Interlocking Joints with Multiple Locks: Torsion-Shear Failure Analysis Using Discrete Element and Equilibrium-Based SiDMACIB Models

Abstract

SiDMACIB (Structurally informed Design of Masonry Assemblages Composed of Interlocking Blocks) is the first numerical model capable of extending the equilibrium problem of limit analysis to interlocking assemblies. Adopting the concave formulation, this model can compute the stress state at the corrugated faces with orthotropic behaviour, such as their combined torsion-shear capacity. Generally speaking, finding the plastic torsion-shear capacity of planar faces shared between conventional blocks is still a fresh topic, while investigating this capacity for interlocking interfaces is particularly rather unexplored. Upon the authors’ previous works that focused on interlocking blocks with a single lock, in this paper, an extension to blocks composed of several locks (multi-lock interfaces) is presented and the SiDMACIB model is upgraded accordingly. For this purpose, the shear-torsion results obtained from the original SiDMACIB formulation are validated and subsequently compared with those derived from distinct element analysis conducted using the 3DEC 7.0 software. Based on this comparison, revisions to the SiDMACIB model are proposed, involving a reduction in the number of locks affecting torsion-shear capacity.

1. Introduction

Advancements in digital fabrication have led to the emerging new generation of masonry discrete assemblies composed of elements with complex geometries and joint shapes, interlocking blocks to each other. These advances can be divided into three main fields, i.e., additive, subtractive, and formative methods [1]. Three-dimensional printing, the most dominant technique among additive manufacturing methods, has now enabled us to implement a wide range of masonry materials such as sand [2], earth [3], clay [4], and concrete [5]. Manufacturing a model layer by layer makes it non-homogeneous, which can sometimes be considered a challenge. Instead, subtractive methods that are mostly known as digital stereotomy allow us to cut relatively homogeneous natural materials like stones. These consist of comprehensive and diverse techniques such as robotic water jet, wire, and blade cutting, as well as robotic carving and milling [6], that allow us to model masonry blocks with highly complex geometries [7]. The most important challenge that subtractive methods face is dealing with material waste like stone remnants. Formative methods, like moulding masonry mortars [8,9], instead, not only enable us to manufacture homogenous models with complex geometries but the material waste issue is also significantly eliminated, mostly when moulds are reusable. However, the casting method is not stand-alone and is mostly linked to additive manufacturing methods implemented to print moulds with complex geometries. This makes the manufacturing process a time-consuming process.

Unlike huge advances in the manufacturing and construction of discrete masonry assemblies with complex geometries, the computer-aided design (CAD) tools supporting designers in modelling structurally sound assemblies are still limited. A large group of research works investigating the structural performance of discrete assemblies with complex geometries are based on computationally expensive methods like finite element (FE) and discrete element (DE) analyses, which are inappropriate for shape exploration at the early stages of design. The authors of this paper have comprehensively reviewed this group of research works in a previous paper [10].

The application of CAD tools to find structurally informed geometries at the early stage of design dates back to the 1970s in Frei Otto’s design [6,11]. The number of these tools increased exponentially in the 1990s, covering a wide range of complex geometries. One of the earliest digital tools to design discrete assemblies with complex geometries was developed by the Block Research Group [12]. Adopting the concave approach of Livesley’s static model [13], this interactive tool allows the designers to model rigid blocks with convex geometry, arbitrarily assembled with contacts that could be points, edges, or 2d faces. Solving a linear equilibrium problem, the designer can simply adjust the geometry of the blocks to finally reach an equilibrated assembly. Solving linear equilibrium problems for an assembly of rigid blocks was later implemented by Wang et al. [14] to design topological interlocking assemblies. Topological interlocking is a term first introduced by Dyskin et al. [15] to refer to various architectured interfaces with the aim of preventing them from displacing within the assembly. The term has been later extended into a wide range of large-scale assemblies [16] to metamaterial structures [17]. Wang et al. [14] selected a specific type of topological assemblies, where the contacts between the blocks are planar, and the specific orientations of these contacts interlock blocks to each other. The developed interactive tool allows the designer to model topological interlocking assemblies with different block geometries and bond patterns. It can also automatically adjust the geometry of the assembly to achieve static equilibrium.

SiDMACIB (Structurally informed Design of Masonry Assemblages Composed of Interlocking Blocks, MSCA_IF, No. 791235), developed by the authors of this paper, was among the first digital tools trying to extend the equilibrium problem using concavity formulation to analyse the structural soundness of interlocking assemblies with non-planar contact faces [18]. Concavity formulation abstracts a contact between two rigid blocks to some points distributed on the contact and tries to find the internal forces at these points equilibrating the external forces applied at the centroids of rigid blocks. SiDMACIB extended this formulation to contact interfaces with orthotropic behaviour (See Section 2). This model can not only calculate the equilibrium problem for an interlocking assembly with non-planar, corrugated interfaces between the blocks but can also, unlike other tools, consider the possibility of fracture of interlocking blocks. Later, Kao et al. [19] adopted the concavity formulation to analyse interlocking assemblies with arbitrary non-planar interfaces through the abstraction of the interface to a large number of contact points distributed on the edges of the face. In that paper, however, blocks were considered to be rigid, and while this assumption for an assembly of convex blocks can be usually acceptable, ignoring the possibility of fracture within the interlocking concave block is not realistic. In fact, considering a discrete assembly composed of rigid blocks dates back to Heyman’s theory [20], according to which the potential failure of such an assembly only takes place at the contact interfaces between the blocks. This theory was later upgraded to include potential failure discontinuities within the blocks where a block might crack. According to this multisurface plasticity model, the failure of a discrete masonry assembly occurs at the contact interfaces between the blocks (joints) and/or at potential cracking discontinuities within the blocks (inner faces) [21,22,23]. Mousavian and Casapulla used this method to model 2D interlocking assemblies with arbitrary interlocking joint shapes [24]. In that paper, only the tangential resistance of the joints and the inner faces can vary while no tensile resistance at both types of failure faces is considered. This method was recently extended to 3D models by Mousavian and Iannuzzo [25], where both tensile and tangential resistances of joints and inner faces are different.

On the other hand, if interlocking blocks are composed of a number of projections (herein called locks), one of the main challenging issues is the assessment of the torsion-shear capacity of the corrugated interfaces. Based on the concavity formulations adopted for conventional planar block interfaces [26], the torsion-shear capacity of a single lock with a rectangular cross-section was recently validated through a series of experimental tests and analytical investigations using limit analysis and distinct element (DE) methods [27,28], the latter via 3DEC numerical modelling software, version 5.0 [29]. This paper aims to extend the analytical exploration to assess the torsion-shear capacity of rectangular interlocking interfaces with multiple locks shared between two stacked blocks, assuming that all the locks of the interface are involved in the overall planar capacity. The results obtained using the novel SiDMACIB model are then compared to the results obtained using the well-established DE analysis (3DEC 7.0, free version [30]), and some proposals to adjust or better calibrate the former formulation are finally presented. 3DEC has been chosen as a reliable candidate to carry out this validation because the experimental tests for single locks presented in [27,28] showed strong agreement with the results obtained by 3DEC.

In the following, Section 2 introduces the SiDMACIB formulation and how an interlocking interface with multiple locks is modelled. Section 3 demonstrates how an assembly of two blocks with an interlocking interface composed of multiple locks is modelled within the 3DEC software for analysing the torsion-shear behaviour. Section 4 compares the results obtained by the SiDMACIB and 3DEC models and proposes some calibrations for the former model. In the end, conclusions are presented in Section 5.

2. SiDMACIB Formulation for the Multi-Lock Interface

2.1. Formulation of the Interlocking Interface

According to the static formulation of limit analysis using the concavity approach, an assembly is modelled as a set of rigid elements with some discontinuity planes in between, where the model can deform and fail [13]. The equilibrium problem is posed to find the stress states at these discontinuities equilibrating the external forces, which are abstracted to internal forces at contact points. The typical internal forces at every contact point are a component normal to the face, which must always be in compression if the discontinuity has zero tensile resistance, and two components tangential to the discontinuity plane, which are perpendicular to each other. For a planar isotropic dry joint, each tangential component must be less than the joint frictional resistance, which equals the coefficient of friction multiplied by the normal force.

SiDMACIB has extended this formulation to interlocking faces with orthotropic behaviour as follows [18]. In this model, two types of discontinuities are taken into account: one consists of strips between the blocks (dry faces) and the other consists of cohesive strips connecting the locks to the main body of the blocks (inner faces) (Figure 1a,b). It is assumed that both types of discontinuities have zero tensile resistance. Instead, the tangential behaviour of the dry and inner faces is different and is governed by Coulomb’s law of friction and the shear strength of the material of which the block is made, respectively. The SiDMACIB model first merges each pair of dry and inner strips representing a lock and then distributes contact points on the centreline CL of each lock, as represented in Figure 1c.

Several options for the number and location of the contact point have been considered and their validity has been reported in [31]. Similar to the conventional concave formulation, three components of internal forces are considered at each contact point i of CL j (Figure 1c). The normal component r _i,j ⁿ should always be in compression; the tangential component parallel to the locks r _i,j ^t1 should be less than the frictional resistance (µ r _i,j ⁿ), where µ is the friction coefficient; the second tangential component _i,j ^t2 must be less than the so-called shear resistance of the contact point, which is a portion p_i,j of the shear resistance T₀ of the whole discontinuity (see Figure 1d for the symbols). T₀ equals τ a b, where a and b are the length and width of the lock, respectively (Figure 1a).

The equilibrium problem then can be formulated as follows:

$$\left\{\begin{array}{c}{C}_{eq}·\overrightarrow{r}+\overrightarrow{E}=0\\ r_{i,j}^n\le 0 \\ \left|r_{i,j}^{t1}\right|\le \mu \left|r_{i,j}^n\right| \\ \left|r_{i,j}^{t2}\right|\le p_{i,j} T_0 \end{array}\right.\forall i\in \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{t} \mathrm{p}\mathrm{o}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{s} \mathrm{o}\mathrm{n} \mathrm{m}\mathrm{e}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{d} \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{p} j; {\sum }_{i=1}{p}_{i,j}=1$$

(1)

where C_eq is the equilibrium coefficient matrix, $\overrightarrow{r}$ is a vector of all normal and tangential components of internal forces, and $\overrightarrow{E}$ is a vector of all external forces EF and torques ET.

As introduced above, this paper aims to implement the torsion-shear resistance of a multi-lock interface shared between two stacked blocks, assuming that all the locks of the interface are involved in the overall planar capacity. In the following section, how this condition is posed in SiDMACIB is presented.

2.2. Multi-Lock Interface Problem

For the sake of consistency with the experimental torsion-shear tests carried out by the authors with others [27,28], the dimensions shown in Figure 2 are defined for stacked blocks with an interlocking joint. Two samples are presented in this figure, one with three and one with five locks. The sample in Figure 2a, in fact, is made of two stacked blocks, the lower one with one central lock and the upper one with two side locks. The sample in Figure 2b instead consists of a lower block with two locks and an upper block with three locks.

The assumed block density is 1400 kg/m³ and an overload of 1000 N is imposed on the upper block to avoid its rocking failure, as shown in Figure 3a. Also, the lower face of the lower block is considered as fixed support. To define the torsion-shear capacity of the interlocking interface, a lateral force normal to the lock (in the Y direction) together with a torque around the Z axis is applied to the centroid of the upper block. Given an eccentricity ex from the lock centroid in the X direction, the maximum lateral force V_y and the corresponding torque, equal to V_y ∙ e_x, can be recorded using the SiDMACIB model, depending on the distribution of the contact point at each interface. To determine this force for a certain eccentricity, the model’s stability is checked using Equation (1) with a lateral force. If the model stays stable, the lateral force is increased gradually until the highest acceptable value is reached.

In fact, four different distributions of contact points on each centreline can be considered, as shown in Figure 3b, for the interlocking block with three locks. The extension of this model to a multiple number of locks is carried out by simply adding contact points on the abstracted planar interface and checking which option for the distribution works better, as demonstrated in Section 4, in comparison with the results obtained using a distinct element model.

3. Distinct Element Model of the Multi-Lock Interface

As mentioned, 3DEC 7.0 [30], which is a numerical modelling code based on the distinct element method (DEM), is herein used to validate the torsion-shear capacities of interlocking blocks obtained by the SiDMACIB model. The DE model is made of discrete particles (bodies) that can move independently including large displacements and rotations. Once particles are moved relative to each other, some contacts between the particles can be lost or newly generated [32].

DEM was initially developed for the simulation of soil modelled as an assembly of rigid and fine particles. The method was later adopted to model masonry assemblies in which masonry blocks are modelled as either rigid or deformable. The most common method to model a deformable body, also used in the 3DEC software, is to subdivide it into finite elements, usually into uniform-strain tetrahedral elements [33,34].

By adopting the assumption of rigid bodies in this paper, each interlocking block can be considered as made of several rigid bodies representing the locks and the main part of the block, as shown in Figure 4a for the three- and five-lock interfaces. The subdivision into smaller parts sketched in this figure is based on the observation that if each block is divided into the main body and the locks, 3DEC displays an error. Considering the abstraction in Figure 1, these bodies are linked by horizontal cohesive interfaces (inner faces coloured red in Figure 4b) with finite shear and tensile strengths, while the horizontal interfaces shared between the blocks are dry faces with zero tensile strength and finite frictional resistance (blue faces in Figure 4b). Note that the provisional yellow parts over the upper blocks are considered fully fixed to avoid potential rocking failure. To avoid the potential sliding interaction between the fixed yellow part and the upper interlocking block, the friction coefficient and the interface between these two entities are set to zero. Later on, it has been observed that the rocking failure does not happen in any of the studied cases.

The vertical faces, instead, are automatically generated when the main bodies of the interlocking blocks are divided. These faces are also categorised as ‘inner’ type, depicted in red.

On the other hand, a simple triangulation method has been selected among others in 3DEC to divide each shared face into 2D sub-contacts, using two different mesh sizes of 0.02 and 0.005 m. Both types of horizontal contact faces (i.e., dry faces between two interlocking blocks and inner faces between the rigid bodies of one interlocking block) are triangulated similarly.

Table 1. Material properties of the block and contact interfaces.

Block Material		Dry Face Material		Inner Face Material
Density	1400 kg/m3	Normal stiffness	1010 N/m	Normal stiffness	1010 N/m
		Shear stiffness	1010 N/m
				Shear stiffness	1010 N/m
		Friction coefficient µ	0.3
				Friction coefficient	0
		Cohesion	0 N/m2	Cohesion τ	8 × 104 N/m2
		Tensile strength	0 N/m2	Tensile strength σ	3.7 × 105 N/m2

shows the material properties of the blocks and the inner and dry faces, already used to model the mentioned experimental tests [27,28]. Note that the properties of the inner interfaces, including their shear and tensile strengths, correspond to the block material properties.

To simulate the torsion-shear interaction at the interlocking interface, the main part of the lower interlocking block is considered to be fixed (depicted in green in Figure 5b), while the top cannot rotate (yellow part in Figure 4a). A non-uniform and linearly distributed force is applied to the whole lateral face of the upper block shown in Figure 5a, involving both the side locks and the main body of this interlocking block. This distributed force is equivalent to a pointed load V_y applied to the block face with the eccentricity ex from the centroid of the face (with e_y = 0). This is equal to the lateral force V_y and torque M_y = V_y ∙ e_x applied to the face centroid.

To model such a distributed force, 3DEC needs the value of the distributed force at the face centroid V_dy,h and the slope of the distributed force K_y, which are [27,28]:

$$V_{dy,h}=\frac{V_y}{ah}\hspace{1em}\hspace{1em}{k}_y=\frac{{12M}_y}{a^{3}h}$$

(2)

where a and h are the length and the height of the lock, respectively.

4. Discussion

In this section, the results in terms of torsion-shear capacity obtained using the SiDMACIB and 3DEC models are presented and compared to each other, in order to propose a reliable calibration of the SiDMACIB model. These are referred to as the samples in Figure 2 of two stacked blocks with three and five locks.

Figure 6 shows the torsion-shear curves obtained for the sample with three locks, using the SiDMACIB model and the four distribution scenarios for the contact points reported in Figure 3, assuming a material shear strength τ = 8 × 10⁴ N/m². These curves show the maximum lateral force that can be applied to the lateral face of the upper block for different eccentricities from the centroid of the face. Knowing the maximum lateral force V_y and the eccentricity e_x, the corresponding maximum torque M_y can be computed.

It is worth highlighting that the four scenarios herein used for the contact points (two points per lock) involve linear curves in favour of safety in terms of torsion-shear capacity, while other options considering three points per lock imply bi-linear and convex curves [31].

In Figure 6, while Options 1 and 4 show the highest and lowest pure torsion, respectively, the computed pure shear force for all the options is equal to 720 N. This value is, in fact, the product of the assigned cohesion τ and the width and the length of the interlocking interface, composed of three inner surfaces.

Figure 7 instead shows the torsion-shear graphs obtained by 3DEC for two different mesh sizes of 0.02 and 0.005 m, the latter being a particularly fine size. The first fundamental observation is that both curves are concave and composed of two discrete linear segments. Parametrizing the curve, the turning point from which the curve’s slope changes is approximately at the two-third length of the curve; after that, the torsion-shear curve of the model with 0.02 m mesh size is slightly steeper than the other. Apparently, it is expected that the results of the model with a finer size are more accurate.

However, the torsion-shear capacities found with SiDMACIB (for all four options) are considerably larger than 3DEC. As mentioned above, the pure shear capacity of all four options of SiDMACIB is 720 N, while this capacity is reduced to 240.76 and 240.83 for the 3DEC models with mesh sizes 0.02 and 0.005 m, respectively. Taking a closer look, it can be observed that the pure shear obtained by SiDMACIB is almost three times larger than the pure shear found with 3DEC. Figure 8 illustrates the distributed shear stresses on the three-lock interfaces when the upper interlocking block is subjected to pure shear forces. It can be noted that while the side lock closer to the subjected lateral forces (directed as shown in Figure 5) experiences failure, the other two locks have not yet initiated failure. This observation aligns with findings previously reported by Jiang et al. [35], who analysed the pure shear strength of interlocking precast concrete blocks utilised in segmental bridges. These authors also presented the sequence of cracking formation for multi-lock joints, both experimentally and numerically.

To demonstrate the variable contribution of locks to the load-bearing capacity of the interlocking interface, a parameter W can be introduced, representing the “weight” of each lock in the capacity of the entire interface. This is achieved by determining the ratio of the uniformly distributed shear stress at the inner interface of each lock to its shear strength and defining the total factor. For the case of pure shear forces applied to the model with three locks in Figure 8, it is evident that these weights are

$$\begin{array}{c}{W}_1=0.99; W_2=0.77; W_3=0\\ {W_{sum}=W}_1+W_2+W_3=1.76\end{array}$$

(3)

These observations mean that, unlike the initial assumption of the SiDMACIB model, the three locks of an interlocking interface are not all involved in the overall planar capacity equally. Based on these outcomes, several scenarios can be considered for adjusting the SiDMACIB formulation. The simplest one involves reducing the number of active locks so that it corresponds to the value of W_sum rounded down to an integer value, in favour of safety. This straightforward modification minimally alters the original SiDMACIB model, with the absolute value acting as a safety factor to prevent overestimation of the load-bearing capacity.

Different weights of locks in load-bearing capacity can also be noted for interlocking blocks subjected to torsion-shear loads, as illustrated in Figure 9. Based on the previous observation, the SiDMACIB numerical model is revised so that an interlocking interface with three locks only includes the side lock in the analysis. Implementing this modification, Figure 10 shows the torsion-shear results considering Options 1 to 4 of the revised SiDMACIB model in terms of continuous curves. It is evident that Option 4 is in good agreement with the torsion-shear results obtained by 3DEC.

More complex scenarios may entail assigning different weights to the control points and/or determining the optimal layout of control points on an interlocking interface when subjected to torsion-shear combinations. However, these intricate modifications can only be pursued after conducting a comprehensive study involving different block sizes and lock numbers, both experimentally and numerically. Such investigations are beyond the scope of this paper.

As mentioned above, to have a more comprehensive observation, a similar analysis is carried out on the sample composed of two stacked blocks with five locks at the interlocking interface. Figure 11 shows the torsion-shear results computed by SiDMACIB (4 options) and 3DEC (two different mesh sizes) using continuous and dashed lines, respectively. Similar to the model with three locks, the results obtained by SiDMACIB are larger than the results found with 3DEC.

Comparing the pure shear capacities obtained by the two methods, those computed by 3DEC with 0.02 and 0.005 m mesh sizes are equal to 408.26 and 410.52 N, respectively, which are slightly less than the pure shear of two locks (480 N). In this case, the calculation of the weight of each lock is determined as follows (Figure 12):

$$\begin{array}{c}{W}_1=0.99; W_2=0.57; W_3=0.68; W_4=0.68; W_5=0\\ W_{sum}=2.92\end{array}$$

(4)

As for the case of pure shear, for the torsion-shear combination (Figure 13), the most important role in the overall performance of the interlocking interface is played by the lock closer to the face at which the distributed force is implemented. Three other central locks have relatively fewer roles in the interlocking capacity, and the last side lock does not participate in this task at all.

Figure 14 shows the results of the modified SiDMACIB model, in which the number of active locks of the interlocking interface is reduced from five to two. These results are illustrated by continuous lines, while the dashed lines represent the torsion-shear results obtained by 3DEC. It can be observed that, same as the model with three locks, Option 4 of the revised SiDMACIB model has the best agreement with the results found with 3DEC.

The implemented modification pattern shows that the torsion-shear capacity of the interlocking interface with three locks should be reduced to one lock and that with five locks can be reduced to two locks. According to this pattern, it is expected that for an interlocking interface with n locks, the torsion-shear capacity of the interface should only include (n − 1)/2 locks in the numerical model. However, this conclusion must be explored more comprehensively in future work, including experimental investigation.

5. Conclusions

This paper evaluates the performance of the SiDMACIB numerical model to analyse the torsion-shear capacity of corrugated interlocking interfaces consisting of several locks. To this aim, two samples made of two stacked interlocking blocks are considered, one involving three locks and the other five locks. The torsion-shear capacities of such interlocking interfaces are obtained using SiDMACIB, an equilibrium-based model, and 3DEC, a software based on the distinct element method. Comparing the results obtained by these two methods, it is observed that the SiDMACIB model overestimates the torsion-shear capacity of the interlocking interface since it includes all the locks in the overall performance. Instead, the 3DEC model indicated that the number of locks participating in this task is much less and usually is reduced to about half the number of locks. Specifically, the lock nearest to the applied force fully engages in load-bearing capacity, whereas the locks farther away contribute minimally tending to zero contribution. The second inquiry involves determining a heuristic for the number of locks involved in the torsion-shear capacity. Investigation using blocks containing three and five locks suggests a heuristic formula of reducing the number of locks involved to half, minus 1/2. Future research endeavours will build upon this methodology to refine the heuristic for greater accuracy. In the future, however, this rough estimation can become more accurate by precisely measuring the level of involvement of each individual lock in the overall performance of the interlocking interface, even using experimental tests.

On the other hand, the 3DEC model implemented in this work to validate the SiDMACIB formulation has been developed based on the rigid approach, in which each block is made of rigid elements connected by cohesive inner faces where the locks may crack, while individual blocks are interconnected by dry faces. Instead, in the future, a similar comparative methodology can adopt the deformable approach of 3DEC, by which the blocks are modelled as deformable solids made of finite elements.