Behavior of unreinforced dry-jointed masonry walls at corners under lateral loads

Abstract

This study is based on the simultaneous application of in-plane and out-of-plane seismic loads on unreinforced dry masonry (URM) walls. The mode of failure under such a type of loading is the separation of the corner portion from the rest of the structure. This type of failure generally occurs in old masonry structures in which the strength of the mortar has deteriorated, and the floor of the structure is weakly coupled with the rest of the structure. Such structures generally lack the box-type behavior of masonry buildings. Single-story dry-stacked masonry models were considered and subjected to bidirectional loading to study such failures. These masonry models were developed and analyzed using the finite-element (FE) software package Abaqus, and results were verified using experimental testing of the half-scaled model. Simplified corner failure mechanisms and limiting acceleration formulation were proposed based on the observed failure pattern.

Keywords

Masonry corner failure seismic limit analysis shake table testing

Introduction

Masonry structures have always been the most widely used type of construction around the globe. The seismic vulnerability of unreinforced masonry structures continues to remain one of the major concerns in earthquake-prone regions (Asteris et al., 2014; D’Ayala, 2013). Various past studies are based on the in-plane (IP) or out-of-plane (OP) stability of masonry walls under seismic loading (Agnihotri et al., 2013; Doherty et al., 2002; Komaraneni et al., 2011; Magenes and Calvi 1997). Generally, the study approach to analyze the unreinforced wall is to first subject it to the IP drift levels and then test in the OP direction under seismic loading (Agnihotri et al., 2013). The analytical approach includes modeling the IP crack in unreinforced dry masonry (URM) walls and then analyzing the OP failure modes (Doherty et al., 2002). There are various methods available to study the unreinforced masonry structure like applied element methods (Malomo et al., 2018), distinct element methods (Caliò and Pantò, 2014; Lemos, 2019), rigid body method (Casolo, 2004), and so on. The comparison of these methods is beyond the scope of this study. Many recent studies deal with reproducing the combined effect of IP and OP forces (Malomo and DeJong, 2021; Vanin et al., 2020). However, the experimental work on combined IP and OP loading effects on URM building walls is limited. Hence, there is a need to study the effect of simultaneous application of both IP and OP forces on masonry structures.

At corners of a building, the mode of failure of the unreinforced masonry structures under combined IP and OP seismic loading is generally due to the loss of equilibrium of a particular portion of the structure. Under such circumstances, a block or a portion of the masonry structure tends to detach from the rest of the structure. The crack patterns that develop in such type of failure depend on factors such as the direction of loading, vertical loads, and size of blocks. Generally, such failures occur when the charecteristic box-type behavior is lacking due to weak joint strength between slab and walls. The deterioration of the mortar strength in the structure leads to such a mode of failure. Previous studies (Casapulla and Maione, 2018, Casapulla et al., 2019; D’Ayala and Speranza, 2003) on this subject area suggest that there are two modes of failure mechanisms that exist in such type of loading conditions in unreinforced masonry structures. The two failure mechanisms are the rocking-sliding mechanism and the horizontal flexure mechanism. The rocking-sliding mechanism involves the combination of sliding and rocking, which occurs about the corner point of the building. The horizontal flexure mechanism occurs when the corner portion of the wall rotates about the developed crack and the corner vertical joint. The failure occurring with a combination of two modes causes the corner failure of the unreinforced masonry. The above-mentioned failure mechanism is commonly observed in past earthquakes, as reported in previous studies (Adanur, 2010). This failure mechanism also exists in buildings made of lightweight concrete briquette (Adanur, 2010).

In this study, the effect of bidirectional (IP and OP) lateral loads on masonry structures was studied. When masonry structures are subjected to such type of loading, damages are generally observed at the corners of the structure where two orthogonal walls meet each other. In such type of failure, a wedge-shaped portion of the corner tends to detach from the rest of the masonry structure by sliding and overturning. Such failure is generally observed in either stand-alone structures or those located at the end of a row of buildings, as there is a lack of restraints in such situations to prevent corner failures (Bokey and Pajgade, 2004; Kawashima et al., 2010; Rai et al., 2012; Shrikhande et al., 2000). Some of such failures observed in past earthquakes are shown in Figures 1a to c.

The main objective of this study is to develop a better understanding of the corner failure mechanism in masonry structures and get an estimate of the corresponding limiting failure accelerations. Finite-element (FE) analysis and experiments were performed on the dry-stacked masonry to study corner failure. These failures are often seen in old masonry structures where the mortar strength is already degraded over time. Masonry with poor quality mortar, such as mud mortar, performs rather poorly during seismic loading (Erdik, 1990), suggesting that their behavior becomes almost equivalent to dry-stacked masonry. In this study, the wall panels of the orthogonal corner walls were of equal size. Due to this symmetry, the loading condition was kept along in the direction of corners, as shown in Figure 2a; this ensured the equal IP and OP forces on the orthogonal walls. The direction of loading makes an angle of θ_f and θ_s with the façade and sidewall, respectively, as shown in Figure 2b.

FE modeling

Masonry structure was modeled in the Abaqus (SIMULIA 2014), an FE software. Brick elements were modeled separately and were later used to create the masonry walls. Brick masonry was modeled using the concrete damaged plasticity (CDP) model in Abaqus, which is used to model concrete-like materials. In compression, a simple trilinear stress–strain curve (Kaushik et al., 2007a and b) was adopted, as shown in Figure 3a, and in tension, the peak tensile strength of the material was considered as 10% of peak compressive strength (Akhaveissy and Desai, 2011), as shown in Figure 3b. These models correspond to the prism strength of masonry. However, in this case, dry-stacked masonry is used, and failure is governed by the separation of bricks rather than crushing; hence it is conservative in adopting the above material models.

The FE model consists of two different units: a slab and a brick. The brick size in the FE model and in experiments was 120 mm × 60 mm × 37.5 mm. Eight-noded C3D8R type element was used with mesh size one-sixth of the size of a brick. The frictional coefficient of 0.6 was introduced between the brick layers as well as between slab and bricks. This frictional coefficient value is in the range of the typical value of the brick-to-brick friction coefficient (D’Ayala and Speranza, 2003), and it was also verified by sliding two bricks on top of each other and measuring the angle of inclination at which the sliding occurs. Both the top and bottom slabs were modeled as homogeneous, isotropic, and elastic concrete materials. The bottom slab was restrained in the vertical direction and allowed to move in both horizontal directions. In order to generate the inertial force in the structure, half-sine wave motion was applied on the bottom slab, as shown in Figure 4. Dynamic Explicit solver in Abaqus was adopted to analyze the structure. In the explicit solver of Abaqus, no additional material damping was introduced in the model. By default, the Abaqus solver includes the numerical damping in the form of Bulk Viscosity.

Experimental setup

Shake table tests were performed on various masonry models to verify the results obtained from the FE analysis. The corner failure mechanism obtained from the FE analysis was verified experimentally and compared with FE results. An experimental test setup was developed at the Structural Engineering Laboratory, IIT Kanpur. The uniaxial shake table used in the study can move the payload at a maximum displacement of ±75 mm and has an operating frequency range of 0–50 Hz (Sinha and Rai, 2009). The complete experimental assembly is shown in Figure 5a.

The masonry structure was placed on the bottom slab, which was tightly screwed to the shake table to avoid the relative movement between the shake table and the model during the loading. Half-scale bricks of size 120 mm × 60 mm × 37.5 mm were stacked in the form of a simple stretcher bond. Each layer consisted of 28 bricks, and 23 layers were stacked over each other. Details of the masonry model are given in Table 1. Test runs were performed by increasing the weight on top of the model. The weight was increased by attaching concrete prisms above the top slab with plaster of Paris, as shown in Figure 5b.

Table 1.

Details of specimens (experimental setup)

Property	Values
Wall size (m)	0.9 × 0.9
Dimension of brick (mm)	120 × 60 × 37.5
Size of slab (m)	1.2 × 1.2 × 0.075
Density of brick (kg/m³)	1750

Instrumentation and nomenclature

Two sensors, an external accelerometer and a linear variable differential transformer (LVDT) were installed on the shake table to monitor its motion variables due to the applied base displacement. Four cameras were installed at various angles around the shake table to observe the displacement and crack pattern of the masonry structure during the loading phase. Figure 6 shows the position of sensors and cameras around the shake table.

Corners are categorized into two sets, “in-line corners” and “orthogonal corners,” as shown in Figure 7a. In order to describe the deformation and cracks in walls and corners, a frame of reference is defined with respect to the observer standing in the middle of the structure facing the direction of inertial forces experienced by the structure, as shown in Figure 7b. The corner and walls in front of the observer were named “front corners” and “front walls,” respectively. Similarly, the corner and walls behind the observer were identified as “rear corner” and “rear walls,” respectively.

Loading protocol

It was desired to determine one cycle response of the structure in which it would experience a forward inertial force and then return to its original place in a very short span of time. The half-sine wave of the time duration of 0.167 s was applied in the form of displacement time history as the lateral load to simulate the impulse loading effect on the masonry model. Loading was applied in four stages: at each stage, maximum displacement amplitude was increased by 5 mm, starting from 10 mm and going up to 25 mm. The comparison of the command signal and observed signal (LVDT response) from the shake table for all the loading steps are shown in Figure 8. Peak acceleration corresponding to each impulse is also shown in Figure 8.

Comparison of crack pattern (FE vs Experimental model)

Two models were considered in the study. In the first model, only one slab was used as a dead load, and in the second model, the dead load was increased by adding concrete prisms over the top slab. The vertical pressure on walls in both models is given in Table 2.

Table 2.

Vertical pressure on masonry models

Model	Vertical pressure (kPa)
Model 1 (one slab)	14.2
Model 2 (increased dead load)	28.5

Model 1 (one slab on top)

The crack pattern observed in Model 1 from FE analysis and experimental results is shown in Figure 9. The deformed shape of the model shown in Figure 9 is after the final step loading, that is, 25 mm peak displacement loading. The arrow in the figure represents the direction of input acceleration, and inertial forces experienced by the structure will be opposite to the direction of the arrow.

It can be observed from the above figure that a wedge-shaped portion of the masonry tends to detach from the rest of the masonry structure at the front corners. In front walls, diagonal cracks were observed in the structure, which became vertical at mid of the wall. The crack pattern in the masonry walls depends on the vertical load acting on the walls, and these cracks in the walls tend to become vertical when the downward load acting on the wall is small. It can be observed from both FE and experimental models that the cracks are vertical around the mid-portion of the wall, which is due to less vertical pressure as there is only one slab on top. The crack pattern of rear corners and rear walls in the FE and experimental models are also shown in Figure 9c and d. In rear walls, the crack pattern was diagonal in masonry walls, and inward bulging of the corner was observed as shown. The overall failure pattern observed from the experimental model was found to be identical to the FE analysis for both front and rear corners.

Model 2 (increased dead load)

The crack pattern observed in Model 2 from FE analysis and experimental results is shown in Figure 10.

A wedge-shaped portion tending to detach from the rest of the structure can be observed in Figure 10a and b. In front walls, the cracks are more inclined as compared with Model 1, as shown in Figure 10a and b; this is because of the increase in the amount of the dead load on the walls due to the increase in weight on top of the structure. The crack pattern in the rear corners is similar in experimental and FE results, as shown in Figure 10c and d. Inward bulging of walls at the location of rear corners can be easily observed in both FE and experimental results. The resulting crack pattern in both the front and rear walls indicates that the rocking-sliding mechanism and the horizontal flexure mechanism are mobilized in the wall. The diagonal cracks developed in the wall indicate the rocking-sliding mechanism, and the rotation of the bricks along the crack indicates the horizontal failure mechanism.

The crack pattern shown in Figures 9 and 10 between FE and experimental is not fully identical, and there are minor discrepancies between the movement of bricks of the FE model from the experimental model. The reason is that during the placement of the top slab on the model, the top surface of the bricks is uneven, which leads to uneven distribution of vertical dead load on the model. The uneven distribution of vertical pressure leads to more movement of the bricks in the experimental model at some locations. Apart from that, there may be some initial minor movements of bricks during the placement of the top slab.

OP and IP displacement of walls

OP and IP displacement in the front and rear walls were measured to compare the deformed state of FE and the experimental model and to get better insight into the deformation pattern of walls. Location with significant IP and OP damages were identified in masonry walls. Measurements were performed on the final deflected shape of the experimental and FE model, that is, after 25 mm peak displacement loading. Comparisons for OP and IP displacement are shown for Model 2, that is, the model with two slabs on top. OP displacement and IP displacement of walls for Model 2 are shown in Figure 11.

OP displacement

OP displacements were measured using a laser distance meter. Readings were taken for all the bricks of 23 layers; however, to show the trend of displacement, only a few readings are shown, that is, from the top three layers and bottom two layers. Layers were named from bottom to top, that is, the bottommost layer is L1, and the topmost layer is L23.

Front walls

OP displacement for the front wall observed for experimental Model 2 and predicted for the FE model is shown in Figure 12. In the figure, it can be observed that two types of failure mechanisms are there in the front wall. The detached part is undergoing IP sliding and rocking and is also undergoing flexure OP mechanism at some locations along the developed cracks. These flexure cracks are indicated by OP movement in Figure 12. OP displacements of all the bricks for all the layers of the front walls in Model 2 were measured from the final deformed shape of the experimental model and compared with the corresponding displacement values from FE results. Figure 13 shows the measurement for OP displacement for the experimental and FE models (Model 2) for the top and bottom layers of one of the front walls. Dashed curves show experimental values, and Abaqus values are shown by the solid curve. Negative readings show the inside movement of bricks in the structure, and positive readings show the outside movement of bricks from the initial alignment of the wall.

The comparison of OP displacement of bricks of front walls between FE and experimental models is shown in Figure 13. The displacement readings for the top layers are shown in Figure 13a, and those of the bottom layers are shown in Figure 13b. In the top portion of the front wall, maximum OP displacement was in the left part of the wall (Figure 13a), and in the bottom portion, maximum OP displacement occurred in the right part (Figure 13b). It can be observed that the OP displacement is smaller in the bottom layers than in the top layers due to larger vertical pressure for the bottom layers. Moreover, it can be observed that the trend of OP displacement of both the walls matches with FE values, that is, the occurrence of maximum OP displacements is observed at the same locations. There is a slight difference between experimental and FE values, especially in the top layers, which could be due to slightly loose contact between the top bricklayer and the bottom surface of the slab in some places due to the uneven surface of bricks.

Rear walls

The observed deflected shape of the rear wall of the experimental model is compared with the prediction of the FE model, as shown in Figure 14. Measurement for the OP displacement of the experimental and FE model for one of the rear walls is shown in Figure 15. It can be observed that the trend of displacement of bricks is similar in the FE and experimental models.

The displacement readings for the top layers are shown in Figure 15a, and those of the bottom layers are shown in Figure 15b. Maximum OP displacement in the top layers was observed in the left portion of the wall where the bricks are moving inside of the structure. Maximum OP displacement in the bottom layers was observed in the right portion of the wall. The locations of maximum OP displacement in the FE and experimental results are slightly different in the top layers, but the overall trend matches.

IP displacement

IP displacement reading of the walls was measured using the image processing technique using a computer application named “Image J” (Schneider et al., 2012) on the deformed image of the wall. Inter-joint space was calculated for the experimental and FE models to estimate the IP displacement. There are seven joints in each layer; the relative displacement between two bricks, that is, the width of each joint was measured in each layer. These joints were numbered from left to right. The width of these joints represented the IP displacement between bricks and was plotted against the joint number. The location and values of IP displacements were compared between the FE and experimental models.

Front walls

A comparison of the IP crack pattern in the experimental model and contours for IP displacement from the FE model for the front wall is shown in Figure 16a and b. As explained earlier, the nature of the cracks in Model 2 is diagonal as compared with Model 1, in which the cracks were vertical. This is due to increased vertical load on top of the structure. In Figure 16, it can be observed that the diagonal cracks are present at both the diagonal corner of the wall, and vertical cracks are observed in the middle portion of the wall. An identical trend of the IP crack pattern was observed between the FE and experimental results.

Measurements for the IP displacement of the experimental and FE model for the front wall are shown in Figure 17. The displacement readings for the top layers are shown in Figure 17a, and those of the bottom layers are shown in Figure 17b. IP cracks in the wall are a combination of diagonal cracks and vertical cracks. From Figure 17a, it can be observed that the width of the crack is the maximum in the left portion at the top layers of the wall. From Figure 17b, it can be observed that the crack width is maximum in the right portion in the bottom layers in both the experimental and FE models.

Rear walls

IP displacements of bricks were also plotted for rear walls, and similar trends were observed from the FE and experimental models, as shown in Figure 18. Diagonal cracks can be observed in both the experimental and FE models. Corresponding measurements of IP displacement for FE and the experimental model for the rear wall are shown in Figure 19. It can be observed from Figure 19a that for the top layers, the FE readings display the maximum IP displacement at the left portion, which is also the case with experimental measurements. In the bottom layers, the maximum IP displacement was observed at the right portion of the wall in both the experimental and FE measurements, as shown in Figure 19b.

Limiting acceleration for corner failure in dry-stacked masonry structure

Corner failure

The failure modes of the masonry structure are governed by IP and OP forces. However, when the walls of the masonry structure are subjected to both IP and OP forces simultaneously, the governing forces that decide the failure of walls depend on the vertical loads on the walls. When the vertical load is much less on the walls, the OP forces dominate, and if the vertical load is substantial, then the IP forces are the governing forces. In this study, two models were studied in which the dead load was varied on the top of the structure. In Model 1, only one slab was placed on top, and in Model 2, the dead load was further increased by adding concrete beams on top. The dead weight on Model 1 is very less; hence it behaves similar to a non-load-bearing structure, and the failure pattern in the walls is also similar to non-load-bearing walls. However, with an increase in the weight on the top in Model 2, a significantly different type of failure pattern was obtained governed by IP forces. The failure consists of IP cracks in two orthogonal walls and their movement as a whole under the effect of inertial forces. This failure mode is referred to as corner failure.

The FE and experimental results showed two types of corner failure in the structure (Model 2), as shown in Figure 20. In the first type of mechanism, due to the effect of inertial forces, the corner wedge was coming out of the rest of the masonry structure, as shown in Figure 20a. In the second type of mechanism, the corner wedge moved inside the masonry structure, as shown in Figure 20b. Both failure modes resulted in more or less diagonal crack patterns in the walls.

There is a chance of overturning of orthogonal front walls, as shown in Figure 20a. Limiting acceleration values for initiation of such type of failure mode were derived through understanding the crack pattern and considering all types of forces experienced by the walls.

Simplified crack pattern

In dry masonry walls, two types of crack formation were observed in FE analysis: one is stepped crack, which is in the form of a staircase pattern, and the second type is interlocking-type crack which is vertical in nature, as shown in Figure 21a.

Initially, under the effect of inertial forces developed in the slab, the bricks beneath the top slab form stepped cracks. Two stepped cracks are connected by another crack termed an interlocking crack. It can be seen from Figure 21a that not all cracks within a section are stepped or interlocked, and these cracks are distributed along the diagonal. However, the crack pattern was assumed to be uniform, as shown in Figure 21b, in order to simplify the process of calculating the limiting acceleration. In this pattern the stepped cracks will originate from the opposite diagonal end and will propagate up to the half-length of the wall and thereafter be joined by the vertical interlocking crack resulting in failure mechanism as shown in Figure 21b (Bansal and Rai, 2017). The angle of the wall panel is named as $α_{p}$ and the angle of stepped crack is named as $α_{b}$ .

Analytical equation for corner failure in load-bearing masonry structure

The crack pattern discussed above was adopted to derive the analytical equation for the overturning of the corners. The crack pattern shown in Figure 22a is the assumed cracked pattern in which all the interlocking cracks are expected to occur in the middle portion of the wall. This makes it easier to calculate the total frictional resistance force against the corner failure.

A schematic diagram of the corner failure mechanism is shown in Figure 22a. The unshaded portion of the model tends to detach from the rest of the masonry structure by overturning about the axis C-C’ The two walls at the corners are named the façade wall and sidewall, represented by the subscript f and s, respectively. Furthermore, the detaching portion of each of the orthogonal walls can be divided into a triangular, a rectangular, and a trapezoidal section; parameters related to these sections are represented by subscripts t, r, and tr, respectively, as shown in Figure 22a. The expression for limiting acceleration for the failure mechanism shown in Figure 22a can be derived by equating the resisting and overturning moment about the rotating axis C-C’ The angle made by the overturning axis with façade and sidewall is represented by $θ_{f}$ and $θ_{s}$ , respectively, as shown in the plan view in Figure 22b. The overturning moment on the failure (wedge) portion due to inertial forces is resisted by the stabilizing moment due to the weight of the side and façade walls, along with the resistance due to the interlocking friction force C in the vertical crack.

The overturning forces experienced by the corner portion are as follows:

Overturning moment of the failure portion about C-C’ due to inertial force is given as follows:

\frac{a_{\lim}}{g} [W_{s} h_{sc} + W_{f} h_{fc}]

(1)

Overturning moment due to lateral load acting on the dead weight on top of masonry is given as follows:

\frac{a_{\lim}}{g} [w_{ds} L_{s} h_{w} + w_{df} L_{f} h_{w}]

(2)

$W_{s}$ and $W_{f}$ are the corresponding weights of the failure portion (unshaded part) of the side and façade walls. $w_{ds}$ and $w_{df}$ represent the vertical dead weight per unit length acting on the walls. The frictional resistance force will consist of the effect of shear resistance between the interlocked bricks, as shown in Figure 22c, as well as the effect of dead load and weight of bricks directly above the interlocked section. The overall equation of shear resistance provided by vertical interlocking cracks in the plane of the wall is given by Equation 3, where n (equal to h_tr/t_b) is the number of bricks in the interlocked section. Here t_b is the thickness of one brick and $W_{b}$ is the weight of one brick. The first part of Equation 3 represents the total frictional force of the interlocked brick portion, as shown in Figure 22c. The second part of the equation is an increase in frictional force due to the weight of bricks just above the interlocking section. The third part of the equation is increase in frictional force due to the contribution of vertical dead load on top of the interlocking portion:

C = μ \frac{n (n + 1)}{2} \frac{W_{b}}{2} + μ n \frac{h_{tr}}{t_{b}} \frac{W_{b}}{2} + μ n w_{df} \frac{l_{b}}{2}

(3)

These forces exist in the plane of walls; hence a component of this force in the direction of loading has been taken in the final equation. The lever arm for the moment created by this frictional force is the vertical distance from the bottom of the structure to the centroid of the force polygon of frictional force. $C_{s}$ and $C_{f}$ represent the total resisting frictional force in the interlocking section in the side and the façade wall, respectively. Therefore, the total resisting moment due to interlocking in the side and façade walls about C-C’ is given as follows:

C_{s} h_{isc} + C_{f} h_{ifc}

(4)

The resisting moment is also provided by the self-weight of the walls and the dead weight of the load on top of the masonry. The total resisting moment in the plane of the wall due to the self-weight of the wall and dead weight on top is given as follows:

W_{f} l_{fc} + w_{df} \frac{L_{f}^{2}}{2}

(5)

Equating the overturning and resisting moment, we get an expression for limiting acceleration (a_lim) as follows:

\frac{a_{\lim}}{g} = \frac{[C_{s} h_{isc} + W_{s} l_{sc} + w_{ds} \frac{L_{s}^{2}}{2}] \cos (θ_{f}) + [C_{f} h_{ifc} + W_{f} l_{fc} + w_{df} \frac{L_{f}^{2}}{2}] \cos (θ_{s})}{[W_{s} h_{sc} + w_{ds} L_{s} h_{w}] + [W_{f} h_{fc} + w_{df} L_{f} h_{w}]}

(6)

The numerator of Equation 5 represents the resisting force, and the denominator represents the overturning force.

Failure acceleration (limit analysis vs FE analysis)

Limit analysis

The limiting failure acceleration equation was developed for front corners as explained earlier. Limiting accelerations were calculated for Model 1 (one slab) and Model 2 (increased dead load) using Equation 5. Various parameters like the weight of failure portion and resisting force were calculated for both models to calculate the limiting acceleration. Numerical values of these parameters are given in Table 3. In all the models, the values of all the input parameters for side and façade walls are equal in magnitude because of the same size of both the walls; hence only one set of values (for façade wall) is provided in Table 3. Limiting acceleration for both models obtained from Equation 5 is given in Table 4. It can be observed from Table 4 that the limiting failure acceleration value for both models are close to each other. However, the difference can be more significant if the weight above the top slab is increased, as it will create a larger overturning moment about the base of the corner.

Table 3.

Input parameters to calculate limiting acceleration

Input parameters	Values
	Model 1	Model 2
$W_{f}$ (Weight of failure portion)	400 N	400 N
$C_{f}$ (Resisting frictional force)	430 N	676 N
$w_{df}$ (Vertical dead load)	855 N/m	1710 N/m

Table 4.

Limiting failure acceleration for both models

Model	Limiting failure acceleration
Model 1 (one slab, 14.2 kPa)	0.55g
Model 1 (increased dead load, 28.5 kPa)	0.52g

FE analysis

IP displacement of nodal points on either side of the cracks was plotted with time to determine the limiting failure acceleration from FE analysis. Failure is assumed to initiate when the relative movement between the adjacent bricks occurs. Nodal points were chosen from the adjacent bricks at the location of the first appearance of the crack, that is, just beneath the top slab where the stepped crack first appears, as shown in Figure 23.

For both models, two nodal points from either side of the IP crack were chosen from the same location as shown in Figure 23, and the IP displacement of these nodal points was plotted with time. A plot of IP displacement for both models is shown in Figure 24. It can be observed from Figure 24 that the relative displacement between the bricks began to increase significantly after the second impulse loading, corresponding to which the value of peak acceleration was 0.54 g. Since this input acceleration is equal to or more than the failure acceleration calculated from limit analysis for all models, therefore, the initiation of failure started after this acceleration level for both the models. It was observed that the failure acceleration obtained from the limiting analysis and the FE analysis are in good agreement with each other.

Comparison and discussion

The limiting acceleration obtained from the formulation of the two studies was compared with the results of the current study. The results are obtained for three models with the varying vertical dead load. The results for all the cases are shown in Table 5. The calculations of the limiting acceleration for all the studies are shown in Appendix 1.

Table 5.

Limiting failure acceleration from different formulation

Model	Limiting failure acceleration from the current study	Limiting failure acceleration (D’Ayala and Speranza, 2003)	Limiting failure acceleration (Casapulla and Maione, 2018)
Model 1 (one slab, 14.2 kPa)	0.55g	0.52g	Not applicable
Model 2 (increased dead load, 28.5 kPa)	0.52g	0.53g	Not applicable
Model 3 (no vertical load)	0.7g	0.50g	0.92g

It can be observed from Table 5 that the limiting acceleration is overestimated for the proposed model and the model by Casapulla (2018) for the case with no vertical dead load (Model 3). This is because it is an upper bound approach, and the walls tend to fail in OP mode before forming the corner failure mechanism. This behavior is also observed in the study by Casapulla and Maione (2018), as the horizontal flexure mechanism has a lower load factor. The case with the presence of vertical roof load shows a similarity to results the from D’Ayala’s (2003) study. As per the formulation of this study, with an increase in the vertical dead load on the walls, there is a corresponding decrease in the limiting failure acceleration due to enhanced overturning moment demands on the walls. However, this is not the case in the D’Ayala and Speranza (2003) formulation, as the weight of the vertical dead load is included in the weight of the wall.

Conclusion

This study investigated the corner failure mode of masonry structures, which has been observed in previous earthquakes, especially in the isolated and independent units. This distinct failure mode of the two orthogonal walls of the corner tends to detach from the rest of the structure. The failure mode was simulated in the shake table testing when the masonry structure was subjected to impulse loading in the diagonal direction. Deformation of the wall was found to be dependent on the vertical dead load. The smaller dead load caused the crack to move in the vertical direction, which makes the wall prone to OP failure prior to the corner failure mechanism formation. In the corner failure mechanism, the OP flexure mechanism was also observed along the crack, which was also seen in a few previous studies. A similar failure mode was also observed in the numerical FE study. Moreover, the experimentally observed pattern of the IP and OP displacement of the bricks in the walls are in good agreement with the prediction of the FE model. An analytical model was developed for estimation of the limiting base acceleration corresponding to the corner failure mode based on the crack pattern observed in experimental and numerical evaluation. The limiting failure acceleration obtained from the analytical model compared well with the FE analysis values. The results presented in this study can be included in the seismic design of masonry structures and devise appropriate measures to minimize the risk of such types of failures by introducing reinforcement techniques against such types of failures.

Footnotes

Appendix 1 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.

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