the wall is unreinforced.
- -
Category 1 buildings [
40].
Two geometries are used in this study for the walls, as shown in
Figure 9 and
Figure 10. The first is a solid wall and the second represents a wall with an opening.
3.4. Details of the Finite Element Model
Figure 11 shows the mesh that is adopted in this study for the models without and with an opening. Three-dimensional, eight-node linear brick elements are used, with the element side equal to 40 mm for both walls. A total number of 4800 elements for the model without the opening and 5600 elements for the model with the opening are used, as shown in
Figure 11.
All four sides on the perimeter of each of the two walls are considered as fixed in three translational degrees of freedom, according to the coordinate system shown in
Figure 11. It is noted that the restraining of the top side of the walls in the Z-direction is attributed to the assumption that an upper slab or roof will provide restraint in that direction.
Concerning the loading of the models, two load steps are used. In an initial, pre-existing step, a vertical pressure of 0.25 MPa is applied to the top side of the structure. In the first load step, a horizontal shear (in-plane) displacement of 10 mm is applied to the top side of the walls. Alternatively, the wall with no horizontal in-plane displacement is also considered. In the second load step, the blast loading is applied.
The simulation is conducted using explicit dynamic analysis. This type of analysis is appropriate since it is able to capture the very short duration of the blast action. It is noted that the explicit dynamic analysis was originally developed to simulate high-speed dynamic events that would otherwise require significant computational resources within implicit codes. For the implementation of this analysis, an automatic time incrementation is used.
For the application of the contact–friction conditions between the masonry blocks, the method of Lagrange multipliers is used. A friction coefficient equal to 0.45 is assigned to the interfaces.
It is noted that for the implementation of the blast load, a charge weight expressed in TNT at the standoff distances of 100 m, 50 m, and 20 m is used. In addition, the effect of the blast weight, as well as the effect of changing the blast charge while keeping the distance constant and changing the standoff distance while keeping the blast charge weight constant, are also investigated. Only the front surface of the walls is loaded (incident surface). In the following sections, results obtained from various parametric investigations, emphasizing the corresponding failure mechanisms, are provided.
6. Conclusions
In this article, the response of masonry walls under static in-plane and blast loads is investigated using non-linear finite element analysis software [
44]. For the simulation of damage in the interfaces between the stone blocks, unilateral contact–friction interfaces are applied to depict opening and sliding failure. In addition, a concrete damage plasticity model is used to describe tensile and compressive damage in the blocks. The proposed scheme is applied to a solid masonry wall and to a wall with an opening (window).
This investigation aims in highlighting potential collapse mechanisms by testing different blast load parameters, namely, the weight of the explosive and the standoff distance between the source of the explosion and the structure. The influence of a horizontal shear displacement in-plane loading at the top of the wall is also investigated.
According to the findings of this study, the failure mode of the wall loaded with both shear in-plane displacement and the blast action can be either in-plane diagonal cracking or out-of-plane flexural failure. The first mode arises when the shear in-plane displacement is the dominant loading, compared with the blasting action, while the second arises when the blast is the dominant loading. For the same material properties and wall dimensions, the weight of the explosive and the standoff distance are the critical parameters, which determine which of the two loading types dominates. In the results section, case studies highlighting both failure modes are discussed for various values of the explosive weight and the standoff distance. A combination of both failure modes can also arise, depending on the values of these parameters.
Another outcome of this work is the fact that the presence of an opening (window) in the wall may reduce the effect of the blast action by decreasing the out-of-plane response of the structure. The reason for this is that due to the opening being located in the middle of the wall, the blast load is not applied to this critical (for out-of-plane flexure) middle part of the surface of the wall. Thus, this study shows that the blast action must occur at a closer standoff distance compared with the solid wall, in order to cause significant damage to the structure.
When no shear displacement in-plane loading is applied, the response is dominated by the out-of-plane flexural deflection, attributed to the blasting action. In this case, lower maximum displacements are obtained compared with the wall loaded with shear displacement and blast actions.
Several future investigations could be used to extend the present work. A potential concept is to study the influence of the area, position, and number of windows on the response of the walls under blast actions. The usage of different initial static loading could modify the results, as was shown in the conducted numerical investigation. Design or re-design based on these findings could also form an interesting research topic. Another concept is related to the implementation of data-driven structural dynamics, introducing machine learning tools, to evaluate the influence of several parameters such as the dimensions of the walls and the blast load parameters on their structural response.