Numerical modeling of confined brick masonry structures with parametric analysis and energy absorption calculation

Abstract

The objective of this paper is to propose a new modeling methodology for numerical analysis of full-scale confined brick masonry structures. Two modeling strategies are used within a single structure, where the in-plane walls are modeled using “simplified micro-modeling” approach and out-of-plane walls are modeled using “macro-modeling” approach. The lateral load capacity is associated with the in-plane shear resistance of masonry elements, therefore more detailed analysis is required for in-plane walls to achieve a comprehensive understanding of the damage mechanism and load transfer. The investigation of the in-plane shear behavior of confined brick masonry structures is of significant importance. Additionally, the proposed hybrid model is validated by comparing the results of experimental studies of confined brick masonry structure. A parametric study is then conducted to investigate the effect of brick and mortar properties on the structural response metrics (e.g. base shear coefficient, effective stiffness, response modification factor, the three performance levels (i.e. Immediate Occupancy, Life Safety and Collapse Prevention limits) and the energy absorption properties). It is observed that these structural response metrics, changed considerably by varying the material properties. Apart from that, the damage behavior and damage pattern are also assessed for the better understanding of effect of these parameters on the response of the structure. The proposed hybrid-modeling approach gives sufficient accuracy in predicting the lateral load behavior as well as the damage mechanism of confined brick masonry structure, subjected to lateral loading.

Keywords

Numerical modeling parametrical analysis confined brick masonry lateral load performance hybrid-modeling energy absorption

Introduction

Based on the post-earthquake damage information, masonry is one of the most vulnerable construction types when subjected to lateral loading (Avila et al., 2018; Colunga et al., 2009). Masonry is an assemblage of bricks and mortar, which possess distinct directional characteristics due to the presence of mortar joints (Brodsky et al., 2018; Magenes, 1998; Mahmoud et al., 1995; Messali and Rots, 2018; Ruiz-García and Negrete, 2009; Zhuge et al., 1996). Since masonry is a composite material with anisotropic behavior, it shows a complex behavior when subjected to bi-axial state of stress (Bosiljkov et al. (2003); Tomazevic (1999)). Several experimental studies have been conducted (for instance, by Yoshimura et al. (2003)) on masonry to evaluate and understand its behavior under lateral loads (Vasconcelos, 2005; Voon and Ingham, 2006). The nonlinear/inelastic behavior of masonry under monotonic loading includes tensile cracking and softening, compressive crushing and softening, and shear transfer through the crack slips, whereas cyclic loading increases the degree of complexity (DISWall Project, 2006–2008; ESECMaSE Project, 2004–2007; Mosele et al., 2008; Steelman and Abrams, 2007). In confined unreinforced masonry structures, the masonry walls are laterally restrained by confining elements (i.e. tie columns and bond beams) to be placed at corners and wall openings. Based on experimental study, ductility of the structure is found to increase by these confining elements and make the structure less vulnerable to earthquake (Ahmed et al., 2018). However, the experimental work that has been done, also persists some shortcomings. Such as the test setups varies with varying researchers and the results sometimes are limited to the conditions to which they have been subjected. Also, the test setups are usually complex, and the actual loading protocol and boundary conditions are difficult to be known exactly.

Thus, in addition to experimental testing, the numerical modeling, principally developed by the finite element method, is ineludible for masonry research. Input vectors for these models can be improved to develop a large number of required numerical simulations. Additionally, one can evaluate different materials and configurations, that is, weak or strong material, with or without opening, slender or short walls, and so on (Abdou, 2005; Alvarez, 2000; Andreaus, 1996; Crisafulli et al., 2000; Ishibashi and Kastumata, 1994). The great number of affecting factors, such as dimensions of the units, width of the mortar joint, arrangement of head and bed joints, anisotropy of both units and mortar and level of the workmanship, make the numerical modeling of masonry extremely difficult. According to several authors (for instance Casolo (2010) and Lourenco (2002)), depending on the level of simplicity and accuracy desired, it is workable to use different modeling methods (Figure 1). The numerical modeling of masonry can be categorized into three groups: a) micro-models (detailed micro-models), b) meso-models (simplified or modified micro-models) and c) macro models (Abdou, 2005; Lourenco, 1996).

Figure 1.

Modeling strategies for masonry structures, Lourenco (2002).

In Micro-modeling approach, unit and mortar are modeled separately and the interface between them is defined. Essentially, the elastic (compressive strength, Young’s modulus and Poisson’s ratio) properties and, electively, inelastic properties of materials are taken into account. Due to the excessive details involved in this type of approach, its implementation is limited only to portion of a masonry structure (e.g. vaults, pediments, and wall; Cennamo, 2011). In Meso-modeling approach, each joint, containing mortar and the two unit-mortar interfaces, is bunched into an average interface, while the units are expanded both ways in order to keep the geometry same. Masonry is thus regarded as a set of elastic blocks surrounded by potential fracture/slip lines at joints (Cennamo, 2011). However, the Macro-modeling approach does not differentiate between individual units and mortar joints and treats the masonry as an anisotropic homogeneous continuum. At the macro-scale level, the characteristic size is the macro-element scale, a part of structures that may be considered with simple equivalent system (e.g. vaults, pediment, walls; Cennamo, 2011).

As discussed above that in micro-modeling approach, the units, mortar joints, and unit-mortar interface are modeled separately, requiring a lot of modeling time and thus resulting in greater computational time. Thus making it not suitable for the large scale structures. However, in the macro-modeling method the units and mortar joints are merged into one material. Conversely, as masonry is not homogeneous material, this approach might not be capable of predicting the local response of the masonry assembly. Also one of the aims of this study is to conduct a parametric study in order to evaluate the effect of unit and interface properties on the global response of the structure, which cannot be achieved using the macro-modeling approach. To solve this problem, meso (or simplified micro) modeling has been developed and majority of the studies conducted on numerical modeling of masonry, such as Shing et al. (1992), Berto et al. (2004), Milani (2008), Stavridis and Shing (2010), and La Mendola et al. (2014) have been performed using this modeling approach. The mortar joints are lumped into the interface as a gap element. Extended units up to half of joint thickness in head and bed joints were modeled as continuum elements.

Therefore, new modeling approach is presented in this paper known as hybrid-modeling approach, where the in-plane walls are modeled using Meso-modeling (or simplified micro-modeling) method while the out-of-plane walls are modeled using the macro-modeling method. Since the lateral loads are totally resisted by the in-plane walls in confined masonry structures, therefore more detailed analysis of these walls is required. However, the out-of-plane walls do not contribute in load distribution and only possess the flexural out-of-plane behavior (Varela-Rivera et al., 2011), so less details regarding the failure are required. By adopting this method, lot of time and energy (computational and modeling) can be saved without compromising the performance evaluation of important components of the structure. It is also important to mention here that since an earthquake ground motion includes two horizontal components, and the walls act as in-plane walls for one component will act as the out-of-plane walls for the component in another (horizontal) direction. Therefore in case of real masonry structures subjected to the earthquake ground motions, how much this method will be effective. While answering this question, one needs to understand the scope of the current study as well as the assumptions made for adopting such technique. While working out the experimental program (Ahmed et al., 2018), it was mentioned that out of two orthogonal directions, the relatively weaker direction was selected for the application of the load as it will be more vulnerable during an earthquake. And as the weaker side always govern the behavior, therefore by knowing the detailed behavior of the structure in the weaker direction, it can be assumed that the approximate behavior of complete structure is known. The same assumption was used in the numerical modeling and therefore the same experimentally tested structure was modeled with much focus given to the relatively weaker side as it will be vulnerable to the damage. Once we know the complete behavior of weaker side with comprehensive parametric analysis, we can better design the masonry structures knowing their detailed behavior. On the other hand, a “reverse approach” can also be adopted, where the current Out-of-Plane walls are modeled by meso modeling and current in-plane walls are modeled by macro modeling and load is applied in direction perpendicular to the current loading direction. In that case, the detailed behavior of the structure in both the orthogonal directions can be assessed with the same efficiency, however it will be out of scope of current study. Thus, the proposed hybrid modeling approach can be used effectively to better understand the behavior of confined masonry structure for cases other than those tested at laboratory. In this paper, a confined masonry structure tested at laboratory under the lateral loading conditions (Ahmed et al., 2018) was modeled using Non-linear Finite Element Method and the results were validated against the actual test results. The effect of several parameters (e.g. strength of brick unit & mortar joint and level of vertical/gravity load) on lateral load capacity, ductility of structure, enhancement of performance levels, and relevant damage pattern are investigated in this study. Additionally, the effects of these parameters on the structural performance levels (i.e. immediate occupancy level, life safety level and collapse prevention level) and energy absorbed under earthquake action are also investigated in this paper.

Brief description of experimental testing and results

As mentioned above, the numerical model was validated against the experimental results of a full-scale test of confined brick masonry structure (Ahmed et al., 2018). The test program included testing of confined brick masonry structure following the typical test setup shown in Figure 2 (Ahmed et al., 2018). The dimensions of the structure were 3048 × 3658 mm and the thickness of wall was kept 229 mm. The walls were made with 1:5 cement-sand mortar in the English bond pattern which is most common construction practice for clay brick masonry in Pakistan. The building height was kept 3353 mm (See Figure 2(a)). The model structure was made over a 305 mm thick reinforced concrete (RC) pad, connected to the strong floor with nut and bolt system. Toothing of 76.2 mm was provided in each layer of masonry at all the corners and also around each opening, which were then filled by concrete. Each tie column consisted of 4 to 12.5 mm bars tied by 9.5 mm stirrups at 229 mm c/c. Lintel beams of 229 mm × 152 mm were used. The beam consisted of 2 to 12.5 mm bars in tension and 2 to 12.5 mm bars as hangers/compression and was tied by 9.5 mm stirrups at 152 mm c/c. Concrete slab with a thickness of 152mm was reinforced in both directions with 12.5 mm-dia. bars spaced at 229 mm c/c. Lateral loading was applied by two actuators, fixed against the strong wall on the side of structure model (see Figure 2(b)). The actuators were attached to the slab on East side of the structure. The model was fixed to the base pad of strong floor with post-tensioning bolts. The detailed description of the model building, testing arrangement, damage pattern and analysis of the experimental results is available (Ahmed et al., 2018).

Figure 2.

(a) A picture of the building model, (b) Test setup, and (c) Structural details (Ahmed et al., 2018).

The damage pattern observed at different drift ratios, is shown in Figure 3. Diagonal hair line cracks started to appear in the lower left corner of pier 2 (pier locations are presented in Figure 2(b), at a drift ratio of 0.093%. Similarly, diagonal cracks started to appear in the upper right corner of the pier 2 and lower right corner of the pier 1. At a drift ratio of 0.16%, hair line cracks started to appear at the middle of pier 2 and extended towards the lintel beam. At a drift ratio of 0.73%, two out of four corner tie-columns cracked at the middle due to flexural stresses. The cracks in the pier 1 opened up to 50 mm (2-inches), resulting in toe crushing phenomena. However, in the pier 2, the diagonal shear cracks propagated through the slab. The detailed description of experimental program and damage analysis can be found in (Ahmed et al., 2018).

Figure 3.

Damage details at different drift levels (Ahmed et al., 2018).

Numerical modeling

For the analysis of masonry structures, subjected to general monotonic or reversed cyclic loading, realistic analytical procedures and constitutive models are required to predict the behavior of the structure precisely and efficiently. To study the behavior of confined brick masonry structure subjected to lateral loading the numerical model was defined through the software ATENA (Červenka et al., 2016). Atena is a non-linear finite element analysis software, designed for earthquake engineering applications. It is preferred here, for being computationally efficient, hence allows for the structural level simulations and also accurate in capturing the global behavior and damage mechanism of the structure. An Updated Lagrangian formulation is used to consider the geometric effects.

In all the past studies mentioned in this paper the masonry assemblage is modeled using a single methodology that is, micro or macro modeling alone. However this study is more focused on the importance of the load resisting mechanism of the structure. Since the experimental program was aimed to apply the lateral load from one side and for the sake of safety the weaker direction was chosen to apply the loads, as the out-of-plane walls contain solid masonry wall on one side and wall with just one window on the other side. So it is well understood that if the earthquake hits the structure from any other side then its response will be much better than the one obtained through this testing. On the basis of this assumption the numerical modeling was also opted in a way that the in-plane load bearing walls were given more importance and their local response was evaluated however the out-of-plane walls were modeled to count their presence in the structure and also to confirm the box behavior of the masonry assembly. The hybrid-modeling approach is thus adopted for the simulation of the structure, where simplified micro-modeling approach is used to model the in-plane walls since it includes the basic failure mechanisms that characterize masonry, enabling the detailed description of resisting mechanisms of the structure. However, macro-modeling approach is used to model the out-of-plane walls. This approach allows for the representation of different materials in the structure. The full-scale experimentally tested confined masonry structure (Ahmed et al., 2018) presented in section 2 of this study is modeled by hybrid-modeling approach using Atena. The modeling details are discussed here.

Material constitutive relationships

A number of proposed constitutive models (Giordano et al., 2002; Lemos, 1995; Lofti and Shing, 1991, 1994; Lourenço, 1996; Lourenço et al., 1995, 1997; Page, 1978; Pegon and Anthoine, 1994) have been used for both micro and macro modeling approaches, including plasticity-based models, smeared-crack models, discrete-crack models, and damage-based models. In micro-modeling approach, all constituent materials with distinct properties, are described independently. Distinct material models are used for concrete (foundation, confining elements and slab), steel reinforcement, brick masonry units, and mortar joints. Experimental tests carried out on materials and masonry assemblages (Ahmed et al., 2018) are used in the description of mechanical properties of the material model. The detailed description of each material model is presented here.

The brick masonry unit behavior is modeled using “3D Non-linear Cementitious 2” material model available in the software ATENA. The tensile and compressive behavior of the material is described with single stress-strain relationship in a coordinate system (Figure 4(a)). Exponential and Parabolic constitutive laws are used to describe the tensile softening and compressive hardening behavior of brick masonry unit, respectively, see Figure 4. The mechanical properties used for the model description are given in Table 1. The compressive strength and elastic modulus were obtained from simple compression test, while tensile strength and modulus of rupture were obtained by 3 point bending test of brick. Masonry joints were modeled using an interface element based on coulomb-friction criteria, see Figure 5. The shear stiffness, cohesion and angle of internal friction for the mortar was obtained through the results of the triplet shear testing to characterize the shear behavior of unit-mortar interface (Ahmed et al., 2018). However, the tensile strength of interface was obtained by bond tensile strength test of masonry prisms (Duplets). The contact joint properties used in the simulations are also given in the Table 1. The K_nn, K_tt denote the initial elastic normal and shear stiffness respectively. Typically for zero thickness interfaces, the value of these stiffnesses correspond to a high consequence number. There are two additional stiffness values denoted in Figure 5 as K_nn ^min and K_tt ^min, used only for numerical purposes after the failure of the element in order to preserve the positive definiteness of the global system of equations. As presented earlier, the out-of-plane walls are modeled using Macro-modeling approach, where the entire wall/pier is modeled as a single macro-element, although the material model used was same as for brick unit, the properties used for that macro-element resembles the masonry prism properties (see Table 2; Ahmed et al., 2018). Concrete used in different elements (Slab, confining elements, and footing) with a compressive strength of 31 MPa, is also modeled using “3D non-linear cementitious2” material model (Ahmed et al., 2018). Steel reinforcement is modeled using a uni-directional elastic-strain hardening response, with yield strength of 280 MPa and ultimate strength of 400 MPa (Ahmed et al., 2018).

Figure 4.

(a) Complete stress-strain behavior and (b) tensile softening of Brick unit (Červenka et al., 2016).

Table 1.

Summary of material properties in micro-element definition.

Topology	Contact—Joint					Brick—Element
Material	3D Interface					3D Non linear cementitious2
Properties	K_nn (MN/m³)	K_tt (MN/m³)	f_t (MPa)	C (MPa)	Ф	E (GPa)	G_F (N/m)	ν	f_t (MPa)	f_c (MPa)
Values	350,000	120,000	0.40	0.52	0.33	23.65	39.67	0.2	1.46	14.40

Figure 5.

(a) Failure surface for interface element, (b) behavior of interface element in shear, and (c) in tension (Červenka et al., 2016).

Table 2.

Summary of material properties in macro-element definition.

Topology	Masonry
Material	3D Non linear cementitious2
Properties	E (GPa)	G_F (N/m)	ν	f_t (MPa)	f_c (MPa)
Values	2.212	36.49	0.2	0.58	3.16

Geometry and meshing

One of the primary objectives of this study is to establish a simple three-dimensional nonlinear model that replicates the original tested structure, capable of capturing the key response features of this kind of quasi-brittle construction. The model employed (1) element CCIsoBrick which is an isoparametric solid element, for concrete and masonry elements and (2) element CCReinforcement a linear 2-node 1D-truss element for the steel rebars. To simulate the bond of reinforcement with concrete, the “perfect connection” option in Atena was used, which enforces full bond. The interface between concrete elements and masonry was modeled as frictional in the tangential direction and “hard contact” in the normal direction. The mortar joints were diffused into the brick units and the brick-mortar interface is modeled by interface elements. To reasonably capture bending response of the out-of-plane walls, shell elements (ahmadshell) are used as there is only flexural behavior required for out-of-plane walls which can be captured easily by using the shell elements. The FE mesh was simply consisted of continuum elements, interface elements and shell elements to represent the brick unit, mortar joint and out of plane macro-elements, respectively. To model RC structures it is common to adopt the finer FE mesh as it is generally thought that a finer mesh will offer a more accurate portrayal of the assembly and therefore more precise results. Studies regarding the response of concrete cylindrical samples subjected to high loading rates, the FE meshes with dimensions as small as 2 to 3 mm was adopted (Cotsovos et al., 2008; Li and Meng, 2003; Thabet and Haldane, 2001). Nevertheless, it should be noted that while modeling the brittle materials—finer FE meshes might not necessarily gives the accurate results. The adoption of huge number of FEs can generate numerical problems (instability) especially in crack-opening and closing procedure employed, as the quantity of possible cracks that will generate or close in each iteration increases considerably with the rise of the Gauss points used in the FE mesh. Thus, a mesh sensitivity study was performed to determine the optimum mesh size suitable for masonry incorporating bricks and mortar. The maximum mesh size was taken as 80 mm for the shell elements and refinements were applied to the column volumes and edges, the concrete slab and its vertical edge, and to the brick units (about 25 mm was used).

Boundary and loading conditions

The boundary conditions have led a central role on the behavior of structures as it decides the predominant failure mechanism subjected to lateral loading. Because of the difficulty of calculating the boundary conditions at laboratory, it is common to assume cantilever structure in the experimental programs. Therefore, to match the actual loading conditions the base of the structure was fixed and the lateral displacement was applied at the slab level in this study. As the focus of this study is on the parametric analysis, therefore only monotonic loading was considered in the numerical analysis. The model is shown in Figure 6. In Figure 6(a) the actual model is shown where all the bricks of in-plane walls are modeled as individual elements connected together with an interface model, while in Figure 6(b) the finite element mesh has been shown.

Figure 6.

Building Model (a) Micro and Macro modeling detail and (b) FE Mesh detail.

Validation of numerical model

The evaluation of the effect of selected parameters on behavior of masonry structure was followed by the calibration of numerical model existing in ATENA (Červenka et al., 2016). As discussed earlier, the mechanical properties of unit-mortar interfaces and their normal and shear stiffness were calibrated, to fit the numerical to experimental results obtained in the tested masonry building. In order to present the match between the numerical simulations and the experimental results, the envelop curve is obtained from the cyclic tests and the secant stiffness (at peak drifts of each cycle during experimental testing). The envelop curve obtained from test results is compared against the Atena simulation in Figure 7. It is observed that the Numerical model captures various features from the tests like ultimate strength and initial stiffness very accurately. The difference between experimental and numerical lateral strength is less than 5%.

Figure 7.

Force displacement comparison of experimental and numerical results.

As shown in Figure 8, the numerical model replicates the main crack pattern developed for confined brick masonry structure, namely diagonal cracking and initial flexural cracking. Also, the damage pattern of Numerical model matches with Experimental pattern very well (Figures 7 and 8). On the basis of comparison between numerical and experimental main results, it can be said that the numerical model is able to reproduce fairly well the experimental Load-deformation curve and damage pattern of confined brick masonry structure under combined vertical and shear loads, and it is suitable to be used in the parametric analysis.

Figure 8.

Damage pattern at ultimate drift ratio of: (a) In-plane North wall, (b) In-plane South wall (Numerical Modeling) and (c) In-plane North wall, and (d) In-plane South wall (Ahmed et al., 2018).

Parametric analysis

Masonry is combination of bricks and mortar, with significant properties. The properties of these materials might greatly affect the global response of the structure. Testing of these materials at the stress-strain level corroborated that the properties of these materials might vary to a significant level, depending on many factors (e.g. type of brick, burning and cooling temperature during the manufacturing of bricks, erectness of brick, water-cement ratio of mortar, cement to sand ratio of mortar, thickness of the mortar joint; Ahmed et al. (2018)). Therefore, it is worth investigating the effect of brick and mortar properties on the response of the structure through a parametric study.

The dimensions (length, width, and height) of the structure, reinforcement details of slab and confining elements, concrete properties and the loading protocol used for parametric study are same as the one used in the numerical analysis of the confined masonry structure described above. The variables of the parametric study are: the level of gravity load (P/W = 0%, 19%, 37%, 56%, 100%, 200%, and 300%, where P is the gravity load applied at the slab and W is the weight of the structure) and properties of unit (brick) and interface (mortar joint). The seven different levels of gravity loading represent walls from seven structures with different height-to-width ratios. A higher level of gravity load corresponds to a slenderer structure where the load on the walls are larger due to higher overturning under the lateral loads. The considered properties of unit and interface are plotted in Figure 9. The unit and interface properties used are based on the response, observed of real materials in several studies (for instance, Narayanan and Sirajuddin, 2013).

Figure 9.

Properties of the considered materials—B and M designates Brick and Mortar respectively.

To capitalize on the effect of the material properties, other properties such as elastic modulus, tensile strength and modulus of rupture for unit, and friction angle and tensile strength for interface, were also enhanced with the enhancement of compressive strength and cohesion of brick and mortar respectively, to make the parametric study more real and reflect the exact on-field conditions. The inelastic interface properties (cohesion, friction angle, and tensile strength) affect the lateral load response of masonry largely (Sarhosis et al., 2015) and therefore only these properties were considered for interface in the parametric study. Properties of four different types of brick and three different types of interface (Figure 9) are given in Tables 3 and 4.

Table 3.

Properties of interface used in parametric analysis.

Material	Interface joint					Description
Properties	K_nn (MN/m³)	K_tt (MN/m³)	C (MPa)	F_t (MPa)	Ф	Description
1	2.5 × 10⁵	1 × 10⁵	0.99	0.75	0.69	Strong mortar
2	2.5 × 10⁵	1 × 10⁵	0.50	0.36	0.35	Intermediate mortar
3	2.5 × 10⁵	1 × 10⁵	0.12	0.10	0.10	Weak mortar

Table 4.

Brick properties used in parametric analysis.

Material	Brick unit					Description
Properties	E (GPa)	G_F (MN/m³)	ν (MPa)	f_t (MPa)	f_c (MPa)	Description
1	24.87	42.72	0.2	1.71	16.0	B16
2	22.54	36.98	0.2	1.48	13.0	B13
3	20.14	31.45	0.2	1.26	10.0	B10
4	16.92	24.6	0.2	0.98	7.0	B7

Seven response metrics at the structural level are evaluated for each of the 84 combinations (7 axial load levels × 4 brick types × 3 interface types (or mortar types for practical application)): base shear coefficient (defined as the ratio of lateral load to the total weight of the structure), effective stiffness, displacement amplification factor, response modification factor and the 3 performance levels (i.e. immediate occupancy, life safety and collapse prevention). The definition of yielding, maximum and ultimate (20% reduction) points, based on ASCE/SEI-41 (ASCE Standard, 2006), are shown on an example envelop curve in Figure 10. The envelope curves are acquired for each of the 84 cases, and the three points namely; yield, maximum and ultimate points are found for these cases. A bilinear curve is first drawn from the envelop curve from which different parameters like stiffness and yield drift are evaluated. The initial stiffness of the structure is defined as the ratio of force corresponding to 75% of the peak lateral load resistance, to the lateral drift at that point. The peak lateral load resistance is defined in Figure 10. The response modification factor (R) and displacement amplification factor (C_d) can be obtained by using equation 1 and 2

R = \sqrt{(2 μ_{D} - 1)}

(1)

C_{d} = \frac{μ_{D}}{\sqrt{(2 μ_{D} - 1)}}

(2)

where μ_D is the displacement ductility, defined as the ratio of the lateral drift at the ultimate point to the yield point. Finally, the performance levels can be defined as per ASCE/SEI-41 (ASCE Standard, 2006). The story drift corresponding to yield point on bilinear elasto-plastic curve is taken as Immediate Occupancy level, to which there is no plastic deformation and strength degradation. The Collapse Prevention level is taken to be the ultimate drift at which the lateral strength dropped by 20%. The drift corresponding to Life Safety level is taken as 75% of the Collapse Prevention level as recommended in the ASCE/SEI-41 (ASCE Standard, 2006) and shown in Figure 10.

Figure 10.

Definition of immediate occupancy (IO), life safety (LS), and collapse prevention (20% reduction) (CP) points on the envelope curve and bilinear curve.

The parametric study results are summarized in Figure 11. The bars are grouped into group of twelve (twelve combinations of Brick and Interface/mortar) for each of the gravity load level. In Figure 11, gravity load levels are mentioned on the top (applied as percentage of total weight of the structure) and for each load level, brick is represented by B (i.e. B7, B10, B13, and B16 stands for 7MPa, 10 MPa, 13 MPa, and 16 MPa respectively as described in Table 4) and mortar is referred as strong, intermediate or weak (as explained in Table 3). The results are then normalized by dividing each group value, by that of structure with the least value. For example, in case of base shear coefficient, the values of each group are divided by the value of weak brick and weak mortar group (i.e. B7-weak mortar). Below the results are explained for each of the structure response metrics: base shear coefficient, effective stiffness, displacement amplification factor, response modification factor and the three performance levels (i.e. immediate occupancy, life safety and collapse prevention).

Figure 11.

Comparison of normalized response metrics: (a) base shear coefficient, (b) effective Stiffness, (c) response modification factor, (d) displacement amplification factor, (e) immediate occupancy level, and (f) life safety/collapse prevention level.

It is seen that the base shear coefficient (which is the ratio of maximum lateral load to the weight of structure) depends on all the three factors that were taken into consideration, that is, brick strength, mortar strength, and gravity load. The base shear coefficient increases with increasing these parameters. For instance, an increase in mortar strength (keeping same brick strength and gravity load level) causes increase in base shear coefficient of the structure that varies from 11% for P/W = 0% group to 19% for P/W = 300% group (see Figure 11). The change in brick strength has a more pronounced effect on base shear coefficient, especially under higher gravity loads. Figure 12 shows the effect of gravity load level on the base shear coefficient of the structure. It can be seen that for same type of brick and mortar, the increase in gravity load level from 0% to 300% causes the increase in base shear coefficient up to 220%. Another conclusion that can be obtained from Figure 12 is that the structure made from weak brick and weak mortar with gravity load level of 300% gives greater base shear coefficient value than the structure made of strong brick and strong mortar but without any gravity load applied on it.

Figure 12.

Effect of gravity load and material properties on Base shear coefficient.

It is observed that the brick strength and gravity load level has almost no effect on the effective stiffness of the structure. It should be noted here that effective stiffness should not be confused with the initial stiffness (see Figure 10) and the initial stiffness of the structure might change by varying brick strength and gravity load. Also, the confinement effect and box behavior of masonry could be the reasons that the effective stiffness does not change by changing the above mentioned parameters. However, the effective stiffness increases with increasing mortar strength. This could be mainly because mortar is the weaker element in masonry and by enhancing its properties causes the global properties of the structure to improve. The increase in effective stiffness of the structure with increasing mortar strength ranges from 101% (for the gravity load level of 0%) to 54% (for the gravity load level of 200%). As mentioned above, the change in effective stiffness of the structure more or less is independent of gravity load level and properties of brick. The mortar properties and gravity load level have a major impact on the response modification factor (R) of the structure. The pattern is uncertain when the gravity load is low, but it becomes more regular for higher gravity loads. For instance, under the gravity load level of 0%, the stronger mortar shows higher values, but when the gravity load level tends to increase (i.e. 19% to 300%) the pattern becomes more regular, and structures with weaker mortar shows higher R factors due to inelasticity governed by the mortar. Thus, following the general trend, it is observed that the increase in R factor of the structure can be as high as 125% due to decrease in mortar strength (see Figure 11). This is mainly due to the fact that, weaker mortar governs the behavior and plasticity of the mortar lead to higher drift value. Thus mortar with high strength results in high effective stiffness and low R factor and vice versa. Figure 13 shows the effect of gravity load level on R factor, by increasing the gravity load level, the R factor decreases for all types of brick and mortar used. The structure becomes more brittle/stiff because the R factor decreases due to increase in the gravity load level. For instance, by increasing the gravity load level from 0% to 300%, the R factor reduces to 37% for the structure with same type of brick and mortar.

Figure 13.

Effect of gravity load and material properties on R factor.

Similar to the case of response modification factor (R), the displacement amplification factor (C_d) also depends on the mortar properties. Although there are exceptions, general trend is that the C_d factor reduced with increasing mortar strength. The increase in C_d factor of the structure can be as high as 113% due to decrease in mortar strength. The brick strength considered here do not significantly affect the displacement amplification factor of the structure. However, the gravity load level affect the C_d here in the same way as we described in the case of R factor. Increasing the gravity load level causes the C_d factor to decrease, hence reducing the plastic displacement of the structure.

It is seen that at higher gravity load, brick strength affects the Immediate Occupancy level and it increases with increasing the brick strength. However, regardless of the level of gravity load, the mortar properties directly influence the immediate occupancy level. The immediate occupancy level increases by decreasing the mortar strength for all the groups. It is observed that the increase in immediate occupancy level of the structure can be as high as 81% due to decreasing the mortar strength. It is mainly because the high strength mortar gives high effective stiffness, therefore the yield displacement would be less, and in contrast, low strength mortar gives low effective stiffness and thus results in higher yield displacement and therefore gives high Immediate Occupancy value. For Life Safety and Collapse Prevention levels the results are shown on a single graph, since life safety is taken as drift level corresponding to the 75% of the drift level of collapse prevention. Therefore, the normalized values for both the performance levels are same and hence shown in one graph. A pattern similar to the response modification factor (R) is observed for life safety/collapse prevention level. It is seen that the mortar strength has the greatest impact on the life safety/collapse prevention levels of the structure. An increase in life safety/collapse prevention level of the structure can be as high as 236% due to decrease in mortar strength (see Figure 11). There are also some exceptions, but general trend is similar where the life safety/collapse prevention value increases by decreasing the mortar strength. It is due to the fact that, the weaker mortar governs the behavior and possess more non-linearity, hence the drift level increases for weaker mortars as compared to the stronger ones. The brick strength and gravity load level also does not have any obvious effect on the life safety/collapse prevention levels. In Figure 14, the effect of gravity load level on the performance levels of the structure is presented. It can be seen that increasing the gravity load level, decreases the ultimate ductility/flexibility of the structure. For instance, for strong brick and strong mortar structure, the performance of the structure reduces to 29% only for the gravity load level of 300%. From above discussion, it can be concluded that under higher gravity load levels the lateral load capacity of the structure increases considerably while its ductility/flexibility reduces significantly, hence the structure becomes more stiff and rigid. While under lower gravity load levels the lateral load capacity decreases but the ductility governs the structure’s behavior.

Figure 14.

Effect of gravity load and material properties on performance levels.

Being a brittle material, the stiffness/strength of masonry decreases, after initiation of crack and its extension and progression towards the middle of solid panel/pier. Mostly, these cracks follow a zigzag pattern and pass through mortar joints, and at some locations pass through the bricks as well, where the compressive strength of brick is relatively less. Figure 15 shows the comparison of damage pattern of in-plane walls of two structures with same brick and mortar properties and different gravity load level. Figure 16 indicates the comparison of damage pattern of in-plane walls of two structures with same gravity load level and mortar properties and different strength of brick. It can be seen that the damage pattern changes by changing the brick strength as well as the gravity load level. In Figure 15(a), where the gravity load level is 19%, the damage in both the left and right piers of front wall is less as compared to the same piers in Figure 15(b), where the gravity load level is 300%. Hence, it can be said that due to increase in gravity load the damage intensity increases

Figure 15.

Damage pattern at different levels of gravity load: (a) P/W = 19% and (b) P/W = 300%.

Figure 16.

Damage pattern with different types of bricks: (a) B-16 and (b) B-7.

Similarly, in Figure 16(a), where the brick is strong, the damage in both the front and back walls is less as compared to Figure 16(b) where brick is weak. Subsequently, in the case of weak brick, the penetration of cracks into bricks, occurs due to increase in tensile stress of units. The damage pattern results are in total agreement with the base shear coefficient graph, where the base shear coefficient increases by increasing the strength of the brick. The parametric study conducted here provides a good understanding of the impact of material load level parameters on the response of confined brick masonry structure with a focus on properties of brick and mortar.

Energy absorption

A quasi-static test results often gives a vibrant indication as to the collapse mechanism, crushing process and properties of energy absorption (Song et al., 2000). The area under the lateral load-deformation curve characterizes the energy that the building will absorb when subjected to an earthquake/lateral loading. Considering this energy is significant, particularly for dynamic loading, and defines the structure’s ductility. The plastic limit analysis can provide no description on the post-peak reduction of the lateral load and the dissipated energy during this process, since the load is constant after the peak, and the energy absorption becomes theoretically infinite. Earthquakes abolish structures by producing waves that propagate through the soil and generate shaking of the structure’s foundation. This energy transfers into the structure and if the structure cannot suitably absorb it, with a proper combination of strength, ductility and flexibility, the structure will fail.

Therefore, structures need to be build in a manner that allows the energy (generated by the earthquake) to be absorbed. The energy absorption is calculated by computing the area under the load-displacement curve up to 75% of the ultimate load after the peak, see Figure 17. In order to evaluate the performance of the structures on the basis of absorbed energy, the parameters discussed in the section 4 were also compared in terms of absorbed energy. Figure 18 presents the comparison of normalized absorbed energy. The results were normalized against the minimum and are presented in the same order to get a clear picture. It is observed that under zero or little gravity load the energy absorption characteristic is uncertain, however by increasing the gravity load the response becomes more regular and enhancing the strength of brick causes the energy absorption of the structure to increase. Conversely, the energy absorption decreases by enhancing the mortar properties because the weaker mortar gives the plane of failure and causing the input energy to be absorbed through this plane of weakness. Also it can be seen that by increasing the gravity load level (i.e. from 100% to 300%) the energy absorption capacity of the structure reduces considerably, hence making the structure more brittle and susceptible of abrupt failure.

Figure 17.

Calculation of energy absorption from load-displacement curve.

Figure 18.

Comparison of normalized energy absorption of structures.

Conclusion

A new modeling methodology known as hybrid-modeling, for analyzing the performance of a full-scale confined brick masonry structure is proposed in this paper. Experimental data from quasi static testing of full scale confined brick masonry structure is utilized in the validation of model. The response of full scale CBM structure in the experimental study is produced and it is observed that the model simulate the lateral load capacity as well as the damage mechanism of the structure very well. It is seen that the proposed numerical tool gives close prophecy of the experimental behavior under given loading and boundary conditions.

The dependence of the structural response metrics (i.e. base shear coefficient, effective stiffness, displacement amplification factor, response modification factor and three performance levels (immediate occupancy, life safety and collapse prevention) and energy absorption), on the properties of brick and mortar is investigated through parametric analysis of full scale CBM structure. It is concluded that the material properties have a huge impact on the structural metrics, mentioned above. The base shear coefficient and effective stiffness may increase more than 128% and 259%, respectively, with increasing the strength of brick and mortar. Alternatively, other structural response metrics such as, response modification factor (R), displacement amplification factor (C_d), Immediate occupancy level and life safety/collapse prevention levels may increase more than 125%, 113%, 81%, and 236%, respectively, with decreasing the mortar strength. A similar response was obtained in case of energy absorption where increasing the brick strength causes the absorbed energy to increase while increasing the mortar strength causes the absorbed energy to decrease. The material properties used in the study (parametric analysis) are not hypothetical, but based on some studies mentioned before. Although the numbers are specific to the structural configuration considered here, the parametric analysis provides a good understanding of the goals that can be achieved in terms of performance exaltation at the structural level by enhancing the material properties, in this case brick and mortar. The effect of gravity load level is also investigated and it is concluded that increasing the gravity load level increases the lateral load resistance of structure considerably but decreases the ductility and hence the performance of the structure. So the effect of gravity load level should be taken into account while designing the structure for seismic forces, because structures with same material properties and subjected to different gravity load levels will perform in different ways (in terms of behavior and performance) during a seismic event, depending on level of gravity loading. The damage pattern for this type of construction is also compared in the study. It is concluded that in relatively weak bricks, the cracks may propagate through the bricks and cause reduction in its lateral load capacity in terms of base shear coefficient and also the stiffness of the structure. However, stronger bricks force the crack propagation, primarily through the mortar joints.

Footnotes

Acknowledgements

The authors are greatly thankful to Higher Education Commission of Pakistan (HEC) for providing funds to purchase the Atena software. The authors are also grateful to Dr. You Dong from Hong Kong Polytechnic University for his support and help.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

H Asfandyar Ahmed

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