In this study, it was considered that the strength of the masonry wall could be increased by using interlocking blocks with a shape that is unlikely to cause stress concentration. To confirm this, the four types of walls shown in Figure 1 were prepared. These are the I-shaped block wall (right angle type), I-shaped block wall (obtuse angle type), hourglass-shaped block wall (linear type) and hourglass-shaped block wall (wavy type). The dimensions of the wall are 18 cm in height, 18 cm in width and 10 cm in depth. The walls are composed of four blocks: upper, middle left, middle right, and lower blocks. Considering that the failure phenomenon will become complicated if the wall consists of many blocks, the wall was constructed with only four blocks to simplify the failure phenomenon.

In a past study Furukawa et al. (2018a), masonry walls were made from burned bricks, and plaster was placed between the adjacent bricks considering the masonry structures in developing counties. However, it was observed that the variations in the material properties and strengths of individual blocks and the variations in mortar bond strength also affected the experimental results.

In order to investigate the pure influence of the block shape, it is necessary to minimize the influence of factors other than the block shape, such as variations in the material properties and strengths of individual blocks. We prepared a formwork for each block, cast at the same time and treated under the same conditions in order to minimize the difference other than the shape. Seventy percent of the material was sand and 30% was Portland cement.

In addition, the plaster was not used between blocks to minimize the influence other than the block shape since it is difficult to realize the constant bond strength throughout the wall. The adjacent blocks are just in contact with each other.

The burned bricks and plaster used in the developing countries were not used to avoid the effect other than the block shape.

Two support conditions were considered, the glued and contact conditions. The glued condition means that the wall was glued to the loading and fixing jigs with plaster. The contact condition means that the wall was just in contact with the loading and fixing jigs. Since the size of the wall is small, it was considered that the effect of the support conditions is significant. So, it was decided to treat both support conditions.

The I-shaped block wall (right angle type) actually has no I-shaped blocks as shown in Figure 1. The top and bottom blocks have the half shape of the I-shaped blocks. If the I-shaped block is cut in half vertically in the middle, then the shape of the top and bottom blocks are obtained. The middle left and the middle right blocks also have the half shape of the I-shaped blocks. If the I-shaped block is cut in half horizontally in the middle, then the shape of the middle left and right blocks are got. The I-shaped block wall (obtuse angle type) has an obtuse angle instead of a right angle. The curve of the hourglass-shaped block (wavy type) is designed using the cosine function.

Table 1 shows the material properties and the strength of the mortar block estimated by the element test using a cylinder test piece made simultaneously with the interlocking blocks. Figure 2 illustrates the experimental setup for element tests. Each element test was conducted three times, and the average values were taken. The compression test shown in Figure 2A was performed to obtain Young’s modulus, Poisson’s ratio and compressive strength. The Young’s modulus was obtained from the vertical stress-strain relationship, in which the stress was obtained by dividing the force by the area. The Poisson’s ratio was obtained by taking the ratio of the horizontal strain to the vertical strain. The maximum stress of the stress-strain relationship was defined as the compressive strength. The tensile strength was obtained by the 4-point bending test shown in Figure 2B. The friction coefficient between the blocks was obtained by the double-sided shear test shown in Figure 2C. The shear stress when the sliding occurs was obtained for three normal stress of 0.1, 0.2, and 0.3 MPa. Friction coefficient was then obtained from the relationship between the shear stress and normal stress.

Diagonal compression tests were performed on four types of masonry walls, and the load-displacement relationship and the failure behavior were examined. As shown in Figure 3A, V-shaped jigs were mounted on the upper and lower loading plates of the universal testing machine, and a masonry wall was installed between the jigs.

There were two types of support conditions between the V-shaped jig and the masonry wall, i.e., glued condition and contact condition. In the glued condition, quick-drying cement was placed between the brick and the jig. The number of test specimens for each support condition is shown in Table 2.

The upper jig was fixed and the lower jig was moved up at a rate of 2.5 × 10^–3 mm/s. The load in the loading (vertical) direction was measured. The experiment of the glued condition was conducted first, and the failure behavior during loading was recorded using a digital camera. In the experiment of the contact condition that was conducted afterward, the failure behavior during loading was measured using two high-resolution digital cameras for image measurement with a digital correlation system (Correlated Solutions, 2020). The digital correlation system uses two high-resolution digital cameras to record the deformation of the surface of an object and tracks the movement of a point of interest in the object. This method enables measurement of strain distribution. A mottled pattern was applied to the masonry wall with a commercially available spray (Figure 3B). The point of the spray was tracked by the digital correlation system. The digital correlation system itself has very high-performance, but the pictures were taken every 5 s due to the limitation of the capacity of the laptop, so the pictures of the very moment when the failure occurred could not be obtained.

The left top figure of Figure 4 shows the load-displacement relationships of I-shaped block walls (right angle type) for glued and contact conditions. Three test specimens showed a similar load-displacement relationship with two peaks and the effect of the support conditions was negligible. The two peaks correspond to the failure of two parts.

The top two figures on the right side of Figure 4 show the normal strain distributions in the horizontal direction and how the test specimen with the contact condition became fractured. The positive strain means the tensile strain. The red color indicates the maximum value and the purple color indicates the smallest value. At the first peak of the load-displacement relationship, the tensile strain concentration occurred in the bottom block as indicated by a gray circle in the top figure on the right side. Then this part fractured. The wall then stabilized and the load increased again. At the second peak of the load-displacement relationship, the tensile strain concentration occurred in the top block as indicated by the gray circle in the second top figure on the right side. Then this part fractured and the total collapse occurred. It was found that the first and the second peaks in the load-displacement relationship correspond to the failure of the bottom and top blocks, respectively. The failure occurred where the tensile strain was concentrated. The tensile strain concentration started from the corner at 270 degrees of the top or bottom block. The failure was caused by the tensile strain concentration at the interlocking part. On the other hand, in the middle blocks, no failure occurred at the corresponding 90-degree corner. This is because strain concentration is more likely to occur at corners larger than 180 degrees than at corners smaller than 180 degrees.

The bottom two figures of Figure 4 indicate how the failure occurred in the glued condition. The digital correlation system was not available when the experiment of the glued condition was conducted. It was found that the influence of the support condition on the failure process of the I-shaped block walls (right angle type) is small and all three test specimens showed almost the same results.

The failure process of the I-shaped block wall (right angle type) is summarized as follows. The load increases as the displacement increases until the first failure occurs. The first failure occurs in either the upper or lower block where the strain concentrates, and the load decreases rapidly. After that, the wall stabilizes again, the load increases again as the displacement increases, and then the other block fractures. After the second failure, the stability is lost, the load decreases to almost 0 and total structural collapse occurs.

The left figure of Figure 5 shows the load-displacement relationship of I-shaped block walls (obtuse angle type) for glued and contact conditions.

Two test specimens with the glued condition showed two peaks even though the second peak is very small. The right bottom figure shows the snapshot of the test specimen just after the second peak. The first peak in the load-displacement relationship occurred just before the failure of the top block and the small second peak occurred just before the failure of the bottom block. Since the wall lost stability after the second failure, the total collapse occurred, and the load reached almost 0.

On the other hand, the test specimen with the contact condition showed only one peak. The right top figure shows the strain distribution of the test specimen in the contact condition. The tensile stress concentration can be seen in the top block. After the fracture of the top block, the total collapse occurred. Since the top and bottom blocks are not glued to the jigs, the model is more unstable. So the total failure occurred just with the fracture of the top block. The wall with the glued condition did not collapse after the fracture of the top block since the top and the bottom blocks are glued to the V-shaped jigs and the model is more stable. The I-shaped block wall (right angle type) with the contact condition did not collapse after the fracture of the bottom block because the stability of the wall was obtained owing to the stronger interlocking between blocks due to the right angle.

Thus, the effect of the support conditions was larger in the I-shaped block wall (obtuse angle type) compared to the I-shaped block walls (right angle type). The effect of the support condition becomes larger as the interlocking effect becomes smaller.

The left top figure of Figure 6 shows the load-displacement relationship of hourglass-shaped block walls (linear type) for glued and contact conditions. In the two types of I-shaped block walls, the initial stiffness (initial slope) was almost the same and the difference in the load-displacement relationships was negligible irrespective of the test specimens. However, in the hourglass-shaped block wall (linear type), the variations in the load-displacement relationships are large even for the test specimens with the same support condition. The load-displacement relationship has several peaks, the largest load (strength) and the final displacement are larger compared to the I-shaped block walls.

The bottom left figure is the snapshot just after the second peak for test specimen No.1 of the glued condition. The first and the second peaks of the load-displacement relationship correspond to the failure of the top and the bottom blocks and the total failure occurred after the second peak.

The bottom middle figure is the snapshot just after the last peak for test specimen No.2 of the glued condition. The first peak of the load-displacement relationship corresponds to the failure of the bottom block and the second and the last peaks correspond to the failure of the top block. The final displacement is quite large.

The top right figure is the snapshot at the largest peak for test specimen No.1 of the contact condition. The load-displacement relationship has only one peak and one top block fractured.

The middle and bottom figures in the right are the strain distributions and the failure process of test specimen No.2 with the contact condition. First, the tensile strain concentration occurred in the bottom brick. The failure occurred where the strain was concentrated, but the tip of the failure was inside the jig, so the total failure did not occur. Then the failure occurred in the top block as shown in the right bottom figure, followed by the total failure.

Compared to the two types of I-shaped block walls, in the hourglass-shaped block walls (linear type) it was easier for dislocation between blocks and rotation around jigs to occur, especially in the contact condition. Therefore, the final displacement was larger. Moreover, the strain concentration in the blocks is less likely to occur due to the smoother interface of blocks, so the maximum load (strength) increased.

The left top figure of Figure 7 shows the load-displacement relationship of hourglass-shaped block walls (wavy type) for glued and contact conditions. Test specimen No.1 for the glued condition had two peaks, corresponding to the failure of the top and bottom blocks as indicated with circles in the left bottom figure. Test specimen No.2 for the glued condition had one peak which corresponds to the failure of the bottom block as indicated with a circle in the middle bottom figure. Test specimens No.1 and No.2 for the contact condition also had one peak which corresponds to the failure of the bottom block. The right middle figure indicates the strain distribution. The strain concentration in the bottom block can be observed.

In this hourglass-shaped block wall (wavy type), the initial stiffness (slope) of the two test specimens for the glued contact is similar but different from that of the two test specimens for the contact condition. This indicates that the influence of the support condition is large if the interlocking block has a smooth shape with less interlocking effect. In the contact condition, the blocks are easier to dislocate and rotate and the final displacement becomes larger.

In this chapter, the 2-dimensional finite element analysis of the diagonal compression test was conducted to understand the failure mechanism. The analysis was performed using the non-linear finite element software, Marc (MSC Software Corporation, 2020).

Figure 8 shows the analytical model of the I-shaped block wall (right angle type). Four mortar blocks and jigs were modeled by two-dimensional plane stress elements. The depth of the element was 10 cm.

The element size of the block was basically about 0.5 cm × 0.5 cm. Although the block elements themselves were elastic, they did not share nodes with other elements and failure can occur between elements.

The upper and lower steel jigs were modeled with seven 1 cm × 1 cm elastic elements and no failure occurs between steel elements.

For the boundary conditions, nine nodes belonging to the upper jig were fixed in both horizontal and vertical directions, and nine nodes belonging to the lower jig were fixed in the horizontal direction but forced displacement of 0.005 mm per step was applied in the vertical direction as shown in Figure 8.

The material properties of the block are shown in Table 4. They are obtained by the elements test. The Young’s modulus and Poisson’s ratio are given to the block elements. No compression failure was considered since no compression failure was observed during the experiment.

There are two types of interface between two adjacent blocks.

One is the interface between two elements belonging to the different blocks. They are just in touch. In this type of interface, the two adjacent elements do not resist against tensile force and only resist against compressive and friction force when they are in contact, and the friction coefficient in Table 4 was used.

The other interface is the one between two elements belonging to the same blocks. They are continuous elements. Two continuous elements resist against tensile and compressive force until the tensile stress exceeds the tensile strength shown in Table 4. Once the tensile stress exceeds the tensile strength, the failure occurs between the elements. After that, these two elements no longer resist against tensile force and only resist against compressive and friction force. The friction coefficient of 0.7 was used. The friction coefficient between the blocks after fracture is set to 0.7 taking into account the reproducibility of the experimental results.

As for the material properties of the steel jig, Young’s modulus of 2.0 × 10⁵ MPa and a Poisson’s ratio of 0.265 were used.

For the interface between the block and the jig, the contact condition was firstly considered. The interface between the adjacent block and jig elements cannot resist against the tensile force but can resist against compressive and friction force. The friction coefficient of 0.4 was adopted.

The top left figure of Figure 9A shows the comparison of the load-displacement relationship between the experiment and the analysis (FEM) for the contact condition. Before the failure, the stiffness (slope) gradually increased in the experiment, but the stiffness is almost constant in the analysis. The stiffness gradually increased in the experiment because the gap between blocks and the gap between the block and the jig were gradually closed and the stiffness gradually increased. After all the gaps closed, the stiffness became almost constant. On the contrary, in the analysis, the analytical model is ideally created with no gaps, so the stiffness is constant from the beginning. To compare the final elastic stiffness just before the failure, the displacement of the analysis was shifted to the right. The stiffness and the load before the failure were almost similar. However, there are two peaks in the experiment while there is only one peak in the analysis.

The left bottom figure is the snapshot of the last step of the analysis when the numerical analysis stopped due to the loss of stability. This figure shows where the failure occurred. In the analysis, both the top and bottom bricks fractured which is the same as the experiment. However, the load-displacement relationship had only one peak since the failure of the top block occurred immediately after the failure of the bottom blocks. In the load-displacement relationship for the analysis, the first peak corresponds to the failure of the bottom block and the area encircled by the red dotted lines corresponds to the failure of the bottom block. The failure of the top blocks occurred immediately after the failure of the bottom block, so the second peak is almost negligible. In the experiment, the strength of each block is not perfectly the same, so the timing when the top and bottom blocks fracture is different.

The right top figure is the normal strain distribution in the horizontal direction of the experiment around the second peak load. The right bottom figure is the maximum principal strain distribution of the analysis before the peak load. Since the timing of the two figures is different, comparison of the strain value is difficult. However, it can be seen that the strain is not evenly distributed but concentrated around the corner with a 270-degree angle of the top and the bottom blocks, and failure occurred at the locations where the strain is concentrated. This tendency can be seen both in the experiment and the analysis. It can be concluded that the block fractured due to the strain concentration at the interlocking parts. In the analysis, the strain distribution is point-symmetric around the center of the wall since the material properties and strength are ideally uniform. In the experiment, the bottom block fractured earlier because the material properties and strength of the block are not uniformly equal.

In both the experiment and analysis, failure occurred at 270-degree corners of the top and bottom blocks. No failure occurred at the corresponding 90-degree corners of the middle blocks. This may be because the strain concentration tends to occur at corners larger than 180 degrees. In the right bottom figure, the strain concentration can also be seen at 270-degree corners of the middle blocks. No failure occurred at the 270-degree corner in the middle blocks because the strain concentration was greater at the 270-degree corners of the top and bottom blocks, and the failure had occurred beforehand.

The top left figure of Figure 9B shows a comparison of the load-displacement relationship between the experiment and the analysis (FEM) for the contact condition. The stiffness and the maximum load are very similar between the experiment and the analysis.

The left bottom figure is the snapshot of the last step of the analysis when the numerical analysis stopped due to the loss of stability. This figure shows where the failure occurred. In the analysis, the bottom block fractured first which corresponds to the first peak of the load-displacement relationship, then the top block fractured which corresponds to the area encircled by red dotted lines. The failure of the top blocks occurred immediately after the failure of the bottom block, so the second peak is almost negligible. After the failure of the top and bottom blocks, the load dropped sharply, then the wall stabilized temporarily as shown by the green dotted lines, and then dropped again. The temporary stability is due to a change in the direction of fracture propagation that depends on the discretization of the elements. On the contrary, in the experiment, only the top block fractured since the stability was lost after the failure of the top block and total collapse occurred.

The right top figure is the normal strain distribution in the horizontal direction of the experiment around the peak. The right bottom figure is the maximum principal strain distribution of the analysis before the peak load. It can be seen that the strain is not evenly distributed but concentrated around the corner at a 225-degree angle of the top and the bottom blocks, and failure occurred where the strain is concentrated. The reason why only the top block fractured in the experiment is considered as follows. The angle of the top block in contact with the bottom jig was not perfectly 90 degrees since there was an error during production. Therefore, the wall was unstable in the experiment after the failure of the top block.

It can be seen that the maximum principal strain of the right angle type is more locally concentrated than the obtuse angle type from the comparison between Figures 9A,B. The local stress concentration can be relaxed by changing the right angle into the obtuse angle.

The top left figure of Figure 9C shows a comparison of the load-displacement relationship between the experiment and the analysis (FEM) for the contact condition. The displacement of the FEM was shifted to the right so that the peak point of Experiment No.2 and FEM becomes similar.

The left bottom figure is the snapshot of the last step of the analysis when the numerical analysis stopped due to the loss of stability. This figure shows where the failure occurred. Also, the dislocation between blocks and rotation of the wall can be observed.

The right top figure is the normal strain distribution in the horizontal direction of the experiment around the peak. The right bottom figure is the maximum principal strain distribution of the analysis before the peak load. It was found that in the both experiment and analysis, the strain was mainly concentrated in the bottom block, and failure occurred in the bottom block.

The top left figure of Figure 9D shows a comparison of the load-displacement relationship between the experiment and the analysis (FEM) for the contact condition. The displacement of the FEM was shifted to the right so that the peak point of Experiment No.2 and FEM becomes similar.

The left bottom figure is the snapshot of the last step of the analysis when the numerical analysis stopped due to the loss of stability. This figure shows where the failure occurred. In this hourglass-shaped block wall (wavy type), the failure did not occur at the interface between blocks but in the area near the jigs. Moreover, the dislocation between blocks and the rotation of blocks around the jigs are observed. Due to this dislocation and rotation, the gap between blocks is generated.

The right top figure is the normal strain distribution in the horizontal direction of the experiment around the peak. The right bottom figure is the maximum principal strain distribution of the analysis before the peak load. In the experiment, slight strain concentration was observed inside the bottom block near the interface with the middle left block. In the analysis, the strain was concentrated in the top and bottom blocks around the jigs rather than around the interfaces between blocks. It was considered that the strain was more strongly concentrated around the jig since the interfaces between blocks are smooth.